Research ArticleHopf Bifurcations of a Stochastic Fractional-OrderVan der Pol System

Xiaojun Liu12 Ling Hong1 and Lixin Yang12

1 State Key Laboratory for Strength and Vibration of Mechanical Structures Xirsquoan Jiaotong University Xirsquoan 710049 China2 School of Mathematics and Statistics Tianshui Normal University Tianshui 741001 China

Correspondence should be addressed to Ling Hong honglingmailxjtueducn

Received 27 March 2014 Revised 21 May 2014 Accepted 21 May 2014 Published 9 June 2014

Academic Editor Juan J Trujillo

Copyright copy 2014 Xiaojun Liu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The Hopf bifurcation of a fractional-order Van der Pol (VDP for short) system with a random parameter is investigated Firstlythe Chebyshev polynomial approximation is applied to study the stochastic fractional-order system Based on the method thestochastic system is reduced to the equivalent deterministic one and then the responses of the stochastic system can be obtained bynumerical methodsThen according to the existence conditions of Hopf bifurcation the critical parameter value of the bifurcationis obtained by theoretical analysis Then numerical simulations are carried out to verify the theoretical results

1 Introduction

Fractional calculus is a topic of more than 300 years oldThe idea of fractional calculus has been known since theregular calculus However its development is very rapid justin recent several decades due to its application in physicsengineering secure communications and so on Meanwhileit has been found that many fractional-order nonlinear sys-tems can demonstrate chaotic behavior such as fractional-order Chua circuit [1] fractional-order Lorenz system [2]and fractional-order Chen system [3] These examples andmany other similar samples perfectly clarify the importanceof consideration and analysis of dynamical systems with frac-tional-order models [4ndash6]

The VDP oscillator represents a nonlinear systemwith aninteresting behavior that exhibits naturally in several appli-cations It has been used for study and design of many mod-els including biological phenomena such as the heartbeatneurons acoustic models and radiation of mobile phonesand as a model of electrical oscillators The deterministicVDP has very rich dynamical behaviors such as saddle-nodebifurcation symmetry-breaking bifurcation period-doubl-ing bifurcation Hopf bifurcation and chaos Recently manyresearchers studied the bifurcation and periodic solutions ofthe VDP system in detail [7ndash11] As research on fractional

differential equations goes up the VDPmodel with fractionalorder was proposed and investigated in [12 13] It was foundthat when the order is less than 1 the fractional-order VDPsystem also has rich dynamics similar to the correspondinginteger-order one

In the real world uncertainty due to measuring mate-rials manufacturing and assembling is inevitable Theseuncertainties usually can be modeled as random parameterswith certain statistics A stochastic system means a systemwith random parameters of given statistics which is a mathe-matical model close to the real worldThere are several meth-ods to deal with the dynamical problems of stochasticsystems The first one is the Monte Carlo method [14] whichis simple and universal but usually involves a great deal ofcomputational effort The second one is the stochastic finiteelement method [15] which involves the least computationbut is usually restricted to systems with random variables ofsmall perturbations only And the third one is orthogonalpolynomials [16ndash20] which is based on the expansion theoryof orthogonal polynomials without the limitation of smallperturbations There are a lot of references about the analysisof stochastic systems via orthogonal polynomials For exam-ple in [21 22] the Chebyshev polynomial approximationwasapplied to analyze the bifurcation and chaos of stochasticDuffing system and stochastic Duffing-Van der Pol system

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 835482 10 pageshttpdxdoiorg1011552014835482

2 Abstract and Applied Analysis

The period-doubling bifurcation for double-well stochasticDuffing system was investigated via Lagurre polynomialapproximation in [23] Recently the synchronization for astochastic fractional-order system with a random parametervia Laguerre polynomial was analyzed in [24] However theliterature on the investigation of a stochastic fractional-orderVDP system is few

Inspired by the above discussion in this paper Hopfbifurcations of a fractional-order VDP system with a randomparameter are studied Firstly the Chebyshev polynomialapproximationmethod is applied to investigate the stochasticfractional-order system Based on this method the stochasticsystem is reduced to the equivalent deterministic one Thenaccording to the existence conditions of Hopf bifurcationthe critical parameter value of the bifurcation is obtainedby theoretical analysis Meanwhile numerical simulationsare performed to verify the theoretical results Besides theHopf bifurcation with the variation of derivative order is alsoobserved by numerical computations

The paper is organized as follows In Section 2 the defi-nitions for the fractional calculus and numerical algorithmsare given In Section 3 the Chebyshev polynomials are intro-duced as the basis of orthogonal polynomial approximationIn Section 4 the transformation of stochastic fractional-order VDP system into its equivalent deterministic one byChebyshev polynomial approximation is shown Section 5 isdevoted to studying the Hopf bifurcation of the stochasticsystem Finally we summarize the results in Section 6

2 Fundamentals of Fractional Derivative

21 Definition Fractional calculus is a generalization ofintegration and differentiation to a noninteger-order inte-grodifferential operator

119886119863119902

119905which is defined by

119886119863119902

119905=

119889119902

119889119905119902119877 (119902) gt 0

1 119877 (119902) = 0

int

119905

119886

(119889120591)minus119902

119877 (119902) lt 0

(1)

where 119902 is the derivative order which can be a complexnumber and 119877(119902) is the real part of 119902 The numbers 119886 and119905 are the limits of the operatorThere are many definitions forfractional derivatives Three most frequently used ones arethe Grunwald-Letnikov definition the Riemann-Liouvilleand the Caputo definitions

The Grunwald-Letnikov definition (GL) derivative withfractional-order 119902 is described by

GL119886

119863119902

119905119891 (119905) = lim

ℎrarr0

1

ℎ119902

[(119905minus119886)ℎ]

sum

119895=0

(minus1)119895

(119902

119895)119891 (119905 minus 119895ℎ) (2)

where the symbol [∙] means the integer part

The Riemann-Liouville (RL) definition of fractionalderivatives is given by

RL119886

119863119902

119905119891 (119905) =

119889119899

119889119905119899

1

Γ (119899 minus 119902)int

119905

119886

119891 (120591)

(119905 minus 120591)119902minus119899+1

119889120591

119899 minus 1 lt 119902 lt 119899

(3)

where Γ(∙) is the gamma functionThe Caputo (119862) fractional derivative is defined as follows

119862

119886119863119902

119905119891 (119905) =

1

Γ (119899 minus 119902)int

119905

119886

(119905 minus 120591)119899minus119902minus1

119891(119899)

(120591) 119889120591

119899 minus 1 lt 119902 lt 119899

(4)

It is well known that the initial conditions for thefractional differential equations with Caputo derivatives takethe same form as for the integer-order ones which is verysuitable for practical problems [25]Therefore we will use theCaputo definition for the fractional derivatives in this paper

22 Numerical Algorithms Obtaining numerical solutionsof a fractional differential equation is not easily comparedwith that for ordinary differential equations There are twoapproximation methods which can frequently be used tonumerical computations on chaos and bifurcations withfractional differential equations One is an improved ver-sion of Adams-Bashforth-Moulton algorithm based on thepredictor-correctors scheme [26] which is a time-domainapproach The other is a method known as frequencydomain approximation [27] based on numerical analysis offractional-order systems in the frequency domain

Simulations for fractional-order systems using the timedomain methods are complicated and due to long memorycharacteristics of these systems require a very long simulationtime but on the other hand it ismore accurate [28]Thereforewe employ the improved predictor-corrector algorithm forfractional-order differential equations in this paper

In order to get the approximate solution of a fractional-order chaotic system by the improved predictor-correctoralgorithm the following equation is considered

119889119902

119909

119889119905119902= 119891 (119905 119909) 0 le 119905 le 119879

119909(119896)

(0) = 119909(119896)

0 119896 = 0 1 lceil119902rceil minus 1

(5)

where lceil119902rceil is just the value 119902 rounded up to the nearest integerand 119909

(119896) is the ordinary 119896th derivative of 119909 Formula (5) isequivalent to the Volterra integral equation

119909 (119905) =

lceil119902rceilminus1

sum

119896=0

119909(119896)

0

119905119896

119896+

1

Γ (119902)int

119905

0

(119905 minus 120591)119902minus1

119891 (120591 119909 (120591)) 119889120591 (6)

Now for the sake of simplicity we assume that we areworkingon a uniform grid 119905

119899= 119899ℎ 119899 = 0 1 119872 with some

integer 119872 and set ℎ = 119879119872 Using the standard quadraturetechniques for the integral in (6) denote 119892(120591) = 119891(120591 119909(120591))

Abstract and Applied Analysis 3

the integral is replaced by the trapezoidal quadrature formulaat point 119905

119899+1

int

119905119899+1

0

(119905119899+1

minus 120591)119902minus1

119892 (120591) 119889120591 asymp int

119905119899+1

0

(119905119899+1

minus 120591)119902minus1

119892119899+1

(120591) 119889120591

(7)

where119892119899+1

is the piecewise linear interpolant for119892with nodesand knots chosen at the 119905

119895 (119895 = 0 1 119899 + 1) After some

elementary calculations the right hand side of (7) gives

int

119905119899+1

0

(119905119899+1

minus 120591)119902minus1

119892119899+1

(120591) 119889120591 =ℎ119902

119902 (119902 + 1)

119899+1

sum

119895=0

120572119895119899+1

119892 (119905119895)

(8)

And if we use the product rectangle rule the right hand of (7)can be written as

int

119905119899+1

0

(119905119899+1

minus 120591)

119902minus1

119892119899+1

(120591) 119889120591 =

119899+1

sum

119895=0

120573119895119899+1

119892 (119905119895) (9)

where120572119895119899+1

=

119899119902+1

minus (119899 minus 119902) (119899 + 1)119902

119895 = 0

(119899 minus 119895 + 2)119902+1

+ (119899 minus 119895)119902+1

minus 2(119899 minus 119895 + 1)119902+1

1 le 119895 le 119899

1 119895 = 119899 + 1

120573119895119899+1

=ℎ119902

119902((119899 minus 119895 + 1)

119902

minus (119899 minus 119895)119902

) 1 le 119895 le 119899

(10)

Then the predictor and corrector formulae for solving (6) aregiven respectively by

119909119875

ℎ(119905119899+1

) =

lceil119902rceilminus1

sum

119896=0

119909(119896)

0

119905119896

119899+1

119896+

1

Γ (119902)

119899

sum

119895=0

120573119895119899+1

119891 (119905119895 119909ℎ(119905119895))

119909ℎ(119905119899+1

) =

lceil119902rceilminus1

sum

119896=0

119909(119896)

0

119905119896

119899+1

119896+

ℎ119902

Γ (119902 + 2)119891 (119905119899+1

119909119875

ℎ(119905119899+1

))

+ℎ119902

Γ (119902 + 2)

119899

sum

119895=0

120572119895119899+1

119891 (119905119895 119909ℎ(119905119895))

(11)

The approximation accuracy of scheme (11) is 119874(ℎmin21+119902

)

3 The Chebyshev Polynomials

In general random parameters in some engineering struc-tures are bounded in nature So the probability densityfunction (PDF for short) model for the bounded randomvariables is taken to be an arch-like PDF of the form [29]

120588 (119906) =

(2

120587)radic1 minus 1199062 |119906| le 1

0 |119906| gt 1

(12)

0 05 10

01

02

03

04

05

06

07

u

minus1 minus05

120588(u)

Figure 1 The arch-like PDF curve for random variable 119906

Figure 1 shows the arch-like PDF curve for boundedrandom variable 119906 As the orthogonal polynomial basis forthis kind of PDF the second kind of Chebyshev polynomialsis the only choice The general expansion for the second kindof Chebyshev polynomials can be described as follows

119867119899(119906) =

[1198992]

sum

119896=0

(minus1)119896 (119899 minus 119896)

119896 (119899 minus 2119896)(2119906)119899minus2119896

(13)

Then we have

1198670(119906) = 1

1198671(119906) = 2119906

1198672(119906) = 4119906

2

minus 1

1198673(119906) = 8119906

3

minus 4119906

1198674(119906) = 16119906

4

minus 121199062

+ 1

sdot sdot sdot

(14)

The recurrent formula for the Chebyshev polynomials of thesecond kind is

119906119867119899(119906) =

1

2[119867119899minus1

(119906) + 119867119899+1

(119906)] (15)

The orthogonality of Chebyshev polynomial of the secondkind can be expressed as

int

1

minus1

2

120587

radic1 minus 1199062119867119894(119906)119867119895(119906) 119889119906 =

1 (119894 = 119895)

0 (119894 = 119895) (16)

Equation (16) represents a weighted orthogonal relationshipThe weighting function is just the same as that for the PDFof the random parameter which obeys arch distribution120588(119906) in (12) and the left-hand side of (16) can be regardedas the expectation of the product 119867

119894(119906)119867119895(119906) Owing to

4 Abstract and Applied Analysis

the orthogonality of Chebyshev polynomial any measurablefunction 119891(119906) isin 119871

2 can be expanded as

119891 (119906) =

infin

sum

119894=0

119888119894119867119894(119906) (17)

where 119888119894= int1

minus1

120588(119906)119891(119906)119867119894(119906)119889119906 This expansion is called an

orthogonal decomposition of random function119891(119906) which isthe theoretical base of orthogonal decomposition methods

4 The Stochastic Fractional-OrderVDP System

In this paper we consider the fractional-order VDP systemwith a random parameter The system can be described bythe following differential equations

119863119902

119909 = 119910

119863119902

119910 = 120583119910 minus 1198861199092

119910 + 119887119909

(18)

where 119902 is the fractional order and 119909 119910 are state variables 119886119887 are deterministic parameters and 120583 is a random parameterand it can be expressed as

120583 = 120583 + 120575119906 (19)

where 120583 and 1205752 are the mean value and standard deviationof 120583 respectively and 120575 is regarded as the intensity of 120583 119906 isa random variable defined on [minus1 1]

It is well known that the stochastic function spacewhich is constituted with the responses of nonlinear dynamicsystems with random parameters has proved to be a Hilbertspace with respect to convergence in the mean squareTherefore the fractional-order system (18) with randomparameters can be reduced into the equivalent deterministicsystem by using of the orthogonal polynomial expansion Itfollows from the orthogonal polynomial approximation thatthe responses of system (18) can be expressed approximatelyby the following series

119909 (119905 119906) =

119873

sum

119894=0

119909119894(119905)119867119894(119906)

119910 (119905 119906) =

119873

sum

119894=0

119910119894(119905)119867119894(119906)

(20)

where119867119894(119906) represents the 119894th Chebyshev polynomial and119873

is the largest order of the polynomialsThen substituting (20)into the system (18) we can get

119863119902

(

119873

sum

119894=0

119909119894(119905)119867119894(119906)) = minus(

119873

sum

119894=0

119910119894(119905)119867119894(119906))

119863119902

(

119873

sum

119894=0

119910119894(119905)119867119894(119906))

= 120583(

119873

sum

119894=0

119910119894(119905)119867119894(119906)) + 120575119906(

119873

sum

119894=0

119910119894(119905)119867119894(119906))

minus 119886(

119873

sum

119894=0

119909119894(119905)119867119894(119906))

2

(

119873

sum

119894=0

119910119894(119905)119867119894(119906))

+ 119887(

119873

sum

119894=0

119909119894(119905)119867119894(119906))

(21)

and with the aid of the recurrent formulas of Chebyshevpolynomial all the nonlinear terms in the system can befurther reduced into linear combination of 119871

119894(119906) as follows

(

119873

sum

119894=0

119909119894(119905)119867119894(119906))

2

(

119873

sum

119894=0

119910119894(119905)119867119894(119906))

2

=

3119873

sum

119894=0

119878119894(119905)119867119894(119906) (22)

where 119878119894(119905) can be obtained with the help of the computer

algebraic systems such as Maple or MATLAB Meanwhilethe term 120575119906(sum

119873

119894=0119910119894(119905)119867119894(119906)) can be reduced to

119906120575(

119873

sum

119894=0

119910119894(119905)119867119894(119906)) = 120575(

119873

sum

119894=0

119910119894(119905)

119867119894minus1

(119906) + 119867119894+1

(119906)

2)

(23)

Substituting (22) and (23) into (21) then we can have

119863119902

(

119873

sum

119894=0

119909119894(119905)119867119894(119906)) = minus(

119873

sum

119894=0

119910119894(119905)119867119894(119906))

119863119902

(

119873

sum

119894=0

119910119894(119905)119867119894(119906)) = 120583(

119873

sum

119894=0

119910119894(119905)119867119894(119906))

+ 120575(

119873

sum

119894=0

119910119894(119905)

119867119894minus1

(119906) + 119867119894+1

(119906)

2)

minus 119886

119873

sum

119894=0

119878119894(119905)119867119894(119906)

+ 119887(

119873

sum

119894=0

119909119894(119905)119867119894(119906))

(24)

Multiply both sides of (24) by 119867119894(119906) 119894 = 0 1 2 119873 in

sequence and take expectation with respect to 119906Then we canfinally obtain the equivalent deterministic system Rememberthat if 119873 rarr infin in (24) then its left is strictly equivalent tothe response of stochastic system (18) Otherwise (24) is justapproximately valid with aminimal residual error Due to therequirement of computational precision we take119873 = 4 in the

Abstract and Applied Analysis 5

following numerical analysisTherefore we get the equivalentdeterministic system approximately as

119863119902

1199090= minus 119910

0

119863119902

1199100= minus 119886119878

0+ 1198871199090+ 1205831199100+

1

21205751199101

119863119902

1199091= minus 119910

1

119863119902

1199101= minus 119886119878

1+ 1198871199091+ 1205831199101+

1

2120575 (1199100+ 1199102)

119863119902

1199092= minus 119910

2

119863119902

1199102= minus 119886119878

2+ 1198871199092+ 1205831199102+

1

2120575 (1199101+ 1199103)

119863119902

1199093= minus 119910

3

119863119902

1199103= minus 119886119878

3+ 1198871199093+ 1205831199103+

1

2120575 (1199102+ 1199104)

119863119902

1199094= minus 119910

4

119863119902

1199104= minus 119886119878

4+ 1198871199094+ 1205831199104+

1

21205751199103

(25)

We can get the numerical solutions 119909119894 119910119894 (119894 = 0 1 2 3 4) of

system (25) by the improved predictor-corrector algorithmTherefore the approximate random responses of the stochas-tic system (18) can be expressed as

119909 (119905 119906) asymp

4

sum

119894=0

119909119894(119905)119867119894(119906)

119910 (119905 119906) asymp

4

sum

119894=0

119910119894(119905)119867119894(119906)

(26)

The ensemble mean responses (EMRs for short) of thestochastic fractional-order system are described as

119864 [119909 (119905 119906)] asymp

4

sum

119894=0

119909119894(119905) 119864 [119867

119894(119906)] = 119909

0(119905)

119864 [119910 (119905 119906)] asymp

4

sum

119894=0

119910119894(119905) 119864 [119867

119894(119906)] = 119910

0(119905)

(27)

Especially for 119906 = 0 namely 120583 = 120583 this sample system issignificant for reference It is usually called a mean parametersystem of which the responses (SRMs for short) may beapproximated as

119909 (119905 0) asymp

4

sum

119894=0

119909119894(119905)119867119894(0) = 119909

0(119905) minus 119909

2(119905) + 119909

4(119905)

119910 (119905 0) asymp

4

sum

119894=0

119910119894(119905)119867119894(0) = 119910

0(119905) minus 119910

2(119905) + 119910

4(119905)

(28)

5 Hopf Bifurcations

A Hopf bifurcation occurs when a periodic solution or limitcycle surrounding an equilibrium point arises or goes awayas a parameter varies In this section the Hopf bifurcation forthe stochastic fractional-order VDP system will be studiedwith theoretical and numerical computations Obviously

both the stochastic VDP and equivalent deterministic sys-tems (18) and (25) only have one equilibrium point namelythe origin (00) The Jacobian matrix 119869 for the linearizedsystem evaluated at the origin is

119869 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

0 minus1 0 0 0 0 0 0 0 0

119887 120583 01

2120575 0 0 0 0 0 0

0 0 0 minus1 0 0 0 0 0 0

1

2120575 0 119887 120583 0

1

2120575 0 0 0 0

0 0 0 0 0 minus1 0 0 0 0

0 0 01

2120575 119887 120583 0

1

2120575 0 0

0 0 0 0 0 0 0 minus1 0 0

0 0 0 0 01

2120575 119887 120583 0

1

2120575

0 0 0 0 0 0 0 0 0 minus1

0 0 0 0 0 0 01

2120575 119887 120583

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(29)

The corresponding characteristic equation is

119891 (120582) = 119886012058210

+ 11988611205829

+ 11988621205828

+ 11988631205827

+ 11988641205826

+ 11988651205825

+ 11988661205824

+ 11988671205823

+ 11988681205822

+ 1198869120582 + 11988610

(30)

where 1198860 1198861 119886

10are the coefficients of (30) and can be

obtained by computing via Maple or MATLAB According tothe bifurcation theory of nonlinear dynamical systems andtaking 119908 as a bifurcation parameter [22] we can obtain theexistence conditions of Hopf bifurcation for the system

(1) The eigenvalues of the linearized system about thefixed point are 120572(119908) plusmn 119894120573(119908)

(2) The real and image parts of the eigenvalues satisfy120572(119908119888) = 0 120573(119908

119888) gt 0 1205721015840(119908

119888) = 0 when 119908 = 119908

119888

(3) The real parts of other eigenvalues are not equal tozeros

For the high order of equivalent system (25) so it is hardto get the threshold value of bifurcation parameter directlyTherefore we can get the threshold via the following theorem

Theorem 1 Let 119891(120582) = 1198860120582119899

+ 1198861120582119899minus1

+ sdot sdot sdot + 119886119899

= 0 be thecharacteristic equation evaluated at the equilibrium point Δ

119899

the n-dimensional Routh-Hurwitz determinant and 119886119894 (119894 =

0 1 119899) the coefficient of the characteristic equation If thereexists a threshold of bifurcation parameter 119908

119888which conforms

to the following terms then the first two existence conditions ofHopf bifurcation are satisfied

(1) Δ119899minus1

(119908119888) = 0 Δ

119899minus2(119908119888) = 0 Δ

119899minus3(119908119888) = 0

(2) 119886119894(119908119888) gt 0 (0 1 119899)

(3) Δ1015840

119899minus1(119908119888) = 0

6 Abstract and Applied Analysis

0 50 100 150 200 250 300

0

025

t

005

01

015

02

minus015

minus01

minus005

x0

(a) The time trajectory of EMR

0 005 01 015 02 025

0

005

01

015

02

y0

x0

minus015

minus01

minus005

minus02minus015 minus01 minus005

(b) The phase diagram of EMR in 1199090minus 1199100plane

0 50 100 150 200 250 300

0

025

t

x(t0)

005

01

015

02

minus015

minus01

minus005

(c) The time trajectory of SRM

0 005 01 015 02 025

0

005

01

015

minus015

minus01

minus005

minus02minus015 minus01 minus005

x(t 0)

y(t0)

(d) The phase diagram of SRM

Figure 2 The time trajectories and phase diagrams for 119906 = minus009

According to Theorem 1 the following parameter rela-tions which can make the equality Δ

9= 0 are derived

120583 = 0 120583 = plusmn120575

2 120583 = plusmn

120575

4 120583 = plusmn

radic3120575

2

120583 = plusmnradic3120575

4 120583 =

120575

4plusmn

radic3120575

4 120583 = minus

120575

4plusmn

radic3120575

4

(31)

Then (31) is substituted into the 8-dimensional and 7-dimensional Routh-Hurwitz determinants Δ

8 Δ7 and only

the following parameters can make Δ8

= 0 Δ7

= 0 satisfied

120583 = plusmn120575

2

120583 = plusmnradic3120575

2

(32)

Meanwhile the relations which can make all the coefficientssatisfied 119886

119894gt 0 (119894 = 0 1 10) are

120583 = plusmnradic3120575

2 (33)

We substitute the relations (33) intoΔ1015840

9(120583) andfind thatΔ1015840

9(120583)

is not equal to 0 therefore according to Theorem 1 the firsttwo existence conditions of Hopf bifurcation are satisfiedWhen (33) is substituted into the characteristic equation allthe eigenvalues of it can be obtained

12058212

= plusmnradic119887119894

12058234

=120575

4minus

radic3120575

4plusmn

1

4

radic41205752 minus 2radic31205752 minus 16119887

12058256

= minusradic3120575

4plusmn

1

4

radic31205752 minus 16119887

Abstract and Applied Analysis 7

0 50 100 150 200 250 300

0

025

t

005

01

015

02

minus015

minus01

minus005

x0

minus02

(a) The time trajectory of EMR

y0

x0

0 005 01

0

002

004

006

008

01

minus01minus005minus01

minus002

minus004

minus006

minus008

(b) The phase diagram of EMR in 1199090minus 1199100plane

0 50 100 150 200 250 300

025

t

005

01

015

02

minus015

minus01

minus005

0

x(t0)

(c) The time trajectory of SRM

0 001 002 003

0

001

002

003

minus003 minus002 minus001minus003

minus002

minus001

x(t 0)

y(t0)

(d) The phase diagram of SRM

Figure 3 The time trajectories and phase diagrams for 119906 = minus001

12058278

= minus120575

4minus

radic3120575

4plusmn

1

4

radic41205752 minus 2radic31205752 minus 16119887

120582910

= minusradic3120575

2plusmn

1

2

radic31205752 minus 4119887

(34)

Obviously when the intensity of random parameter 120575 = 0 thereal parts of all the other eigenvalues are less than 0 except1205821and 120582

2 The characteristic equation does not have zero

solution for 119887 = 0 Therefore all the existence conditions ofHopf bifurcation are satisfied When 120583

119888= minusradic31205752 the Hopf

bifurcation occurs around the equilibrium (0 0)The parameters are taken as 119886 = 1 119887 = 1 and 120575 =

01 and the derivative order is 119902 = 0995 And initialconditions are [119909

0(0) 119909

4(0)] = (02 02 02 02)

119879 and[1199100(0) 119910

4(0)] = (minus02 minus02 minus02 minus02)

119879 According tothe above analysis we can get that the critical parameter valueof the bifurcation is 120583

119888= minusradic31205752 asymp minus00866 which implies

that if 120583 lt 120583119888 the system will converge to the equilibrium

otherwise it will converge to a limit cycle The parametervalues are chosen as 120583 = minus009 and 119906 = minus001 respectivelyThen we obtain the time trajectories and phase diagrams forthe EMR and SRM of the stochastic system by simulationcomputations see Figures 2 and 3 It is clearly seen that thesystem converges to the fixed point (0 0) for 120583 = minus009 andto a limit cycle for 119906 = minus001 which implies that the Hopfbifurcation occurs in the stochastic system (18) Meanwhilethe SRM and EMR of phase plots coincide well enoughIt means that the Chebyshev polynomial approximationmethod is effective for the stochastic fractional-order system(18)

When the others parameters are fixed and 120583 = minus002 wealso found that Hopf bifurcation occurs with the variationof the derivative order The stochastic system converges tothe equilibrium when 119902 le 098 and to a limit cycle for 119902 ge

099 which means that the Hopf bifurcation occurs whenthe derivative order 119902 = 099 see Figure 4 from which it

8 Abstract and Applied Analysis

0 005 01 015 02 025

0

005

01

015

02

y0

x0

minus015

minus01

minus005

minus02minus015 minus01 minus005

(a) 119902 = 098

0 005 01

0

002

004

006

008

01

minus002

minus004

minus006

minus008

minus005

y0

x0

minus01minus01

(b) 119902 = 099

Figure 4 The phase diagrams for different values of order 119902

0 100 200 300 400 500 600 700 800

0

005

01

015

02

025

minus005

minus015

minus01

minus02

t

x0

(a)

0

0y0

x0

005

01

015

02

minus005

minus015

minus01

minus02005 01 015 02 025minus005minus015 minus01minus02

(b)

Figure 5 The time trajectory and phase diagram of EMR when 119905 isin [0 800] (a) The time trajectory (b) The phase diagram in 1199090minus 1199100plane

is obtained that the derivative order can also be taken as abifurcation parameter for fractional-order system

The simulation time is taken as 119905 = 800 one integrationstep of time is taken as Δ119905 = 0001 derivative order 119902 = 097and divide the time interval [0 800] into 8 equal parts Thetime trajectory and phase diagram for 119905 = 800 are shownin Figure 5 from which it appears that the 119909

0converges to a

fixed point as 119905 ge 300 However the time trajectory is zoomedin when 119905 belongs to the interval [300 400] see Figure 6(a)fromwhich we can see that the trajectory gradually decreasesbut does not converge to a point directly Secondly the timetrajectory is zoomed in when 119905 isin [400 500] see Figure 6(b)Meanwhile we also obtain the trajectories for 119905 isin [500 600]119905 isin [600 700] and 119905 isin [700 800] which are shown in Figures6(c)ndash6(e) respectively Comparing these figures it is seenthat the amplitude of 119909

0decreases as the time 119905 increases

Based on these observations we can suggest that the EMRsof the stochastic system (18) have an extremely long transientprocess while converging to the equilibrium as 119905 rarr infin

6 Conclusions

In this paper the Hopf bifurcation of a fractional-order VDPsystem with a random parameter is analyzed An equivalentdeterministic system is obtained with the approximationprinciple of Chebyshev orthogonal polynomials Based onbifurcation theory of nonlinear dynamic systems the criticalparameter value of the bifurcation is obtained by theoreticalanalysis Meanwhile numerical simulations are performedto verify the theoretical results Besides the derivative ordercan also be taken as a bifurcation parameter and the Hopfbifurcation is observed with the variation of the order

Abstract and Applied Analysis 9

300 320 340 360 380 400

012345times10minus4

minus1

minus2

minus3

minus4

minus5

t

x0

(a) 119905 isin [300 400]

400 420 440 460 480 500

0

05

1

15times10minus4

minus1

minus15

minus05

t

x0

(b) 119905 isin [400 500]

500 520 540 560 580 600

0

1

2

3

4times10minus5

minus1

minus2

minus3

minus4

t

x0

(c) 119905 isin [500 600]

600 620 640 660 680 700

0

2

4

6

8times10minus6

minus2

minus4

minus6

minus8

minus10

t

x0

(d) 119905 isin [600 700]

700 720 740 760 780 800t

0

2

05

1

15

times10minus6

minus2

minus25

minus15

minus1

minus05

x0

(e) 119905 isin [700 800]

Figure 6 The time trajectories of EMR 1199090for different time intervals

10 Abstract and Applied Analysis

It is worth mentioning that the orthogonal polyno-mial approximation is an effective method for integer-orderstochastic systems but the universality of the method forfractional-order ones still remains an open question Due tothe absences of analytical methods the threshold value ofbifurcation when the derivative order is taken as a bifurcationparameter cannot be obtained by theoretical analysis but onlythe numerical experiments

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the National Natural ScienceFoundation of China (NSFC) under Grant nos 1133200811172223 and 11172224

References

[1] C P Li W H Deng and D Xu ldquoChaos synchronization ofthe Chua system with a fractional orderrdquo Physica A StatisticalMechanics and Its Applications vol 360 no 2 pp 171ndash185 2006

[2] Z M Ge and C Y Ou ldquoChaos in a fractional order modifiedDuffing systemrdquo Chaos Solitons amp Fractals vol 34 no 2 pp262ndash291 2007

[3] J G Lu and G Chen ldquoA note on the fractional-order Chensystemrdquo Chaos Solitons and Fractals vol 27 no 3 pp 685ndash6882006

[4] H Sun W Chen and Y Chen ldquoVariable-order fractional dif-ferential operators in anomalous diffusion modelingrdquo PhysicaA Statistical Mechanics and Its Applications vol 388 no 21 pp4586ndash4592 2009

[5] Y Liang and W Chen ldquoA survey on computing Levy stabledistributions and a new MATLAB toolboxrdquo Signal Processingvol 93 no 1 pp 242ndash251 2013

[6] Y J Liang and W Chen ldquoReliability analysis for sluice gateanti-sliding stability using Levy stable distributionsrdquo SignalProcessing 2014

[7] V Venkatasubramanian ldquoSingularity induced bifurcation andthe van der Pol oscillatorrdquo IEEE Transactions on Circuits andSystems I Fundamental Theory and Applications vol 41 no 11pp 765ndash769 1994

[8] R Mettin U Parlitz andW Lauterborn ldquoBifurcation structureof the driven van der Pol oscillatorrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol3 no 6 pp 1529ndash1555 1993

[9] U Parlitz and W Lauterborn ldquoPeriod-doubling cascades anddevils staircases of the driven van der Pol oscillatorrdquo PhysicalReview A vol 36 no 3 pp 1428ndash1434 1987

[10] A Buonomo ldquoThe periodic solution of van der Polrsquos equationrdquoSIAM Journal on AppliedMathematics vol 59 no 1 pp 156ndash1711998

[11] J-X Xu and J Jiang ldquoThe global bifurcation characteristics ofthe forced van der Pol oscillatorrdquo Chaos Solitons and Fractalsvol 7 no 1 pp 3ndash19 1996

[12] R S Barbosa J A T MacHado B M Vinagre and A JCaldern ldquoAnalysis of the Van der Pol oscillator containing

derivatives of fractional orderrdquo Journal of Vibration and Controlvol 13 no 9-10 pp 1291ndash1301 2007

[13] J-H Chen andW-C Chen ldquoChaotic dynamics of the fraction-ally damped van der Pol equationrdquo Chaos Solitons and Fractalsvol 35 no 1 pp 188ndash198 2008

[14] M Shinozuka ldquoProbability modeling of concrete structuresrdquoASCE Journal of the EngineeringMechanics Division vol 98 no6 pp 1433ndash1451 1972

[15] R Ghamem and P Spans Stochastic Finite Element A SpectralApproach Springer Berlin Germany 1991

[16] C L Pettit and P S Beran ldquoSpectral and multiresolutionWiener expansions of oscillatory stochastic processesrdquo Journalof Sound and Vibration vol 294 no 4 pp 752ndash779 2006

[17] O P Le Maıtre O M Knio H N Najm and R G GhanemldquoUncertainty propagation using Wiener-Haar expansionsrdquoJournal of Computational Physics vol 197 no 1 pp 28ndash57 2004

[18] O P Le Maıtre H N Najm R G Ghanem and O M KnioldquoMulti-resolution analysis ofWiener-type uncertainty propaga-tion schemesrdquo Journal of Computational Physics vol 197 no 2pp 502ndash531 2004

[19] D Xiu and G E Karniadakis ldquoModeling uncertainty in steadystate diffusion problems via generalized polynomial chaosrdquoComputer Methods in Applied Mechanics and Engineering vol191 no 43 pp 4927ndash4948 2002

[20] X Wan and G E Karniadakis ldquoAn adaptive multi-elementgeneralized polynomial chaosmethod for stochastic differentialequationsrdquo Journal of Computational Physics vol 209 no 2 pp617ndash642 2005

[21] X L Leng C L Wu X P Ma G Meng and T Fang ldquoBifur-cation and chaos analysis of stochastic Duffing system underharmonic excitationsrdquo Nonlinear Dynamics vol 42 no 2 pp185ndash198 2005

[22] S-J MaW XuW Li and T Fang ldquoAnalysis of stochastic bifur-cation and chaos in stochastic Duffing-van der Pol system viaChebyshev polynomial approximationrdquo Chinese Physics vol 15no 6 article 017 pp 1231ndash1238 2006

[23] S J MaW Xu and T Fang ldquoAnalysis of period-doubling bifur-cation in double-well stochastic Duffing system via Laguerrepolynomial approximationrdquoNonlinear Dynamics vol 52 no 3pp 289ndash299 2008

[24] S-J Ma Q Shen and J Hou ldquoModified projective synchro-nization of stochastic fractional order chaotic systems withuncertain parametersrdquo Nonlinear Dynamics vol 73 no 1-2 pp93ndash100 2013

[25] X Liu L Hong and L Yang ldquoFractional-order complex Tsystem bifurcations chaos control and synchronizationrdquoNon-linear Dynamics vol 75 no 3 pp 589ndash602 2014

[26] K Diethelm N J Ford and A D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 3ndash22 2002

[27] A Charef H H Sun Y-Y Tsao and B Onaral ldquoFractal systemas represented by singularity functionrdquo IEEE Transactions onAutomatic Control vol 37 no 9 pp 1465ndash1470 1992

[28] M S Tavazoei andM Haeri ldquoLimitations of frequency domainapproximation for detecting chaos in fractional order systemsrdquoNonlinear Analysis Theory Methods amp Applications vol 69 no4 pp 1299ndash1320 2008

[29] P Borwein and T Erdelyi Polynomials and Polynomial Inequal-ities Springer New York NY USA 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

The period-doubling bifurcation for double-well stochasticDuffing system was investigated via Lagurre polynomialapproximation in [23] Recently the synchronization for astochastic fractional-order system with a random parametervia Laguerre polynomial was analyzed in [24] However theliterature on the investigation of a stochastic fractional-orderVDP system is few

Inspired by the above discussion in this paper Hopfbifurcations of a fractional-order VDP system with a randomparameter are studied Firstly the Chebyshev polynomialapproximationmethod is applied to investigate the stochasticfractional-order system Based on this method the stochasticsystem is reduced to the equivalent deterministic one Thenaccording to the existence conditions of Hopf bifurcationthe critical parameter value of the bifurcation is obtainedby theoretical analysis Meanwhile numerical simulationsare performed to verify the theoretical results Besides theHopf bifurcation with the variation of derivative order is alsoobserved by numerical computations

The paper is organized as follows In Section 2 the defi-nitions for the fractional calculus and numerical algorithmsare given In Section 3 the Chebyshev polynomials are intro-duced as the basis of orthogonal polynomial approximationIn Section 4 the transformation of stochastic fractional-order VDP system into its equivalent deterministic one byChebyshev polynomial approximation is shown Section 5 isdevoted to studying the Hopf bifurcation of the stochasticsystem Finally we summarize the results in Section 6

2 Fundamentals of Fractional Derivative

21 Definition Fractional calculus is a generalization ofintegration and differentiation to a noninteger-order inte-grodifferential operator

119886119863119902

119905which is defined by

119886119863119902

119905=

119889119902

119889119905119902119877 (119902) gt 0

1 119877 (119902) = 0

int

119905

119886

(119889120591)minus119902

119877 (119902) lt 0

(1)

where 119902 is the derivative order which can be a complexnumber and 119877(119902) is the real part of 119902 The numbers 119886 and119905 are the limits of the operatorThere are many definitions forfractional derivatives Three most frequently used ones arethe Grunwald-Letnikov definition the Riemann-Liouvilleand the Caputo definitions

The Grunwald-Letnikov definition (GL) derivative withfractional-order 119902 is described by

GL119886

119863119902

119905119891 (119905) = lim

ℎrarr0

1

ℎ119902

[(119905minus119886)ℎ]

sum

119895=0

(minus1)119895

(119902

119895)119891 (119905 minus 119895ℎ) (2)

where the symbol [∙] means the integer part

The Riemann-Liouville (RL) definition of fractionalderivatives is given by

RL119886

119863119902

119905119891 (119905) =

119889119899

119889119905119899

1

Γ (119899 minus 119902)int

119905

119886

119891 (120591)

(119905 minus 120591)119902minus119899+1

119889120591

119899 minus 1 lt 119902 lt 119899

(3)

where Γ(∙) is the gamma functionThe Caputo (119862) fractional derivative is defined as follows

119862

119886119863119902

119905119891 (119905) =

1

Γ (119899 minus 119902)int

119905

119886

(119905 minus 120591)119899minus119902minus1

119891(119899)

(120591) 119889120591

119899 minus 1 lt 119902 lt 119899

(4)

It is well known that the initial conditions for thefractional differential equations with Caputo derivatives takethe same form as for the integer-order ones which is verysuitable for practical problems [25]Therefore we will use theCaputo definition for the fractional derivatives in this paper

22 Numerical Algorithms Obtaining numerical solutionsof a fractional differential equation is not easily comparedwith that for ordinary differential equations There are twoapproximation methods which can frequently be used tonumerical computations on chaos and bifurcations withfractional differential equations One is an improved ver-sion of Adams-Bashforth-Moulton algorithm based on thepredictor-correctors scheme [26] which is a time-domainapproach The other is a method known as frequencydomain approximation [27] based on numerical analysis offractional-order systems in the frequency domain

Simulations for fractional-order systems using the timedomain methods are complicated and due to long memorycharacteristics of these systems require a very long simulationtime but on the other hand it ismore accurate [28]Thereforewe employ the improved predictor-corrector algorithm forfractional-order differential equations in this paper

In order to get the approximate solution of a fractional-order chaotic system by the improved predictor-correctoralgorithm the following equation is considered

119889119902

119909

119889119905119902= 119891 (119905 119909) 0 le 119905 le 119879

119909(119896)

(0) = 119909(119896)

0 119896 = 0 1 lceil119902rceil minus 1

(5)

where lceil119902rceil is just the value 119902 rounded up to the nearest integerand 119909

(119896) is the ordinary 119896th derivative of 119909 Formula (5) isequivalent to the Volterra integral equation

119909 (119905) =

lceil119902rceilminus1

sum

119896=0

119909(119896)

0

119905119896

119896+

1

Γ (119902)int

119905

0

(119905 minus 120591)119902minus1

119891 (120591 119909 (120591)) 119889120591 (6)

Now for the sake of simplicity we assume that we areworkingon a uniform grid 119905

119899= 119899ℎ 119899 = 0 1 119872 with some

integer 119872 and set ℎ = 119879119872 Using the standard quadraturetechniques for the integral in (6) denote 119892(120591) = 119891(120591 119909(120591))

Abstract and Applied Analysis 3

the integral is replaced by the trapezoidal quadrature formulaat point 119905

119899+1

int

119905119899+1

0

(119905119899+1

minus 120591)119902minus1

119892 (120591) 119889120591 asymp int

119905119899+1

0

(119905119899+1

minus 120591)119902minus1

119892119899+1

(120591) 119889120591

(7)

where119892119899+1

is the piecewise linear interpolant for119892with nodesand knots chosen at the 119905

119895 (119895 = 0 1 119899 + 1) After some

elementary calculations the right hand side of (7) gives

int

119905119899+1

0

(119905119899+1

minus 120591)119902minus1

119892119899+1

(120591) 119889120591 =ℎ119902

119902 (119902 + 1)

119899+1

sum

119895=0

120572119895119899+1

119892 (119905119895)

(8)

And if we use the product rectangle rule the right hand of (7)can be written as

int

119905119899+1

0

(119905119899+1

minus 120591)

119902minus1

119892119899+1

(120591) 119889120591 =

119899+1

sum

119895=0

120573119895119899+1

119892 (119905119895) (9)

where120572119895119899+1

=

119899119902+1

minus (119899 minus 119902) (119899 + 1)119902

119895 = 0

(119899 minus 119895 + 2)119902+1

+ (119899 minus 119895)119902+1

minus 2(119899 minus 119895 + 1)119902+1

1 le 119895 le 119899

1 119895 = 119899 + 1

120573119895119899+1

=ℎ119902

119902((119899 minus 119895 + 1)

119902

minus (119899 minus 119895)119902

) 1 le 119895 le 119899

(10)

Then the predictor and corrector formulae for solving (6) aregiven respectively by

119909119875

ℎ(119905119899+1

) =

lceil119902rceilminus1

sum

119896=0

119909(119896)

0

119905119896

119899+1

119896+

1

Γ (119902)

119899

sum

119895=0

120573119895119899+1

119891 (119905119895 119909ℎ(119905119895))

119909ℎ(119905119899+1

) =

lceil119902rceilminus1

sum

119896=0

119909(119896)

0

119905119896

119899+1

119896+

ℎ119902

Γ (119902 + 2)119891 (119905119899+1

119909119875

ℎ(119905119899+1

))

+ℎ119902

Γ (119902 + 2)

119899

sum

119895=0

120572119895119899+1

119891 (119905119895 119909ℎ(119905119895))

(11)

The approximation accuracy of scheme (11) is 119874(ℎmin21+119902

)

3 The Chebyshev Polynomials

In general random parameters in some engineering struc-tures are bounded in nature So the probability densityfunction (PDF for short) model for the bounded randomvariables is taken to be an arch-like PDF of the form [29]

120588 (119906) =

(2

120587)radic1 minus 1199062 |119906| le 1

0 |119906| gt 1

(12)

0 05 10

01

02

03

04

05

06

07

u

minus1 minus05

120588(u)

Figure 1 The arch-like PDF curve for random variable 119906

Figure 1 shows the arch-like PDF curve for boundedrandom variable 119906 As the orthogonal polynomial basis forthis kind of PDF the second kind of Chebyshev polynomialsis the only choice The general expansion for the second kindof Chebyshev polynomials can be described as follows

119867119899(119906) =

[1198992]

sum

119896=0

(minus1)119896 (119899 minus 119896)

119896 (119899 minus 2119896)(2119906)119899minus2119896

(13)

Then we have

1198670(119906) = 1

1198671(119906) = 2119906

1198672(119906) = 4119906

2

minus 1

1198673(119906) = 8119906

3

minus 4119906

1198674(119906) = 16119906

4

minus 121199062

+ 1

sdot sdot sdot

(14)

The recurrent formula for the Chebyshev polynomials of thesecond kind is

119906119867119899(119906) =

1

2[119867119899minus1

(119906) + 119867119899+1

(119906)] (15)

The orthogonality of Chebyshev polynomial of the secondkind can be expressed as

int

1

minus1

2

120587

radic1 minus 1199062119867119894(119906)119867119895(119906) 119889119906 =

1 (119894 = 119895)

0 (119894 = 119895) (16)

Equation (16) represents a weighted orthogonal relationshipThe weighting function is just the same as that for the PDFof the random parameter which obeys arch distribution120588(119906) in (12) and the left-hand side of (16) can be regardedas the expectation of the product 119867

119894(119906)119867119895(119906) Owing to

4 Abstract and Applied Analysis

the orthogonality of Chebyshev polynomial any measurablefunction 119891(119906) isin 119871

2 can be expanded as

119891 (119906) =

infin

sum

119894=0

119888119894119867119894(119906) (17)

where 119888119894= int1

minus1

120588(119906)119891(119906)119867119894(119906)119889119906 This expansion is called an

orthogonal decomposition of random function119891(119906) which isthe theoretical base of orthogonal decomposition methods

4 The Stochastic Fractional-OrderVDP System

In this paper we consider the fractional-order VDP systemwith a random parameter The system can be described bythe following differential equations

119863119902

119909 = 119910

119863119902

119910 = 120583119910 minus 1198861199092

119910 + 119887119909

(18)

where 119902 is the fractional order and 119909 119910 are state variables 119886119887 are deterministic parameters and 120583 is a random parameterand it can be expressed as

120583 = 120583 + 120575119906 (19)

where 120583 and 1205752 are the mean value and standard deviationof 120583 respectively and 120575 is regarded as the intensity of 120583 119906 isa random variable defined on [minus1 1]

It is well known that the stochastic function spacewhich is constituted with the responses of nonlinear dynamicsystems with random parameters has proved to be a Hilbertspace with respect to convergence in the mean squareTherefore the fractional-order system (18) with randomparameters can be reduced into the equivalent deterministicsystem by using of the orthogonal polynomial expansion Itfollows from the orthogonal polynomial approximation thatthe responses of system (18) can be expressed approximatelyby the following series

119909 (119905 119906) =

119873

sum

119894=0

119909119894(119905)119867119894(119906)

119910 (119905 119906) =

119873

sum

119894=0

119910119894(119905)119867119894(119906)

(20)

where119867119894(119906) represents the 119894th Chebyshev polynomial and119873

is the largest order of the polynomialsThen substituting (20)into the system (18) we can get

119863119902

(

119873

sum

119894=0

119909119894(119905)119867119894(119906)) = minus(

119873

sum

119894=0

119910119894(119905)119867119894(119906))

119863119902

(

119873

sum

119894=0

119910119894(119905)119867119894(119906))

= 120583(

119873

sum

119894=0

119910119894(119905)119867119894(119906)) + 120575119906(

119873

sum

119894=0

119910119894(119905)119867119894(119906))

minus 119886(

119873

sum

119894=0

119909119894(119905)119867119894(119906))

2

(

119873

sum

119894=0

119910119894(119905)119867119894(119906))

+ 119887(

119873

sum

119894=0

119909119894(119905)119867119894(119906))

(21)

and with the aid of the recurrent formulas of Chebyshevpolynomial all the nonlinear terms in the system can befurther reduced into linear combination of 119871

119894(119906) as follows

(

119873

sum

119894=0

119909119894(119905)119867119894(119906))

2

(

119873

sum

119894=0

119910119894(119905)119867119894(119906))

2

=

3119873

sum

119894=0

119878119894(119905)119867119894(119906) (22)

where 119878119894(119905) can be obtained with the help of the computer

algebraic systems such as Maple or MATLAB Meanwhilethe term 120575119906(sum

119873

119894=0119910119894(119905)119867119894(119906)) can be reduced to

119906120575(

119873

sum

119894=0

119910119894(119905)119867119894(119906)) = 120575(

119873

sum

119894=0

119910119894(119905)

119867119894minus1

(119906) + 119867119894+1

(119906)

2)

(23)

Substituting (22) and (23) into (21) then we can have

119863119902

(

119873

sum

119894=0

119909119894(119905)119867119894(119906)) = minus(

119873

sum

119894=0

119910119894(119905)119867119894(119906))

119863119902

(

119873

sum

119894=0

119910119894(119905)119867119894(119906)) = 120583(

119873

sum

119894=0

119910119894(119905)119867119894(119906))

+ 120575(

119873

sum

119894=0

119910119894(119905)

119867119894minus1

(119906) + 119867119894+1

(119906)

2)

minus 119886

119873

sum

119894=0

119878119894(119905)119867119894(119906)

+ 119887(

119873

sum

119894=0

119909119894(119905)119867119894(119906))

(24)

Multiply both sides of (24) by 119867119894(119906) 119894 = 0 1 2 119873 in

sequence and take expectation with respect to 119906Then we canfinally obtain the equivalent deterministic system Rememberthat if 119873 rarr infin in (24) then its left is strictly equivalent tothe response of stochastic system (18) Otherwise (24) is justapproximately valid with aminimal residual error Due to therequirement of computational precision we take119873 = 4 in the

Abstract and Applied Analysis 5

following numerical analysisTherefore we get the equivalentdeterministic system approximately as

119863119902

1199090= minus 119910

0

119863119902

1199100= minus 119886119878

0+ 1198871199090+ 1205831199100+

1

21205751199101

119863119902

1199091= minus 119910

1

119863119902

1199101= minus 119886119878

1+ 1198871199091+ 1205831199101+

1

2120575 (1199100+ 1199102)

119863119902

1199092= minus 119910

2

119863119902

1199102= minus 119886119878

2+ 1198871199092+ 1205831199102+

1

2120575 (1199101+ 1199103)

119863119902

1199093= minus 119910

3

119863119902

1199103= minus 119886119878

3+ 1198871199093+ 1205831199103+

1

2120575 (1199102+ 1199104)

119863119902

1199094= minus 119910

4

119863119902

1199104= minus 119886119878

4+ 1198871199094+ 1205831199104+

1

21205751199103

(25)

We can get the numerical solutions 119909119894 119910119894 (119894 = 0 1 2 3 4) of

system (25) by the improved predictor-corrector algorithmTherefore the approximate random responses of the stochas-tic system (18) can be expressed as

119909 (119905 119906) asymp

4

sum

119894=0

119909119894(119905)119867119894(119906)

119910 (119905 119906) asymp

4

sum

119894=0

119910119894(119905)119867119894(119906)

(26)

The ensemble mean responses (EMRs for short) of thestochastic fractional-order system are described as

119864 [119909 (119905 119906)] asymp

4

sum

119894=0

119909119894(119905) 119864 [119867

119894(119906)] = 119909

0(119905)

119864 [119910 (119905 119906)] asymp

4

sum

119894=0

119910119894(119905) 119864 [119867

119894(119906)] = 119910

0(119905)

(27)

Especially for 119906 = 0 namely 120583 = 120583 this sample system issignificant for reference It is usually called a mean parametersystem of which the responses (SRMs for short) may beapproximated as

119909 (119905 0) asymp

4

sum

119894=0

119909119894(119905)119867119894(0) = 119909

0(119905) minus 119909

2(119905) + 119909

4(119905)

119910 (119905 0) asymp

4

sum

119894=0

119910119894(119905)119867119894(0) = 119910

0(119905) minus 119910

2(119905) + 119910

4(119905)

(28)

5 Hopf Bifurcations

A Hopf bifurcation occurs when a periodic solution or limitcycle surrounding an equilibrium point arises or goes awayas a parameter varies In this section the Hopf bifurcation forthe stochastic fractional-order VDP system will be studiedwith theoretical and numerical computations Obviously

both the stochastic VDP and equivalent deterministic sys-tems (18) and (25) only have one equilibrium point namelythe origin (00) The Jacobian matrix 119869 for the linearizedsystem evaluated at the origin is

119869 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

0 minus1 0 0 0 0 0 0 0 0

119887 120583 01

2120575 0 0 0 0 0 0

0 0 0 minus1 0 0 0 0 0 0

1

2120575 0 119887 120583 0

1

2120575 0 0 0 0

0 0 0 0 0 minus1 0 0 0 0

0 0 01

2120575 119887 120583 0

1

2120575 0 0

0 0 0 0 0 0 0 minus1 0 0

0 0 0 0 01

2120575 119887 120583 0

1

2120575

0 0 0 0 0 0 0 0 0 minus1

0 0 0 0 0 0 01

2120575 119887 120583

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(29)

The corresponding characteristic equation is

119891 (120582) = 119886012058210

+ 11988611205829

+ 11988621205828

+ 11988631205827

+ 11988641205826

+ 11988651205825

+ 11988661205824

+ 11988671205823

+ 11988681205822

+ 1198869120582 + 11988610

(30)

where 1198860 1198861 119886

10are the coefficients of (30) and can be

obtained by computing via Maple or MATLAB According tothe bifurcation theory of nonlinear dynamical systems andtaking 119908 as a bifurcation parameter [22] we can obtain theexistence conditions of Hopf bifurcation for the system

(1) The eigenvalues of the linearized system about thefixed point are 120572(119908) plusmn 119894120573(119908)

(2) The real and image parts of the eigenvalues satisfy120572(119908119888) = 0 120573(119908

119888) gt 0 1205721015840(119908

119888) = 0 when 119908 = 119908

119888

(3) The real parts of other eigenvalues are not equal tozeros

For the high order of equivalent system (25) so it is hardto get the threshold value of bifurcation parameter directlyTherefore we can get the threshold via the following theorem

Theorem 1 Let 119891(120582) = 1198860120582119899

+ 1198861120582119899minus1

+ sdot sdot sdot + 119886119899

= 0 be thecharacteristic equation evaluated at the equilibrium point Δ

119899

the n-dimensional Routh-Hurwitz determinant and 119886119894 (119894 =

0 1 119899) the coefficient of the characteristic equation If thereexists a threshold of bifurcation parameter 119908

119888which conforms

to the following terms then the first two existence conditions ofHopf bifurcation are satisfied

(1) Δ119899minus1

(119908119888) = 0 Δ

119899minus2(119908119888) = 0 Δ

119899minus3(119908119888) = 0

(2) 119886119894(119908119888) gt 0 (0 1 119899)

(3) Δ1015840

119899minus1(119908119888) = 0

6 Abstract and Applied Analysis

0 50 100 150 200 250 300

0

025

t

005

01

015

02

minus015

minus01

minus005

x0

(a) The time trajectory of EMR

0 005 01 015 02 025

0

005

01

015

02

y0

x0

minus015

minus01

minus005

minus02minus015 minus01 minus005

(b) The phase diagram of EMR in 1199090minus 1199100plane

0 50 100 150 200 250 300

0

025

t

x(t0)

005

01

015

02

minus015

minus01

minus005

(c) The time trajectory of SRM

0 005 01 015 02 025

0

005

01

015

minus015

minus01

minus005

minus02minus015 minus01 minus005

x(t 0)

y(t0)

(d) The phase diagram of SRM

Figure 2 The time trajectories and phase diagrams for 119906 = minus009

According to Theorem 1 the following parameter rela-tions which can make the equality Δ

9= 0 are derived

120583 = 0 120583 = plusmn120575

2 120583 = plusmn

120575

4 120583 = plusmn

radic3120575

2

120583 = plusmnradic3120575

4 120583 =

120575

4plusmn

radic3120575

4 120583 = minus

120575

4plusmn

radic3120575

4

(31)

Then (31) is substituted into the 8-dimensional and 7-dimensional Routh-Hurwitz determinants Δ

8 Δ7 and only

the following parameters can make Δ8

= 0 Δ7

= 0 satisfied

120583 = plusmn120575

2

120583 = plusmnradic3120575

2

(32)

Meanwhile the relations which can make all the coefficientssatisfied 119886

119894gt 0 (119894 = 0 1 10) are

120583 = plusmnradic3120575

2 (33)

We substitute the relations (33) intoΔ1015840

9(120583) andfind thatΔ1015840

9(120583)

is not equal to 0 therefore according to Theorem 1 the firsttwo existence conditions of Hopf bifurcation are satisfiedWhen (33) is substituted into the characteristic equation allthe eigenvalues of it can be obtained

12058212

= plusmnradic119887119894

12058234

=120575

4minus

radic3120575

4plusmn

1

4

radic41205752 minus 2radic31205752 minus 16119887

12058256

= minusradic3120575

4plusmn

1

4

radic31205752 minus 16119887

Abstract and Applied Analysis 7

0 50 100 150 200 250 300

0

025

t

005

01

015

02

minus015

minus01

minus005

x0

minus02

(a) The time trajectory of EMR

y0

x0

0 005 01

0

002

004

006

008

01

minus01minus005minus01

minus002

minus004

minus006

minus008

(b) The phase diagram of EMR in 1199090minus 1199100plane

0 50 100 150 200 250 300

025

t

005

01

015

02

minus015

minus01

minus005

0

x(t0)

(c) The time trajectory of SRM

0 001 002 003

0

001

002

003

minus003 minus002 minus001minus003

minus002

minus001

x(t 0)

y(t0)

(d) The phase diagram of SRM

Figure 3 The time trajectories and phase diagrams for 119906 = minus001

12058278

= minus120575

4minus

radic3120575

4plusmn

1

4

radic41205752 minus 2radic31205752 minus 16119887

120582910

= minusradic3120575

2plusmn

1

2

radic31205752 minus 4119887

(34)

Obviously when the intensity of random parameter 120575 = 0 thereal parts of all the other eigenvalues are less than 0 except1205821and 120582

2 The characteristic equation does not have zero

solution for 119887 = 0 Therefore all the existence conditions ofHopf bifurcation are satisfied When 120583

119888= minusradic31205752 the Hopf

bifurcation occurs around the equilibrium (0 0)The parameters are taken as 119886 = 1 119887 = 1 and 120575 =

01 and the derivative order is 119902 = 0995 And initialconditions are [119909

0(0) 119909

4(0)] = (02 02 02 02)

119879 and[1199100(0) 119910

4(0)] = (minus02 minus02 minus02 minus02)

119879 According tothe above analysis we can get that the critical parameter valueof the bifurcation is 120583

119888= minusradic31205752 asymp minus00866 which implies

that if 120583 lt 120583119888 the system will converge to the equilibrium

otherwise it will converge to a limit cycle The parametervalues are chosen as 120583 = minus009 and 119906 = minus001 respectivelyThen we obtain the time trajectories and phase diagrams forthe EMR and SRM of the stochastic system by simulationcomputations see Figures 2 and 3 It is clearly seen that thesystem converges to the fixed point (0 0) for 120583 = minus009 andto a limit cycle for 119906 = minus001 which implies that the Hopfbifurcation occurs in the stochastic system (18) Meanwhilethe SRM and EMR of phase plots coincide well enoughIt means that the Chebyshev polynomial approximationmethod is effective for the stochastic fractional-order system(18)

When the others parameters are fixed and 120583 = minus002 wealso found that Hopf bifurcation occurs with the variationof the derivative order The stochastic system converges tothe equilibrium when 119902 le 098 and to a limit cycle for 119902 ge

099 which means that the Hopf bifurcation occurs whenthe derivative order 119902 = 099 see Figure 4 from which it

8 Abstract and Applied Analysis

0 005 01 015 02 025

0

005

01

015

02

y0

x0

minus015

minus01

minus005

minus02minus015 minus01 minus005

(a) 119902 = 098

0 005 01

0

002

004

006

008

01

minus002

minus004

minus006

minus008

minus005

y0

x0

minus01minus01

(b) 119902 = 099

Figure 4 The phase diagrams for different values of order 119902

0 100 200 300 400 500 600 700 800

0

005

01

015

02

025

minus005

minus015

minus01

minus02

t

x0

(a)

0

0y0

x0

005

01

015

02

minus005

minus015

minus01

minus02005 01 015 02 025minus005minus015 minus01minus02

(b)

Figure 5 The time trajectory and phase diagram of EMR when 119905 isin [0 800] (a) The time trajectory (b) The phase diagram in 1199090minus 1199100plane

is obtained that the derivative order can also be taken as abifurcation parameter for fractional-order system

The simulation time is taken as 119905 = 800 one integrationstep of time is taken as Δ119905 = 0001 derivative order 119902 = 097and divide the time interval [0 800] into 8 equal parts Thetime trajectory and phase diagram for 119905 = 800 are shownin Figure 5 from which it appears that the 119909

0converges to a

fixed point as 119905 ge 300 However the time trajectory is zoomedin when 119905 belongs to the interval [300 400] see Figure 6(a)fromwhich we can see that the trajectory gradually decreasesbut does not converge to a point directly Secondly the timetrajectory is zoomed in when 119905 isin [400 500] see Figure 6(b)Meanwhile we also obtain the trajectories for 119905 isin [500 600]119905 isin [600 700] and 119905 isin [700 800] which are shown in Figures6(c)ndash6(e) respectively Comparing these figures it is seenthat the amplitude of 119909

0decreases as the time 119905 increases

Based on these observations we can suggest that the EMRsof the stochastic system (18) have an extremely long transientprocess while converging to the equilibrium as 119905 rarr infin

6 Conclusions

In this paper the Hopf bifurcation of a fractional-order VDPsystem with a random parameter is analyzed An equivalentdeterministic system is obtained with the approximationprinciple of Chebyshev orthogonal polynomials Based onbifurcation theory of nonlinear dynamic systems the criticalparameter value of the bifurcation is obtained by theoreticalanalysis Meanwhile numerical simulations are performedto verify the theoretical results Besides the derivative ordercan also be taken as a bifurcation parameter and the Hopfbifurcation is observed with the variation of the order

Abstract and Applied Analysis 9

300 320 340 360 380 400

012345times10minus4

minus1

minus2

minus3

minus4

minus5

t

x0

(a) 119905 isin [300 400]

400 420 440 460 480 500

0

05

1

15times10minus4

minus1

minus15

minus05

t

x0

(b) 119905 isin [400 500]

500 520 540 560 580 600

0

1

2

3

4times10minus5

minus1

minus2

minus3

minus4

t

x0

(c) 119905 isin [500 600]

600 620 640 660 680 700

0

2

4

6

8times10minus6

minus2

minus4

minus6

minus8

minus10

t

x0

(d) 119905 isin [600 700]

700 720 740 760 780 800t

0

2

05

1

15

times10minus6

minus2

minus25

minus15

minus1

minus05

x0

(e) 119905 isin [700 800]

Figure 6 The time trajectories of EMR 1199090for different time intervals

10 Abstract and Applied Analysis

It is worth mentioning that the orthogonal polyno-mial approximation is an effective method for integer-orderstochastic systems but the universality of the method forfractional-order ones still remains an open question Due tothe absences of analytical methods the threshold value ofbifurcation when the derivative order is taken as a bifurcationparameter cannot be obtained by theoretical analysis but onlythe numerical experiments

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the National Natural ScienceFoundation of China (NSFC) under Grant nos 1133200811172223 and 11172224

References

[1] C P Li W H Deng and D Xu ldquoChaos synchronization ofthe Chua system with a fractional orderrdquo Physica A StatisticalMechanics and Its Applications vol 360 no 2 pp 171ndash185 2006

[2] Z M Ge and C Y Ou ldquoChaos in a fractional order modifiedDuffing systemrdquo Chaos Solitons amp Fractals vol 34 no 2 pp262ndash291 2007

[3] J G Lu and G Chen ldquoA note on the fractional-order Chensystemrdquo Chaos Solitons and Fractals vol 27 no 3 pp 685ndash6882006

[4] H Sun W Chen and Y Chen ldquoVariable-order fractional dif-ferential operators in anomalous diffusion modelingrdquo PhysicaA Statistical Mechanics and Its Applications vol 388 no 21 pp4586ndash4592 2009

[5] Y Liang and W Chen ldquoA survey on computing Levy stabledistributions and a new MATLAB toolboxrdquo Signal Processingvol 93 no 1 pp 242ndash251 2013

[6] Y J Liang and W Chen ldquoReliability analysis for sluice gateanti-sliding stability using Levy stable distributionsrdquo SignalProcessing 2014

[7] V Venkatasubramanian ldquoSingularity induced bifurcation andthe van der Pol oscillatorrdquo IEEE Transactions on Circuits andSystems I Fundamental Theory and Applications vol 41 no 11pp 765ndash769 1994

[8] R Mettin U Parlitz andW Lauterborn ldquoBifurcation structureof the driven van der Pol oscillatorrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol3 no 6 pp 1529ndash1555 1993

[9] U Parlitz and W Lauterborn ldquoPeriod-doubling cascades anddevils staircases of the driven van der Pol oscillatorrdquo PhysicalReview A vol 36 no 3 pp 1428ndash1434 1987

[10] A Buonomo ldquoThe periodic solution of van der Polrsquos equationrdquoSIAM Journal on AppliedMathematics vol 59 no 1 pp 156ndash1711998

[11] J-X Xu and J Jiang ldquoThe global bifurcation characteristics ofthe forced van der Pol oscillatorrdquo Chaos Solitons and Fractalsvol 7 no 1 pp 3ndash19 1996

[12] R S Barbosa J A T MacHado B M Vinagre and A JCaldern ldquoAnalysis of the Van der Pol oscillator containing

derivatives of fractional orderrdquo Journal of Vibration and Controlvol 13 no 9-10 pp 1291ndash1301 2007

[13] J-H Chen andW-C Chen ldquoChaotic dynamics of the fraction-ally damped van der Pol equationrdquo Chaos Solitons and Fractalsvol 35 no 1 pp 188ndash198 2008

[14] M Shinozuka ldquoProbability modeling of concrete structuresrdquoASCE Journal of the EngineeringMechanics Division vol 98 no6 pp 1433ndash1451 1972

[15] R Ghamem and P Spans Stochastic Finite Element A SpectralApproach Springer Berlin Germany 1991

[16] C L Pettit and P S Beran ldquoSpectral and multiresolutionWiener expansions of oscillatory stochastic processesrdquo Journalof Sound and Vibration vol 294 no 4 pp 752ndash779 2006

[17] O P Le Maıtre O M Knio H N Najm and R G GhanemldquoUncertainty propagation using Wiener-Haar expansionsrdquoJournal of Computational Physics vol 197 no 1 pp 28ndash57 2004

[18] O P Le Maıtre H N Najm R G Ghanem and O M KnioldquoMulti-resolution analysis ofWiener-type uncertainty propaga-tion schemesrdquo Journal of Computational Physics vol 197 no 2pp 502ndash531 2004

[19] D Xiu and G E Karniadakis ldquoModeling uncertainty in steadystate diffusion problems via generalized polynomial chaosrdquoComputer Methods in Applied Mechanics and Engineering vol191 no 43 pp 4927ndash4948 2002

[20] X Wan and G E Karniadakis ldquoAn adaptive multi-elementgeneralized polynomial chaosmethod for stochastic differentialequationsrdquo Journal of Computational Physics vol 209 no 2 pp617ndash642 2005

[21] X L Leng C L Wu X P Ma G Meng and T Fang ldquoBifur-cation and chaos analysis of stochastic Duffing system underharmonic excitationsrdquo Nonlinear Dynamics vol 42 no 2 pp185ndash198 2005

[22] S-J MaW XuW Li and T Fang ldquoAnalysis of stochastic bifur-cation and chaos in stochastic Duffing-van der Pol system viaChebyshev polynomial approximationrdquo Chinese Physics vol 15no 6 article 017 pp 1231ndash1238 2006

[23] S J MaW Xu and T Fang ldquoAnalysis of period-doubling bifur-cation in double-well stochastic Duffing system via Laguerrepolynomial approximationrdquoNonlinear Dynamics vol 52 no 3pp 289ndash299 2008

[24] S-J Ma Q Shen and J Hou ldquoModified projective synchro-nization of stochastic fractional order chaotic systems withuncertain parametersrdquo Nonlinear Dynamics vol 73 no 1-2 pp93ndash100 2013

[25] X Liu L Hong and L Yang ldquoFractional-order complex Tsystem bifurcations chaos control and synchronizationrdquoNon-linear Dynamics vol 75 no 3 pp 589ndash602 2014

[26] K Diethelm N J Ford and A D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 3ndash22 2002

[27] A Charef H H Sun Y-Y Tsao and B Onaral ldquoFractal systemas represented by singularity functionrdquo IEEE Transactions onAutomatic Control vol 37 no 9 pp 1465ndash1470 1992

[28] M S Tavazoei andM Haeri ldquoLimitations of frequency domainapproximation for detecting chaos in fractional order systemsrdquoNonlinear Analysis Theory Methods amp Applications vol 69 no4 pp 1299ndash1320 2008

[29] P Borwein and T Erdelyi Polynomials and Polynomial Inequal-ities Springer New York NY USA 1995

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

the integral is replaced by the trapezoidal quadrature formulaat point 119905

119899+1

int

119905119899+1

0

(119905119899+1

minus 120591)119902minus1

119892 (120591) 119889120591 asymp int

119905119899+1

0

(119905119899+1

minus 120591)119902minus1

119892119899+1

(120591) 119889120591

(7)

where119892119899+1

is the piecewise linear interpolant for119892with nodesand knots chosen at the 119905

119895 (119895 = 0 1 119899 + 1) After some

elementary calculations the right hand side of (7) gives

int

119905119899+1

0

(119905119899+1

minus 120591)119902minus1

119892119899+1

(120591) 119889120591 =ℎ119902

119902 (119902 + 1)

119899+1

sum

119895=0

120572119895119899+1

119892 (119905119895)

(8)

And if we use the product rectangle rule the right hand of (7)can be written as

int

119905119899+1

0

(119905119899+1

minus 120591)

119902minus1

119892119899+1

(120591) 119889120591 =

119899+1

sum

119895=0

120573119895119899+1

119892 (119905119895) (9)

where120572119895119899+1

=

119899119902+1

minus (119899 minus 119902) (119899 + 1)119902

119895 = 0

(119899 minus 119895 + 2)119902+1

+ (119899 minus 119895)119902+1

minus 2(119899 minus 119895 + 1)119902+1

1 le 119895 le 119899

1 119895 = 119899 + 1

120573119895119899+1

=ℎ119902

119902((119899 minus 119895 + 1)

119902

minus (119899 minus 119895)119902

) 1 le 119895 le 119899

(10)

Then the predictor and corrector formulae for solving (6) aregiven respectively by

119909119875

ℎ(119905119899+1

) =

lceil119902rceilminus1

sum

119896=0

119909(119896)

0

119905119896

119899+1

119896+

1

Γ (119902)

119899

sum

119895=0

120573119895119899+1

119891 (119905119895 119909ℎ(119905119895))

119909ℎ(119905119899+1

) =

lceil119902rceilminus1

sum

119896=0

119909(119896)

0

119905119896

119899+1

119896+

ℎ119902

Γ (119902 + 2)119891 (119905119899+1

119909119875

ℎ(119905119899+1

))

+ℎ119902

Γ (119902 + 2)

119899

sum

119895=0

120572119895119899+1

119891 (119905119895 119909ℎ(119905119895))

(11)

The approximation accuracy of scheme (11) is 119874(ℎmin21+119902

)

3 The Chebyshev Polynomials

In general random parameters in some engineering struc-tures are bounded in nature So the probability densityfunction (PDF for short) model for the bounded randomvariables is taken to be an arch-like PDF of the form [29]

120588 (119906) =

(2

120587)radic1 minus 1199062 |119906| le 1

0 |119906| gt 1

(12)

0 05 10

01

02

03

04

05

06

07

u

minus1 minus05

120588(u)

Figure 1 The arch-like PDF curve for random variable 119906

Figure 1 shows the arch-like PDF curve for boundedrandom variable 119906 As the orthogonal polynomial basis forthis kind of PDF the second kind of Chebyshev polynomialsis the only choice The general expansion for the second kindof Chebyshev polynomials can be described as follows

119867119899(119906) =

[1198992]

sum

119896=0

(minus1)119896 (119899 minus 119896)

119896 (119899 minus 2119896)(2119906)119899minus2119896

(13)

Then we have

1198670(119906) = 1

1198671(119906) = 2119906

1198672(119906) = 4119906

2

minus 1

1198673(119906) = 8119906

3

minus 4119906

1198674(119906) = 16119906

4

minus 121199062

+ 1

sdot sdot sdot

(14)

The recurrent formula for the Chebyshev polynomials of thesecond kind is

119906119867119899(119906) =

1

2[119867119899minus1

(119906) + 119867119899+1

(119906)] (15)

The orthogonality of Chebyshev polynomial of the secondkind can be expressed as

int

1

minus1

2

120587

radic1 minus 1199062119867119894(119906)119867119895(119906) 119889119906 =

1 (119894 = 119895)

0 (119894 = 119895) (16)

Equation (16) represents a weighted orthogonal relationshipThe weighting function is just the same as that for the PDFof the random parameter which obeys arch distribution120588(119906) in (12) and the left-hand side of (16) can be regardedas the expectation of the product 119867

119894(119906)119867119895(119906) Owing to

4 Abstract and Applied Analysis

the orthogonality of Chebyshev polynomial any measurablefunction 119891(119906) isin 119871

2 can be expanded as

119891 (119906) =

infin

sum

119894=0

119888119894119867119894(119906) (17)

where 119888119894= int1

minus1

120588(119906)119891(119906)119867119894(119906)119889119906 This expansion is called an

orthogonal decomposition of random function119891(119906) which isthe theoretical base of orthogonal decomposition methods

4 The Stochastic Fractional-OrderVDP System

In this paper we consider the fractional-order VDP systemwith a random parameter The system can be described bythe following differential equations

119863119902

119909 = 119910

119863119902

119910 = 120583119910 minus 1198861199092

119910 + 119887119909

(18)

where 119902 is the fractional order and 119909 119910 are state variables 119886119887 are deterministic parameters and 120583 is a random parameterand it can be expressed as

120583 = 120583 + 120575119906 (19)

where 120583 and 1205752 are the mean value and standard deviationof 120583 respectively and 120575 is regarded as the intensity of 120583 119906 isa random variable defined on [minus1 1]

It is well known that the stochastic function spacewhich is constituted with the responses of nonlinear dynamicsystems with random parameters has proved to be a Hilbertspace with respect to convergence in the mean squareTherefore the fractional-order system (18) with randomparameters can be reduced into the equivalent deterministicsystem by using of the orthogonal polynomial expansion Itfollows from the orthogonal polynomial approximation thatthe responses of system (18) can be expressed approximatelyby the following series

119909 (119905 119906) =

119873

sum

119894=0

119909119894(119905)119867119894(119906)

119910 (119905 119906) =

119873

sum

119894=0

119910119894(119905)119867119894(119906)

(20)

where119867119894(119906) represents the 119894th Chebyshev polynomial and119873

is the largest order of the polynomialsThen substituting (20)into the system (18) we can get

119863119902

(

119873

sum

119894=0

119909119894(119905)119867119894(119906)) = minus(

119873

sum

119894=0

119910119894(119905)119867119894(119906))

119863119902

(

119873

sum

119894=0

119910119894(119905)119867119894(119906))

= 120583(

119873

sum

119894=0

119910119894(119905)119867119894(119906)) + 120575119906(

119873

sum

119894=0

119910119894(119905)119867119894(119906))

minus 119886(

119873

sum

119894=0

119909119894(119905)119867119894(119906))

2

(

119873

sum

119894=0

119910119894(119905)119867119894(119906))

+ 119887(

119873

sum

119894=0

119909119894(119905)119867119894(119906))

(21)

and with the aid of the recurrent formulas of Chebyshevpolynomial all the nonlinear terms in the system can befurther reduced into linear combination of 119871

119894(119906) as follows

(

119873

sum

119894=0

119909119894(119905)119867119894(119906))

2

(

119873

sum

119894=0

119910119894(119905)119867119894(119906))

2

=

3119873

sum

119894=0

119878119894(119905)119867119894(119906) (22)

where 119878119894(119905) can be obtained with the help of the computer

algebraic systems such as Maple or MATLAB Meanwhilethe term 120575119906(sum

119873

119894=0119910119894(119905)119867119894(119906)) can be reduced to

119906120575(

119873

sum

119894=0

119910119894(119905)119867119894(119906)) = 120575(

119873

sum

119894=0

119910119894(119905)

119867119894minus1

(119906) + 119867119894+1

(119906)

2)

(23)

Substituting (22) and (23) into (21) then we can have

119863119902

(

119873

sum

119894=0

119909119894(119905)119867119894(119906)) = minus(

119873

sum

119894=0

119910119894(119905)119867119894(119906))

119863119902

(

119873

sum

119894=0

119910119894(119905)119867119894(119906)) = 120583(

119873

sum

119894=0

119910119894(119905)119867119894(119906))

+ 120575(

119873

sum

119894=0

119910119894(119905)

119867119894minus1

(119906) + 119867119894+1

(119906)

2)

minus 119886

119873

sum

119894=0

119878119894(119905)119867119894(119906)

+ 119887(

119873

sum

119894=0

119909119894(119905)119867119894(119906))

(24)

Multiply both sides of (24) by 119867119894(119906) 119894 = 0 1 2 119873 in

sequence and take expectation with respect to 119906Then we canfinally obtain the equivalent deterministic system Rememberthat if 119873 rarr infin in (24) then its left is strictly equivalent tothe response of stochastic system (18) Otherwise (24) is justapproximately valid with aminimal residual error Due to therequirement of computational precision we take119873 = 4 in the

Abstract and Applied Analysis 5

following numerical analysisTherefore we get the equivalentdeterministic system approximately as

119863119902

1199090= minus 119910

0

119863119902

1199100= minus 119886119878

0+ 1198871199090+ 1205831199100+

1

21205751199101

119863119902

1199091= minus 119910

1

119863119902

1199101= minus 119886119878

1+ 1198871199091+ 1205831199101+

1

2120575 (1199100+ 1199102)

119863119902

1199092= minus 119910

2

119863119902

1199102= minus 119886119878

2+ 1198871199092+ 1205831199102+

1

2120575 (1199101+ 1199103)

119863119902

1199093= minus 119910

3

119863119902

1199103= minus 119886119878

3+ 1198871199093+ 1205831199103+

1

2120575 (1199102+ 1199104)

119863119902

1199094= minus 119910

4

119863119902

1199104= minus 119886119878

4+ 1198871199094+ 1205831199104+

1

21205751199103

(25)

We can get the numerical solutions 119909119894 119910119894 (119894 = 0 1 2 3 4) of

system (25) by the improved predictor-corrector algorithmTherefore the approximate random responses of the stochas-tic system (18) can be expressed as

119909 (119905 119906) asymp

4

sum

119894=0

119909119894(119905)119867119894(119906)

119910 (119905 119906) asymp

4

sum

119894=0

119910119894(119905)119867119894(119906)

(26)

The ensemble mean responses (EMRs for short) of thestochastic fractional-order system are described as

119864 [119909 (119905 119906)] asymp

4

sum

119894=0

119909119894(119905) 119864 [119867

119894(119906)] = 119909

0(119905)

119864 [119910 (119905 119906)] asymp

4

sum

119894=0

119910119894(119905) 119864 [119867

119894(119906)] = 119910

0(119905)

(27)

Especially for 119906 = 0 namely 120583 = 120583 this sample system issignificant for reference It is usually called a mean parametersystem of which the responses (SRMs for short) may beapproximated as

119909 (119905 0) asymp

4

sum

119894=0

119909119894(119905)119867119894(0) = 119909

0(119905) minus 119909

2(119905) + 119909

4(119905)

119910 (119905 0) asymp

4

sum

119894=0

119910119894(119905)119867119894(0) = 119910

0(119905) minus 119910

2(119905) + 119910

4(119905)

(28)

5 Hopf Bifurcations

A Hopf bifurcation occurs when a periodic solution or limitcycle surrounding an equilibrium point arises or goes awayas a parameter varies In this section the Hopf bifurcation forthe stochastic fractional-order VDP system will be studiedwith theoretical and numerical computations Obviously

both the stochastic VDP and equivalent deterministic sys-tems (18) and (25) only have one equilibrium point namelythe origin (00) The Jacobian matrix 119869 for the linearizedsystem evaluated at the origin is

119869 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

0 minus1 0 0 0 0 0 0 0 0

119887 120583 01

2120575 0 0 0 0 0 0

0 0 0 minus1 0 0 0 0 0 0

1

2120575 0 119887 120583 0

1

2120575 0 0 0 0

0 0 0 0 0 minus1 0 0 0 0

0 0 01

2120575 119887 120583 0

1

2120575 0 0

0 0 0 0 0 0 0 minus1 0 0

0 0 0 0 01

2120575 119887 120583 0

1

2120575

0 0 0 0 0 0 0 0 0 minus1

0 0 0 0 0 0 01

2120575 119887 120583

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(29)

The corresponding characteristic equation is

119891 (120582) = 119886012058210

+ 11988611205829

+ 11988621205828

+ 11988631205827

+ 11988641205826

+ 11988651205825

+ 11988661205824

+ 11988671205823

+ 11988681205822

+ 1198869120582 + 11988610

(30)

where 1198860 1198861 119886

10are the coefficients of (30) and can be

obtained by computing via Maple or MATLAB According tothe bifurcation theory of nonlinear dynamical systems andtaking 119908 as a bifurcation parameter [22] we can obtain theexistence conditions of Hopf bifurcation for the system

(1) The eigenvalues of the linearized system about thefixed point are 120572(119908) plusmn 119894120573(119908)

(2) The real and image parts of the eigenvalues satisfy120572(119908119888) = 0 120573(119908

119888) gt 0 1205721015840(119908

119888) = 0 when 119908 = 119908

119888

(3) The real parts of other eigenvalues are not equal tozeros

For the high order of equivalent system (25) so it is hardto get the threshold value of bifurcation parameter directlyTherefore we can get the threshold via the following theorem

Theorem 1 Let 119891(120582) = 1198860120582119899

+ 1198861120582119899minus1

+ sdot sdot sdot + 119886119899

= 0 be thecharacteristic equation evaluated at the equilibrium point Δ

119899

the n-dimensional Routh-Hurwitz determinant and 119886119894 (119894 =

0 1 119899) the coefficient of the characteristic equation If thereexists a threshold of bifurcation parameter 119908

119888which conforms

to the following terms then the first two existence conditions ofHopf bifurcation are satisfied

(1) Δ119899minus1

(119908119888) = 0 Δ

119899minus2(119908119888) = 0 Δ

119899minus3(119908119888) = 0

(2) 119886119894(119908119888) gt 0 (0 1 119899)

(3) Δ1015840

119899minus1(119908119888) = 0

6 Abstract and Applied Analysis

0 50 100 150 200 250 300

0

025

t

005

01

015

02

minus015

minus01

minus005

x0

(a) The time trajectory of EMR

0 005 01 015 02 025

0

005

01

015

02

y0

x0

minus015

minus01

minus005

minus02minus015 minus01 minus005

(b) The phase diagram of EMR in 1199090minus 1199100plane

0 50 100 150 200 250 300

0

025

t

x(t0)

005

01

015

02

minus015

minus01

minus005

(c) The time trajectory of SRM

0 005 01 015 02 025

0

005

01

015

minus015

minus01

minus005

minus02minus015 minus01 minus005

x(t 0)

y(t0)

(d) The phase diagram of SRM

Figure 2 The time trajectories and phase diagrams for 119906 = minus009

According to Theorem 1 the following parameter rela-tions which can make the equality Δ

9= 0 are derived

120583 = 0 120583 = plusmn120575

2 120583 = plusmn

120575

4 120583 = plusmn

radic3120575

2

120583 = plusmnradic3120575

4 120583 =

120575

4plusmn

radic3120575

4 120583 = minus

120575

4plusmn

radic3120575

4

(31)

Then (31) is substituted into the 8-dimensional and 7-dimensional Routh-Hurwitz determinants Δ

8 Δ7 and only

the following parameters can make Δ8

= 0 Δ7

= 0 satisfied

120583 = plusmn120575

2

120583 = plusmnradic3120575

2

(32)

Meanwhile the relations which can make all the coefficientssatisfied 119886

119894gt 0 (119894 = 0 1 10) are

120583 = plusmnradic3120575

2 (33)

We substitute the relations (33) intoΔ1015840

9(120583) andfind thatΔ1015840

9(120583)

is not equal to 0 therefore according to Theorem 1 the firsttwo existence conditions of Hopf bifurcation are satisfiedWhen (33) is substituted into the characteristic equation allthe eigenvalues of it can be obtained

12058212

= plusmnradic119887119894

12058234

=120575

4minus

radic3120575

4plusmn

1

4

radic41205752 minus 2radic31205752 minus 16119887

12058256

= minusradic3120575

4plusmn

1

4

radic31205752 minus 16119887

Abstract and Applied Analysis 7

0 50 100 150 200 250 300

0

025

t

005

01

015

02

minus015

minus01

minus005

x0

minus02

(a) The time trajectory of EMR

y0

x0

0 005 01

0

002

004

006

008

01

minus01minus005minus01

minus002

minus004

minus006

minus008

(b) The phase diagram of EMR in 1199090minus 1199100plane

0 50 100 150 200 250 300

025

t

005

01

015

02

minus015

minus01

minus005

0

x(t0)

(c) The time trajectory of SRM

0 001 002 003

0

001

002

003

minus003 minus002 minus001minus003

minus002

minus001

x(t 0)

y(t0)

(d) The phase diagram of SRM

Figure 3 The time trajectories and phase diagrams for 119906 = minus001

12058278

= minus120575

4minus

radic3120575

4plusmn

1

4

radic41205752 minus 2radic31205752 minus 16119887

120582910

= minusradic3120575

2plusmn

1

2

radic31205752 minus 4119887

(34)

Obviously when the intensity of random parameter 120575 = 0 thereal parts of all the other eigenvalues are less than 0 except1205821and 120582

2 The characteristic equation does not have zero

solution for 119887 = 0 Therefore all the existence conditions ofHopf bifurcation are satisfied When 120583

119888= minusradic31205752 the Hopf

bifurcation occurs around the equilibrium (0 0)The parameters are taken as 119886 = 1 119887 = 1 and 120575 =

01 and the derivative order is 119902 = 0995 And initialconditions are [119909

0(0) 119909

4(0)] = (02 02 02 02)

119879 and[1199100(0) 119910

4(0)] = (minus02 minus02 minus02 minus02)

119879 According tothe above analysis we can get that the critical parameter valueof the bifurcation is 120583

119888= minusradic31205752 asymp minus00866 which implies

that if 120583 lt 120583119888 the system will converge to the equilibrium

otherwise it will converge to a limit cycle The parametervalues are chosen as 120583 = minus009 and 119906 = minus001 respectivelyThen we obtain the time trajectories and phase diagrams forthe EMR and SRM of the stochastic system by simulationcomputations see Figures 2 and 3 It is clearly seen that thesystem converges to the fixed point (0 0) for 120583 = minus009 andto a limit cycle for 119906 = minus001 which implies that the Hopfbifurcation occurs in the stochastic system (18) Meanwhilethe SRM and EMR of phase plots coincide well enoughIt means that the Chebyshev polynomial approximationmethod is effective for the stochastic fractional-order system(18)

When the others parameters are fixed and 120583 = minus002 wealso found that Hopf bifurcation occurs with the variationof the derivative order The stochastic system converges tothe equilibrium when 119902 le 098 and to a limit cycle for 119902 ge

099 which means that the Hopf bifurcation occurs whenthe derivative order 119902 = 099 see Figure 4 from which it

8 Abstract and Applied Analysis

0 005 01 015 02 025

0

005

01

015

02

y0

x0

minus015

minus01

minus005

minus02minus015 minus01 minus005

(a) 119902 = 098

0 005 01

0

002

004

006

008

01

minus002

minus004

minus006

minus008

minus005

y0

x0

minus01minus01

(b) 119902 = 099

Figure 4 The phase diagrams for different values of order 119902

0 100 200 300 400 500 600 700 800

0

005

01

015

02

025

minus005

minus015

minus01

minus02

t

x0

(a)

0

0y0

x0

005

01

015

02

minus005

minus015

minus01

minus02005 01 015 02 025minus005minus015 minus01minus02

(b)

Figure 5 The time trajectory and phase diagram of EMR when 119905 isin [0 800] (a) The time trajectory (b) The phase diagram in 1199090minus 1199100plane

is obtained that the derivative order can also be taken as abifurcation parameter for fractional-order system

The simulation time is taken as 119905 = 800 one integrationstep of time is taken as Δ119905 = 0001 derivative order 119902 = 097and divide the time interval [0 800] into 8 equal parts Thetime trajectory and phase diagram for 119905 = 800 are shownin Figure 5 from which it appears that the 119909

0converges to a

fixed point as 119905 ge 300 However the time trajectory is zoomedin when 119905 belongs to the interval [300 400] see Figure 6(a)fromwhich we can see that the trajectory gradually decreasesbut does not converge to a point directly Secondly the timetrajectory is zoomed in when 119905 isin [400 500] see Figure 6(b)Meanwhile we also obtain the trajectories for 119905 isin [500 600]119905 isin [600 700] and 119905 isin [700 800] which are shown in Figures6(c)ndash6(e) respectively Comparing these figures it is seenthat the amplitude of 119909

0decreases as the time 119905 increases

Based on these observations we can suggest that the EMRsof the stochastic system (18) have an extremely long transientprocess while converging to the equilibrium as 119905 rarr infin

6 Conclusions

In this paper the Hopf bifurcation of a fractional-order VDPsystem with a random parameter is analyzed An equivalentdeterministic system is obtained with the approximationprinciple of Chebyshev orthogonal polynomials Based onbifurcation theory of nonlinear dynamic systems the criticalparameter value of the bifurcation is obtained by theoreticalanalysis Meanwhile numerical simulations are performedto verify the theoretical results Besides the derivative ordercan also be taken as a bifurcation parameter and the Hopfbifurcation is observed with the variation of the order

Abstract and Applied Analysis 9

300 320 340 360 380 400

012345times10minus4

minus1

minus2

minus3

minus4

minus5

t

x0

(a) 119905 isin [300 400]

400 420 440 460 480 500

0

05

1

15times10minus4

minus1

minus15

minus05

t

x0

(b) 119905 isin [400 500]

500 520 540 560 580 600

0

1

2

3

4times10minus5

minus1

minus2

minus3

minus4

t

x0

(c) 119905 isin [500 600]

600 620 640 660 680 700

0

2

4

6

8times10minus6

minus2

minus4

minus6

minus8

minus10

t

x0

(d) 119905 isin [600 700]

700 720 740 760 780 800t

0

2

05

1

15

times10minus6

minus2

minus25

minus15

minus1

minus05

x0

(e) 119905 isin [700 800]

Figure 6 The time trajectories of EMR 1199090for different time intervals

10 Abstract and Applied Analysis

It is worth mentioning that the orthogonal polyno-mial approximation is an effective method for integer-orderstochastic systems but the universality of the method forfractional-order ones still remains an open question Due tothe absences of analytical methods the threshold value ofbifurcation when the derivative order is taken as a bifurcationparameter cannot be obtained by theoretical analysis but onlythe numerical experiments

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the National Natural ScienceFoundation of China (NSFC) under Grant nos 1133200811172223 and 11172224

References

[1] C P Li W H Deng and D Xu ldquoChaos synchronization ofthe Chua system with a fractional orderrdquo Physica A StatisticalMechanics and Its Applications vol 360 no 2 pp 171ndash185 2006

[2] Z M Ge and C Y Ou ldquoChaos in a fractional order modifiedDuffing systemrdquo Chaos Solitons amp Fractals vol 34 no 2 pp262ndash291 2007

[3] J G Lu and G Chen ldquoA note on the fractional-order Chensystemrdquo Chaos Solitons and Fractals vol 27 no 3 pp 685ndash6882006

[4] H Sun W Chen and Y Chen ldquoVariable-order fractional dif-ferential operators in anomalous diffusion modelingrdquo PhysicaA Statistical Mechanics and Its Applications vol 388 no 21 pp4586ndash4592 2009

[5] Y Liang and W Chen ldquoA survey on computing Levy stabledistributions and a new MATLAB toolboxrdquo Signal Processingvol 93 no 1 pp 242ndash251 2013

[6] Y J Liang and W Chen ldquoReliability analysis for sluice gateanti-sliding stability using Levy stable distributionsrdquo SignalProcessing 2014

[7] V Venkatasubramanian ldquoSingularity induced bifurcation andthe van der Pol oscillatorrdquo IEEE Transactions on Circuits andSystems I Fundamental Theory and Applications vol 41 no 11pp 765ndash769 1994

[8] R Mettin U Parlitz andW Lauterborn ldquoBifurcation structureof the driven van der Pol oscillatorrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol3 no 6 pp 1529ndash1555 1993

[9] U Parlitz and W Lauterborn ldquoPeriod-doubling cascades anddevils staircases of the driven van der Pol oscillatorrdquo PhysicalReview A vol 36 no 3 pp 1428ndash1434 1987

[10] A Buonomo ldquoThe periodic solution of van der Polrsquos equationrdquoSIAM Journal on AppliedMathematics vol 59 no 1 pp 156ndash1711998

[11] J-X Xu and J Jiang ldquoThe global bifurcation characteristics ofthe forced van der Pol oscillatorrdquo Chaos Solitons and Fractalsvol 7 no 1 pp 3ndash19 1996

[12] R S Barbosa J A T MacHado B M Vinagre and A JCaldern ldquoAnalysis of the Van der Pol oscillator containing

derivatives of fractional orderrdquo Journal of Vibration and Controlvol 13 no 9-10 pp 1291ndash1301 2007

[13] J-H Chen andW-C Chen ldquoChaotic dynamics of the fraction-ally damped van der Pol equationrdquo Chaos Solitons and Fractalsvol 35 no 1 pp 188ndash198 2008

[14] M Shinozuka ldquoProbability modeling of concrete structuresrdquoASCE Journal of the EngineeringMechanics Division vol 98 no6 pp 1433ndash1451 1972

[15] R Ghamem and P Spans Stochastic Finite Element A SpectralApproach Springer Berlin Germany 1991

[16] C L Pettit and P S Beran ldquoSpectral and multiresolutionWiener expansions of oscillatory stochastic processesrdquo Journalof Sound and Vibration vol 294 no 4 pp 752ndash779 2006

[17] O P Le Maıtre O M Knio H N Najm and R G GhanemldquoUncertainty propagation using Wiener-Haar expansionsrdquoJournal of Computational Physics vol 197 no 1 pp 28ndash57 2004

[18] O P Le Maıtre H N Najm R G Ghanem and O M KnioldquoMulti-resolution analysis ofWiener-type uncertainty propaga-tion schemesrdquo Journal of Computational Physics vol 197 no 2pp 502ndash531 2004

[19] D Xiu and G E Karniadakis ldquoModeling uncertainty in steadystate diffusion problems via generalized polynomial chaosrdquoComputer Methods in Applied Mechanics and Engineering vol191 no 43 pp 4927ndash4948 2002

[20] X Wan and G E Karniadakis ldquoAn adaptive multi-elementgeneralized polynomial chaosmethod for stochastic differentialequationsrdquo Journal of Computational Physics vol 209 no 2 pp617ndash642 2005

[21] X L Leng C L Wu X P Ma G Meng and T Fang ldquoBifur-cation and chaos analysis of stochastic Duffing system underharmonic excitationsrdquo Nonlinear Dynamics vol 42 no 2 pp185ndash198 2005

[22] S-J MaW XuW Li and T Fang ldquoAnalysis of stochastic bifur-cation and chaos in stochastic Duffing-van der Pol system viaChebyshev polynomial approximationrdquo Chinese Physics vol 15no 6 article 017 pp 1231ndash1238 2006

[23] S J MaW Xu and T Fang ldquoAnalysis of period-doubling bifur-cation in double-well stochastic Duffing system via Laguerrepolynomial approximationrdquoNonlinear Dynamics vol 52 no 3pp 289ndash299 2008

[24] S-J Ma Q Shen and J Hou ldquoModified projective synchro-nization of stochastic fractional order chaotic systems withuncertain parametersrdquo Nonlinear Dynamics vol 73 no 1-2 pp93ndash100 2013

[25] X Liu L Hong and L Yang ldquoFractional-order complex Tsystem bifurcations chaos control and synchronizationrdquoNon-linear Dynamics vol 75 no 3 pp 589ndash602 2014

[26] K Diethelm N J Ford and A D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 3ndash22 2002

[27] A Charef H H Sun Y-Y Tsao and B Onaral ldquoFractal systemas represented by singularity functionrdquo IEEE Transactions onAutomatic Control vol 37 no 9 pp 1465ndash1470 1992

[28] M S Tavazoei andM Haeri ldquoLimitations of frequency domainapproximation for detecting chaos in fractional order systemsrdquoNonlinear Analysis Theory Methods amp Applications vol 69 no4 pp 1299ndash1320 2008

[29] P Borwein and T Erdelyi Polynomials and Polynomial Inequal-ities Springer New York NY USA 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

the orthogonality of Chebyshev polynomial any measurablefunction 119891(119906) isin 119871

2 can be expanded as

119891 (119906) =

infin

sum

119894=0

119888119894119867119894(119906) (17)

where 119888119894= int1

minus1

120588(119906)119891(119906)119867119894(119906)119889119906 This expansion is called an

orthogonal decomposition of random function119891(119906) which isthe theoretical base of orthogonal decomposition methods

4 The Stochastic Fractional-OrderVDP System

In this paper we consider the fractional-order VDP systemwith a random parameter The system can be described bythe following differential equations

119863119902

119909 = 119910

119863119902

119910 = 120583119910 minus 1198861199092

119910 + 119887119909

(18)

where 119902 is the fractional order and 119909 119910 are state variables 119886119887 are deterministic parameters and 120583 is a random parameterand it can be expressed as

120583 = 120583 + 120575119906 (19)

where 120583 and 1205752 are the mean value and standard deviationof 120583 respectively and 120575 is regarded as the intensity of 120583 119906 isa random variable defined on [minus1 1]

It is well known that the stochastic function spacewhich is constituted with the responses of nonlinear dynamicsystems with random parameters has proved to be a Hilbertspace with respect to convergence in the mean squareTherefore the fractional-order system (18) with randomparameters can be reduced into the equivalent deterministicsystem by using of the orthogonal polynomial expansion Itfollows from the orthogonal polynomial approximation thatthe responses of system (18) can be expressed approximatelyby the following series

119909 (119905 119906) =

119873

sum

119894=0

119909119894(119905)119867119894(119906)

119910 (119905 119906) =

119873

sum

119894=0

119910119894(119905)119867119894(119906)

(20)

where119867119894(119906) represents the 119894th Chebyshev polynomial and119873

is the largest order of the polynomialsThen substituting (20)into the system (18) we can get

119863119902

(

119873

sum

119894=0

119909119894(119905)119867119894(119906)) = minus(

119873

sum

119894=0

119910119894(119905)119867119894(119906))

119863119902

(

119873

sum

119894=0

119910119894(119905)119867119894(119906))

= 120583(

119873

sum

119894=0

119910119894(119905)119867119894(119906)) + 120575119906(

119873

sum

119894=0

119910119894(119905)119867119894(119906))

minus 119886(

119873

sum

119894=0

119909119894(119905)119867119894(119906))

2

(

119873

sum

119894=0

119910119894(119905)119867119894(119906))

+ 119887(

119873

sum

119894=0

119909119894(119905)119867119894(119906))

(21)

and with the aid of the recurrent formulas of Chebyshevpolynomial all the nonlinear terms in the system can befurther reduced into linear combination of 119871

119894(119906) as follows

(

119873

sum

119894=0

119909119894(119905)119867119894(119906))

2

(

119873

sum

119894=0

119910119894(119905)119867119894(119906))

2

=

3119873

sum

119894=0

119878119894(119905)119867119894(119906) (22)

where 119878119894(119905) can be obtained with the help of the computer

algebraic systems such as Maple or MATLAB Meanwhilethe term 120575119906(sum

119873

119894=0119910119894(119905)119867119894(119906)) can be reduced to

119906120575(

119873

sum

119894=0

119910119894(119905)119867119894(119906)) = 120575(

119873

sum

119894=0

119910119894(119905)

119867119894minus1

(119906) + 119867119894+1

(119906)

2)

(23)

Substituting (22) and (23) into (21) then we can have

119863119902

(

119873

sum

119894=0

119909119894(119905)119867119894(119906)) = minus(

119873

sum

119894=0

119910119894(119905)119867119894(119906))

119863119902

(

119873

sum

119894=0

119910119894(119905)119867119894(119906)) = 120583(

119873

sum

119894=0

119910119894(119905)119867119894(119906))

+ 120575(

119873

sum

119894=0

119910119894(119905)

119867119894minus1

(119906) + 119867119894+1

(119906)

2)

minus 119886

119873

sum

119894=0

119878119894(119905)119867119894(119906)

+ 119887(

119873

sum

119894=0

119909119894(119905)119867119894(119906))

(24)

Multiply both sides of (24) by 119867119894(119906) 119894 = 0 1 2 119873 in

sequence and take expectation with respect to 119906Then we canfinally obtain the equivalent deterministic system Rememberthat if 119873 rarr infin in (24) then its left is strictly equivalent tothe response of stochastic system (18) Otherwise (24) is justapproximately valid with aminimal residual error Due to therequirement of computational precision we take119873 = 4 in the

Abstract and Applied Analysis 5

following numerical analysisTherefore we get the equivalentdeterministic system approximately as

119863119902

1199090= minus 119910

0

119863119902

1199100= minus 119886119878

0+ 1198871199090+ 1205831199100+

1

21205751199101

119863119902

1199091= minus 119910

1

119863119902

1199101= minus 119886119878

1+ 1198871199091+ 1205831199101+

1

2120575 (1199100+ 1199102)

119863119902

1199092= minus 119910

2

119863119902

1199102= minus 119886119878

2+ 1198871199092+ 1205831199102+

1

2120575 (1199101+ 1199103)

119863119902

1199093= minus 119910

3

119863119902

1199103= minus 119886119878

3+ 1198871199093+ 1205831199103+

1

2120575 (1199102+ 1199104)

119863119902

1199094= minus 119910

4

119863119902

1199104= minus 119886119878

4+ 1198871199094+ 1205831199104+

1

21205751199103

(25)

We can get the numerical solutions 119909119894 119910119894 (119894 = 0 1 2 3 4) of

system (25) by the improved predictor-corrector algorithmTherefore the approximate random responses of the stochas-tic system (18) can be expressed as

119909 (119905 119906) asymp

4

sum

119894=0

119909119894(119905)119867119894(119906)

119910 (119905 119906) asymp

4

sum

119894=0

119910119894(119905)119867119894(119906)

(26)

The ensemble mean responses (EMRs for short) of thestochastic fractional-order system are described as

119864 [119909 (119905 119906)] asymp

4

sum

119894=0

119909119894(119905) 119864 [119867

119894(119906)] = 119909

0(119905)

119864 [119910 (119905 119906)] asymp

4

sum

119894=0

119910119894(119905) 119864 [119867

119894(119906)] = 119910

0(119905)

(27)

Especially for 119906 = 0 namely 120583 = 120583 this sample system issignificant for reference It is usually called a mean parametersystem of which the responses (SRMs for short) may beapproximated as

119909 (119905 0) asymp

4

sum

119894=0

119909119894(119905)119867119894(0) = 119909

0(119905) minus 119909

2(119905) + 119909

4(119905)

119910 (119905 0) asymp

4

sum

119894=0

119910119894(119905)119867119894(0) = 119910

0(119905) minus 119910

2(119905) + 119910

4(119905)

(28)

5 Hopf Bifurcations

A Hopf bifurcation occurs when a periodic solution or limitcycle surrounding an equilibrium point arises or goes awayas a parameter varies In this section the Hopf bifurcation forthe stochastic fractional-order VDP system will be studiedwith theoretical and numerical computations Obviously

both the stochastic VDP and equivalent deterministic sys-tems (18) and (25) only have one equilibrium point namelythe origin (00) The Jacobian matrix 119869 for the linearizedsystem evaluated at the origin is

119869 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

0 minus1 0 0 0 0 0 0 0 0

119887 120583 01

2120575 0 0 0 0 0 0

0 0 0 minus1 0 0 0 0 0 0

1

2120575 0 119887 120583 0

1

2120575 0 0 0 0

0 0 0 0 0 minus1 0 0 0 0

0 0 01

2120575 119887 120583 0

1

2120575 0 0

0 0 0 0 0 0 0 minus1 0 0

0 0 0 0 01

2120575 119887 120583 0

1

2120575

0 0 0 0 0 0 0 0 0 minus1

0 0 0 0 0 0 01

2120575 119887 120583

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(29)

The corresponding characteristic equation is

119891 (120582) = 119886012058210

+ 11988611205829

+ 11988621205828

+ 11988631205827

+ 11988641205826

+ 11988651205825

+ 11988661205824

+ 11988671205823

+ 11988681205822

+ 1198869120582 + 11988610

(30)

where 1198860 1198861 119886

10are the coefficients of (30) and can be

obtained by computing via Maple or MATLAB According tothe bifurcation theory of nonlinear dynamical systems andtaking 119908 as a bifurcation parameter [22] we can obtain theexistence conditions of Hopf bifurcation for the system

(1) The eigenvalues of the linearized system about thefixed point are 120572(119908) plusmn 119894120573(119908)

(2) The real and image parts of the eigenvalues satisfy120572(119908119888) = 0 120573(119908

119888) gt 0 1205721015840(119908

119888) = 0 when 119908 = 119908

119888

(3) The real parts of other eigenvalues are not equal tozeros

For the high order of equivalent system (25) so it is hardto get the threshold value of bifurcation parameter directlyTherefore we can get the threshold via the following theorem

Theorem 1 Let 119891(120582) = 1198860120582119899

+ 1198861120582119899minus1

+ sdot sdot sdot + 119886119899

= 0 be thecharacteristic equation evaluated at the equilibrium point Δ

119899

the n-dimensional Routh-Hurwitz determinant and 119886119894 (119894 =

0 1 119899) the coefficient of the characteristic equation If thereexists a threshold of bifurcation parameter 119908

119888which conforms

to the following terms then the first two existence conditions ofHopf bifurcation are satisfied

(1) Δ119899minus1

(119908119888) = 0 Δ

119899minus2(119908119888) = 0 Δ

119899minus3(119908119888) = 0

(2) 119886119894(119908119888) gt 0 (0 1 119899)

(3) Δ1015840

119899minus1(119908119888) = 0

6 Abstract and Applied Analysis

0 50 100 150 200 250 300

0

025

t

005

01

015

02

minus015

minus01

minus005

x0

(a) The time trajectory of EMR

0 005 01 015 02 025

0

005

01

015

02

y0

x0

minus015

minus01

minus005

minus02minus015 minus01 minus005

(b) The phase diagram of EMR in 1199090minus 1199100plane

0 50 100 150 200 250 300

0

025

t

x(t0)

005

01

015

02

minus015

minus01

minus005

(c) The time trajectory of SRM

0 005 01 015 02 025

0

005

01

015

minus015

minus01

minus005

minus02minus015 minus01 minus005

x(t 0)

y(t0)

(d) The phase diagram of SRM

Figure 2 The time trajectories and phase diagrams for 119906 = minus009

According to Theorem 1 the following parameter rela-tions which can make the equality Δ

9= 0 are derived

120583 = 0 120583 = plusmn120575

2 120583 = plusmn

120575

4 120583 = plusmn

radic3120575

2

120583 = plusmnradic3120575

4 120583 =

120575

4plusmn

radic3120575

4 120583 = minus

120575

4plusmn

radic3120575

4

(31)

Then (31) is substituted into the 8-dimensional and 7-dimensional Routh-Hurwitz determinants Δ

8 Δ7 and only

the following parameters can make Δ8

= 0 Δ7

= 0 satisfied

120583 = plusmn120575

2

120583 = plusmnradic3120575

2

(32)

Meanwhile the relations which can make all the coefficientssatisfied 119886

119894gt 0 (119894 = 0 1 10) are

120583 = plusmnradic3120575

2 (33)

We substitute the relations (33) intoΔ1015840

9(120583) andfind thatΔ1015840

9(120583)

is not equal to 0 therefore according to Theorem 1 the firsttwo existence conditions of Hopf bifurcation are satisfiedWhen (33) is substituted into the characteristic equation allthe eigenvalues of it can be obtained

12058212

= plusmnradic119887119894

12058234

=120575

4minus

radic3120575

4plusmn

1

4

radic41205752 minus 2radic31205752 minus 16119887

12058256

= minusradic3120575

4plusmn

1

4

radic31205752 minus 16119887

Abstract and Applied Analysis 7

0 50 100 150 200 250 300

0

025

t

005

01

015

02

minus015

minus01

minus005

x0

minus02

(a) The time trajectory of EMR

y0

x0

0 005 01

0

002

004

006

008

01

minus01minus005minus01

minus002

minus004

minus006

minus008

(b) The phase diagram of EMR in 1199090minus 1199100plane

0 50 100 150 200 250 300

025

t

005

01

015

02

minus015

minus01

minus005

0

x(t0)

(c) The time trajectory of SRM

0 001 002 003

0

001

002

003

minus003 minus002 minus001minus003

minus002

minus001

x(t 0)

y(t0)

(d) The phase diagram of SRM

Figure 3 The time trajectories and phase diagrams for 119906 = minus001

12058278

= minus120575

4minus

radic3120575

4plusmn

1

4

radic41205752 minus 2radic31205752 minus 16119887

120582910

= minusradic3120575

2plusmn

1

2

radic31205752 minus 4119887

(34)

Obviously when the intensity of random parameter 120575 = 0 thereal parts of all the other eigenvalues are less than 0 except1205821and 120582

2 The characteristic equation does not have zero

solution for 119887 = 0 Therefore all the existence conditions ofHopf bifurcation are satisfied When 120583

119888= minusradic31205752 the Hopf

bifurcation occurs around the equilibrium (0 0)The parameters are taken as 119886 = 1 119887 = 1 and 120575 =

01 and the derivative order is 119902 = 0995 And initialconditions are [119909

0(0) 119909

4(0)] = (02 02 02 02)

119879 and[1199100(0) 119910

4(0)] = (minus02 minus02 minus02 minus02)

119879 According tothe above analysis we can get that the critical parameter valueof the bifurcation is 120583

119888= minusradic31205752 asymp minus00866 which implies

that if 120583 lt 120583119888 the system will converge to the equilibrium

otherwise it will converge to a limit cycle The parametervalues are chosen as 120583 = minus009 and 119906 = minus001 respectivelyThen we obtain the time trajectories and phase diagrams forthe EMR and SRM of the stochastic system by simulationcomputations see Figures 2 and 3 It is clearly seen that thesystem converges to the fixed point (0 0) for 120583 = minus009 andto a limit cycle for 119906 = minus001 which implies that the Hopfbifurcation occurs in the stochastic system (18) Meanwhilethe SRM and EMR of phase plots coincide well enoughIt means that the Chebyshev polynomial approximationmethod is effective for the stochastic fractional-order system(18)

When the others parameters are fixed and 120583 = minus002 wealso found that Hopf bifurcation occurs with the variationof the derivative order The stochastic system converges tothe equilibrium when 119902 le 098 and to a limit cycle for 119902 ge

099 which means that the Hopf bifurcation occurs whenthe derivative order 119902 = 099 see Figure 4 from which it

8 Abstract and Applied Analysis

0 005 01 015 02 025

0

005

01

015

02

y0

x0

minus015

minus01

minus005

minus02minus015 minus01 minus005

(a) 119902 = 098

0 005 01

0

002

004

006

008

01

minus002

minus004

minus006

minus008

minus005

y0

x0

minus01minus01

(b) 119902 = 099

Figure 4 The phase diagrams for different values of order 119902

0 100 200 300 400 500 600 700 800

0

005

01

015

02

025

minus005

minus015

minus01

minus02

t

x0

(a)

0

0y0

x0

005

01

015

02

minus005

minus015

minus01

minus02005 01 015 02 025minus005minus015 minus01minus02

(b)

Figure 5 The time trajectory and phase diagram of EMR when 119905 isin [0 800] (a) The time trajectory (b) The phase diagram in 1199090minus 1199100plane

is obtained that the derivative order can also be taken as abifurcation parameter for fractional-order system

The simulation time is taken as 119905 = 800 one integrationstep of time is taken as Δ119905 = 0001 derivative order 119902 = 097and divide the time interval [0 800] into 8 equal parts Thetime trajectory and phase diagram for 119905 = 800 are shownin Figure 5 from which it appears that the 119909

0converges to a

fixed point as 119905 ge 300 However the time trajectory is zoomedin when 119905 belongs to the interval [300 400] see Figure 6(a)fromwhich we can see that the trajectory gradually decreasesbut does not converge to a point directly Secondly the timetrajectory is zoomed in when 119905 isin [400 500] see Figure 6(b)Meanwhile we also obtain the trajectories for 119905 isin [500 600]119905 isin [600 700] and 119905 isin [700 800] which are shown in Figures6(c)ndash6(e) respectively Comparing these figures it is seenthat the amplitude of 119909

0decreases as the time 119905 increases

Based on these observations we can suggest that the EMRsof the stochastic system (18) have an extremely long transientprocess while converging to the equilibrium as 119905 rarr infin

6 Conclusions

In this paper the Hopf bifurcation of a fractional-order VDPsystem with a random parameter is analyzed An equivalentdeterministic system is obtained with the approximationprinciple of Chebyshev orthogonal polynomials Based onbifurcation theory of nonlinear dynamic systems the criticalparameter value of the bifurcation is obtained by theoreticalanalysis Meanwhile numerical simulations are performedto verify the theoretical results Besides the derivative ordercan also be taken as a bifurcation parameter and the Hopfbifurcation is observed with the variation of the order

Abstract and Applied Analysis 9

300 320 340 360 380 400

012345times10minus4

minus1

minus2

minus3

minus4

minus5

t

x0

(a) 119905 isin [300 400]

400 420 440 460 480 500

0

05

1

15times10minus4

minus1

minus15

minus05

t

x0

(b) 119905 isin [400 500]

500 520 540 560 580 600

0

1

2

3

4times10minus5

minus1

minus2

minus3

minus4

t

x0

(c) 119905 isin [500 600]

600 620 640 660 680 700

0

2

4

6

8times10minus6

minus2

minus4

minus6

minus8

minus10

t

x0

(d) 119905 isin [600 700]

700 720 740 760 780 800t

0

2

05

1

15

times10minus6

minus2

minus25

minus15

minus1

minus05

x0

(e) 119905 isin [700 800]

Figure 6 The time trajectories of EMR 1199090for different time intervals

10 Abstract and Applied Analysis

It is worth mentioning that the orthogonal polyno-mial approximation is an effective method for integer-orderstochastic systems but the universality of the method forfractional-order ones still remains an open question Due tothe absences of analytical methods the threshold value ofbifurcation when the derivative order is taken as a bifurcationparameter cannot be obtained by theoretical analysis but onlythe numerical experiments

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the National Natural ScienceFoundation of China (NSFC) under Grant nos 1133200811172223 and 11172224

References

[1] C P Li W H Deng and D Xu ldquoChaos synchronization ofthe Chua system with a fractional orderrdquo Physica A StatisticalMechanics and Its Applications vol 360 no 2 pp 171ndash185 2006

[2] Z M Ge and C Y Ou ldquoChaos in a fractional order modifiedDuffing systemrdquo Chaos Solitons amp Fractals vol 34 no 2 pp262ndash291 2007

[3] J G Lu and G Chen ldquoA note on the fractional-order Chensystemrdquo Chaos Solitons and Fractals vol 27 no 3 pp 685ndash6882006

[4] H Sun W Chen and Y Chen ldquoVariable-order fractional dif-ferential operators in anomalous diffusion modelingrdquo PhysicaA Statistical Mechanics and Its Applications vol 388 no 21 pp4586ndash4592 2009

[5] Y Liang and W Chen ldquoA survey on computing Levy stabledistributions and a new MATLAB toolboxrdquo Signal Processingvol 93 no 1 pp 242ndash251 2013

[6] Y J Liang and W Chen ldquoReliability analysis for sluice gateanti-sliding stability using Levy stable distributionsrdquo SignalProcessing 2014

[7] V Venkatasubramanian ldquoSingularity induced bifurcation andthe van der Pol oscillatorrdquo IEEE Transactions on Circuits andSystems I Fundamental Theory and Applications vol 41 no 11pp 765ndash769 1994

[8] R Mettin U Parlitz andW Lauterborn ldquoBifurcation structureof the driven van der Pol oscillatorrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol3 no 6 pp 1529ndash1555 1993

[9] U Parlitz and W Lauterborn ldquoPeriod-doubling cascades anddevils staircases of the driven van der Pol oscillatorrdquo PhysicalReview A vol 36 no 3 pp 1428ndash1434 1987

[10] A Buonomo ldquoThe periodic solution of van der Polrsquos equationrdquoSIAM Journal on AppliedMathematics vol 59 no 1 pp 156ndash1711998

[11] J-X Xu and J Jiang ldquoThe global bifurcation characteristics ofthe forced van der Pol oscillatorrdquo Chaos Solitons and Fractalsvol 7 no 1 pp 3ndash19 1996

[12] R S Barbosa J A T MacHado B M Vinagre and A JCaldern ldquoAnalysis of the Van der Pol oscillator containing

derivatives of fractional orderrdquo Journal of Vibration and Controlvol 13 no 9-10 pp 1291ndash1301 2007

[13] J-H Chen andW-C Chen ldquoChaotic dynamics of the fraction-ally damped van der Pol equationrdquo Chaos Solitons and Fractalsvol 35 no 1 pp 188ndash198 2008

[14] M Shinozuka ldquoProbability modeling of concrete structuresrdquoASCE Journal of the EngineeringMechanics Division vol 98 no6 pp 1433ndash1451 1972

[15] R Ghamem and P Spans Stochastic Finite Element A SpectralApproach Springer Berlin Germany 1991

[16] C L Pettit and P S Beran ldquoSpectral and multiresolutionWiener expansions of oscillatory stochastic processesrdquo Journalof Sound and Vibration vol 294 no 4 pp 752ndash779 2006

[17] O P Le Maıtre O M Knio H N Najm and R G GhanemldquoUncertainty propagation using Wiener-Haar expansionsrdquoJournal of Computational Physics vol 197 no 1 pp 28ndash57 2004

[18] O P Le Maıtre H N Najm R G Ghanem and O M KnioldquoMulti-resolution analysis ofWiener-type uncertainty propaga-tion schemesrdquo Journal of Computational Physics vol 197 no 2pp 502ndash531 2004

[19] D Xiu and G E Karniadakis ldquoModeling uncertainty in steadystate diffusion problems via generalized polynomial chaosrdquoComputer Methods in Applied Mechanics and Engineering vol191 no 43 pp 4927ndash4948 2002

[20] X Wan and G E Karniadakis ldquoAn adaptive multi-elementgeneralized polynomial chaosmethod for stochastic differentialequationsrdquo Journal of Computational Physics vol 209 no 2 pp617ndash642 2005

[21] X L Leng C L Wu X P Ma G Meng and T Fang ldquoBifur-cation and chaos analysis of stochastic Duffing system underharmonic excitationsrdquo Nonlinear Dynamics vol 42 no 2 pp185ndash198 2005

[22] S-J MaW XuW Li and T Fang ldquoAnalysis of stochastic bifur-cation and chaos in stochastic Duffing-van der Pol system viaChebyshev polynomial approximationrdquo Chinese Physics vol 15no 6 article 017 pp 1231ndash1238 2006

[23] S J MaW Xu and T Fang ldquoAnalysis of period-doubling bifur-cation in double-well stochastic Duffing system via Laguerrepolynomial approximationrdquoNonlinear Dynamics vol 52 no 3pp 289ndash299 2008

[24] S-J Ma Q Shen and J Hou ldquoModified projective synchro-nization of stochastic fractional order chaotic systems withuncertain parametersrdquo Nonlinear Dynamics vol 73 no 1-2 pp93ndash100 2013

[25] X Liu L Hong and L Yang ldquoFractional-order complex Tsystem bifurcations chaos control and synchronizationrdquoNon-linear Dynamics vol 75 no 3 pp 589ndash602 2014

[26] K Diethelm N J Ford and A D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 3ndash22 2002

[27] A Charef H H Sun Y-Y Tsao and B Onaral ldquoFractal systemas represented by singularity functionrdquo IEEE Transactions onAutomatic Control vol 37 no 9 pp 1465ndash1470 1992

[28] M S Tavazoei andM Haeri ldquoLimitations of frequency domainapproximation for detecting chaos in fractional order systemsrdquoNonlinear Analysis Theory Methods amp Applications vol 69 no4 pp 1299ndash1320 2008

[29] P Borwein and T Erdelyi Polynomials and Polynomial Inequal-ities Springer New York NY USA 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

following numerical analysisTherefore we get the equivalentdeterministic system approximately as

119863119902

1199090= minus 119910

0

119863119902

1199100= minus 119886119878

0+ 1198871199090+ 1205831199100+

1

21205751199101

119863119902

1199091= minus 119910

1

119863119902

1199101= minus 119886119878

1+ 1198871199091+ 1205831199101+

1

2120575 (1199100+ 1199102)

119863119902

1199092= minus 119910

2

119863119902

1199102= minus 119886119878

2+ 1198871199092+ 1205831199102+

1

2120575 (1199101+ 1199103)

119863119902

1199093= minus 119910

3

119863119902

1199103= minus 119886119878

3+ 1198871199093+ 1205831199103+

1

2120575 (1199102+ 1199104)

119863119902

1199094= minus 119910

4

119863119902

1199104= minus 119886119878

4+ 1198871199094+ 1205831199104+

1

21205751199103

(25)

We can get the numerical solutions 119909119894 119910119894 (119894 = 0 1 2 3 4) of

system (25) by the improved predictor-corrector algorithmTherefore the approximate random responses of the stochas-tic system (18) can be expressed as

119909 (119905 119906) asymp

4

sum

119894=0

119909119894(119905)119867119894(119906)

119910 (119905 119906) asymp

4

sum

119894=0

119910119894(119905)119867119894(119906)

(26)

The ensemble mean responses (EMRs for short) of thestochastic fractional-order system are described as

119864 [119909 (119905 119906)] asymp

4

sum

119894=0

119909119894(119905) 119864 [119867

119894(119906)] = 119909

0(119905)

119864 [119910 (119905 119906)] asymp

4

sum

119894=0

119910119894(119905) 119864 [119867

119894(119906)] = 119910

0(119905)

(27)

Especially for 119906 = 0 namely 120583 = 120583 this sample system issignificant for reference It is usually called a mean parametersystem of which the responses (SRMs for short) may beapproximated as

119909 (119905 0) asymp

4

sum

119894=0

119909119894(119905)119867119894(0) = 119909

0(119905) minus 119909

2(119905) + 119909

4(119905)

119910 (119905 0) asymp

4

sum

119894=0

119910119894(119905)119867119894(0) = 119910

0(119905) minus 119910

2(119905) + 119910

4(119905)

(28)

5 Hopf Bifurcations

A Hopf bifurcation occurs when a periodic solution or limitcycle surrounding an equilibrium point arises or goes awayas a parameter varies In this section the Hopf bifurcation forthe stochastic fractional-order VDP system will be studiedwith theoretical and numerical computations Obviously

both the stochastic VDP and equivalent deterministic sys-tems (18) and (25) only have one equilibrium point namelythe origin (00) The Jacobian matrix 119869 for the linearizedsystem evaluated at the origin is

119869 =

[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[

[

0 minus1 0 0 0 0 0 0 0 0

119887 120583 01

2120575 0 0 0 0 0 0

0 0 0 minus1 0 0 0 0 0 0

1

2120575 0 119887 120583 0

1

2120575 0 0 0 0

0 0 0 0 0 minus1 0 0 0 0

0 0 01

2120575 119887 120583 0

1

2120575 0 0

0 0 0 0 0 0 0 minus1 0 0

0 0 0 0 01

2120575 119887 120583 0

1

2120575

0 0 0 0 0 0 0 0 0 minus1

0 0 0 0 0 0 01

2120575 119887 120583

]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]

]

(29)

The corresponding characteristic equation is

119891 (120582) = 119886012058210

+ 11988611205829

+ 11988621205828

+ 11988631205827

+ 11988641205826

+ 11988651205825

+ 11988661205824

+ 11988671205823

+ 11988681205822

+ 1198869120582 + 11988610

(30)

where 1198860 1198861 119886

10are the coefficients of (30) and can be

obtained by computing via Maple or MATLAB According tothe bifurcation theory of nonlinear dynamical systems andtaking 119908 as a bifurcation parameter [22] we can obtain theexistence conditions of Hopf bifurcation for the system

(1) The eigenvalues of the linearized system about thefixed point are 120572(119908) plusmn 119894120573(119908)

(2) The real and image parts of the eigenvalues satisfy120572(119908119888) = 0 120573(119908

119888) gt 0 1205721015840(119908

119888) = 0 when 119908 = 119908

119888

(3) The real parts of other eigenvalues are not equal tozeros

For the high order of equivalent system (25) so it is hardto get the threshold value of bifurcation parameter directlyTherefore we can get the threshold via the following theorem

Theorem 1 Let 119891(120582) = 1198860120582119899

+ 1198861120582119899minus1

+ sdot sdot sdot + 119886119899

= 0 be thecharacteristic equation evaluated at the equilibrium point Δ

119899

the n-dimensional Routh-Hurwitz determinant and 119886119894 (119894 =

0 1 119899) the coefficient of the characteristic equation If thereexists a threshold of bifurcation parameter 119908

119888which conforms

to the following terms then the first two existence conditions ofHopf bifurcation are satisfied

(1) Δ119899minus1

(119908119888) = 0 Δ

119899minus2(119908119888) = 0 Δ

119899minus3(119908119888) = 0

(2) 119886119894(119908119888) gt 0 (0 1 119899)

(3) Δ1015840

119899minus1(119908119888) = 0

6 Abstract and Applied Analysis

0 50 100 150 200 250 300

0

025

t

005

01

015

02

minus015

minus01

minus005

x0

(a) The time trajectory of EMR

0 005 01 015 02 025

0

005

01

015

02

y0

x0

minus015

minus01

minus005

minus02minus015 minus01 minus005

(b) The phase diagram of EMR in 1199090minus 1199100plane

0 50 100 150 200 250 300

0

025

t

x(t0)

005

01

015

02

minus015

minus01

minus005

(c) The time trajectory of SRM

0 005 01 015 02 025

0

005

01

015

minus015

minus01

minus005

minus02minus015 minus01 minus005

x(t 0)

y(t0)

(d) The phase diagram of SRM

Figure 2 The time trajectories and phase diagrams for 119906 = minus009

According to Theorem 1 the following parameter rela-tions which can make the equality Δ

9= 0 are derived

120583 = 0 120583 = plusmn120575

2 120583 = plusmn

120575

4 120583 = plusmn

radic3120575

2

120583 = plusmnradic3120575

4 120583 =

120575

4plusmn

radic3120575

4 120583 = minus

120575

4plusmn

radic3120575

4

(31)

Then (31) is substituted into the 8-dimensional and 7-dimensional Routh-Hurwitz determinants Δ

8 Δ7 and only

the following parameters can make Δ8

= 0 Δ7

= 0 satisfied

120583 = plusmn120575

2

120583 = plusmnradic3120575

2

(32)

Meanwhile the relations which can make all the coefficientssatisfied 119886

119894gt 0 (119894 = 0 1 10) are

120583 = plusmnradic3120575

2 (33)

We substitute the relations (33) intoΔ1015840

9(120583) andfind thatΔ1015840

9(120583)

is not equal to 0 therefore according to Theorem 1 the firsttwo existence conditions of Hopf bifurcation are satisfiedWhen (33) is substituted into the characteristic equation allthe eigenvalues of it can be obtained

12058212

= plusmnradic119887119894

12058234

=120575

4minus

radic3120575

4plusmn

1

4

radic41205752 minus 2radic31205752 minus 16119887

12058256

= minusradic3120575

4plusmn

1

4

radic31205752 minus 16119887

Abstract and Applied Analysis 7

0 50 100 150 200 250 300

0

025

t

005

01

015

02

minus015

minus01

minus005

x0

minus02

(a) The time trajectory of EMR

y0

x0

0 005 01

0

002

004

006

008

01

minus01minus005minus01

minus002

minus004

minus006

minus008

(b) The phase diagram of EMR in 1199090minus 1199100plane

0 50 100 150 200 250 300

025

t

005

01

015

02

minus015

minus01

minus005

0

x(t0)

(c) The time trajectory of SRM

0 001 002 003

0

001

002

003

minus003 minus002 minus001minus003

minus002

minus001

x(t 0)

y(t0)

(d) The phase diagram of SRM

Figure 3 The time trajectories and phase diagrams for 119906 = minus001

12058278

= minus120575

4minus

radic3120575

4plusmn

1

4

radic41205752 minus 2radic31205752 minus 16119887

120582910

= minusradic3120575

2plusmn

1

2

radic31205752 minus 4119887

(34)

Obviously when the intensity of random parameter 120575 = 0 thereal parts of all the other eigenvalues are less than 0 except1205821and 120582

2 The characteristic equation does not have zero

solution for 119887 = 0 Therefore all the existence conditions ofHopf bifurcation are satisfied When 120583

119888= minusradic31205752 the Hopf

bifurcation occurs around the equilibrium (0 0)The parameters are taken as 119886 = 1 119887 = 1 and 120575 =

01 and the derivative order is 119902 = 0995 And initialconditions are [119909

0(0) 119909

4(0)] = (02 02 02 02)

119879 and[1199100(0) 119910

4(0)] = (minus02 minus02 minus02 minus02)

119879 According tothe above analysis we can get that the critical parameter valueof the bifurcation is 120583

119888= minusradic31205752 asymp minus00866 which implies

that if 120583 lt 120583119888 the system will converge to the equilibrium

otherwise it will converge to a limit cycle The parametervalues are chosen as 120583 = minus009 and 119906 = minus001 respectivelyThen we obtain the time trajectories and phase diagrams forthe EMR and SRM of the stochastic system by simulationcomputations see Figures 2 and 3 It is clearly seen that thesystem converges to the fixed point (0 0) for 120583 = minus009 andto a limit cycle for 119906 = minus001 which implies that the Hopfbifurcation occurs in the stochastic system (18) Meanwhilethe SRM and EMR of phase plots coincide well enoughIt means that the Chebyshev polynomial approximationmethod is effective for the stochastic fractional-order system(18)

When the others parameters are fixed and 120583 = minus002 wealso found that Hopf bifurcation occurs with the variationof the derivative order The stochastic system converges tothe equilibrium when 119902 le 098 and to a limit cycle for 119902 ge

099 which means that the Hopf bifurcation occurs whenthe derivative order 119902 = 099 see Figure 4 from which it

8 Abstract and Applied Analysis

0 005 01 015 02 025

0

005

01

015

02

y0

x0

minus015

minus01

minus005

minus02minus015 minus01 minus005

(a) 119902 = 098

0 005 01

0

002

004

006

008

01

minus002

minus004

minus006

minus008

minus005

y0

x0

minus01minus01

(b) 119902 = 099

Figure 4 The phase diagrams for different values of order 119902

0 100 200 300 400 500 600 700 800

0

005

01

015

02

025

minus005

minus015

minus01

minus02

t

x0

(a)

0

0y0

x0

005

01

015

02

minus005

minus015

minus01

minus02005 01 015 02 025minus005minus015 minus01minus02

(b)

Figure 5 The time trajectory and phase diagram of EMR when 119905 isin [0 800] (a) The time trajectory (b) The phase diagram in 1199090minus 1199100plane

is obtained that the derivative order can also be taken as abifurcation parameter for fractional-order system

The simulation time is taken as 119905 = 800 one integrationstep of time is taken as Δ119905 = 0001 derivative order 119902 = 097and divide the time interval [0 800] into 8 equal parts Thetime trajectory and phase diagram for 119905 = 800 are shownin Figure 5 from which it appears that the 119909

0converges to a

fixed point as 119905 ge 300 However the time trajectory is zoomedin when 119905 belongs to the interval [300 400] see Figure 6(a)fromwhich we can see that the trajectory gradually decreasesbut does not converge to a point directly Secondly the timetrajectory is zoomed in when 119905 isin [400 500] see Figure 6(b)Meanwhile we also obtain the trajectories for 119905 isin [500 600]119905 isin [600 700] and 119905 isin [700 800] which are shown in Figures6(c)ndash6(e) respectively Comparing these figures it is seenthat the amplitude of 119909

0decreases as the time 119905 increases

Based on these observations we can suggest that the EMRsof the stochastic system (18) have an extremely long transientprocess while converging to the equilibrium as 119905 rarr infin

6 Conclusions

In this paper the Hopf bifurcation of a fractional-order VDPsystem with a random parameter is analyzed An equivalentdeterministic system is obtained with the approximationprinciple of Chebyshev orthogonal polynomials Based onbifurcation theory of nonlinear dynamic systems the criticalparameter value of the bifurcation is obtained by theoreticalanalysis Meanwhile numerical simulations are performedto verify the theoretical results Besides the derivative ordercan also be taken as a bifurcation parameter and the Hopfbifurcation is observed with the variation of the order

Abstract and Applied Analysis 9

300 320 340 360 380 400

012345times10minus4

minus1

minus2

minus3

minus4

minus5

t

x0

(a) 119905 isin [300 400]

400 420 440 460 480 500

0

05

1

15times10minus4

minus1

minus15

minus05

t

x0

(b) 119905 isin [400 500]

500 520 540 560 580 600

0

1

2

3

4times10minus5

minus1

minus2

minus3

minus4

t

x0

(c) 119905 isin [500 600]

600 620 640 660 680 700

0

2

4

6

8times10minus6

minus2

minus4

minus6

minus8

minus10

t

x0

(d) 119905 isin [600 700]

700 720 740 760 780 800t

0

2

05

1

15

times10minus6

minus2

minus25

minus15

minus1

minus05

x0

(e) 119905 isin [700 800]

Figure 6 The time trajectories of EMR 1199090for different time intervals

10 Abstract and Applied Analysis

It is worth mentioning that the orthogonal polyno-mial approximation is an effective method for integer-orderstochastic systems but the universality of the method forfractional-order ones still remains an open question Due tothe absences of analytical methods the threshold value ofbifurcation when the derivative order is taken as a bifurcationparameter cannot be obtained by theoretical analysis but onlythe numerical experiments

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the National Natural ScienceFoundation of China (NSFC) under Grant nos 1133200811172223 and 11172224

References

[1] C P Li W H Deng and D Xu ldquoChaos synchronization ofthe Chua system with a fractional orderrdquo Physica A StatisticalMechanics and Its Applications vol 360 no 2 pp 171ndash185 2006

[2] Z M Ge and C Y Ou ldquoChaos in a fractional order modifiedDuffing systemrdquo Chaos Solitons amp Fractals vol 34 no 2 pp262ndash291 2007

[3] J G Lu and G Chen ldquoA note on the fractional-order Chensystemrdquo Chaos Solitons and Fractals vol 27 no 3 pp 685ndash6882006

[4] H Sun W Chen and Y Chen ldquoVariable-order fractional dif-ferential operators in anomalous diffusion modelingrdquo PhysicaA Statistical Mechanics and Its Applications vol 388 no 21 pp4586ndash4592 2009

[5] Y Liang and W Chen ldquoA survey on computing Levy stabledistributions and a new MATLAB toolboxrdquo Signal Processingvol 93 no 1 pp 242ndash251 2013

[6] Y J Liang and W Chen ldquoReliability analysis for sluice gateanti-sliding stability using Levy stable distributionsrdquo SignalProcessing 2014

[7] V Venkatasubramanian ldquoSingularity induced bifurcation andthe van der Pol oscillatorrdquo IEEE Transactions on Circuits andSystems I Fundamental Theory and Applications vol 41 no 11pp 765ndash769 1994

[8] R Mettin U Parlitz andW Lauterborn ldquoBifurcation structureof the driven van der Pol oscillatorrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol3 no 6 pp 1529ndash1555 1993

[9] U Parlitz and W Lauterborn ldquoPeriod-doubling cascades anddevils staircases of the driven van der Pol oscillatorrdquo PhysicalReview A vol 36 no 3 pp 1428ndash1434 1987

[10] A Buonomo ldquoThe periodic solution of van der Polrsquos equationrdquoSIAM Journal on AppliedMathematics vol 59 no 1 pp 156ndash1711998

[11] J-X Xu and J Jiang ldquoThe global bifurcation characteristics ofthe forced van der Pol oscillatorrdquo Chaos Solitons and Fractalsvol 7 no 1 pp 3ndash19 1996

[12] R S Barbosa J A T MacHado B M Vinagre and A JCaldern ldquoAnalysis of the Van der Pol oscillator containing

derivatives of fractional orderrdquo Journal of Vibration and Controlvol 13 no 9-10 pp 1291ndash1301 2007

[13] J-H Chen andW-C Chen ldquoChaotic dynamics of the fraction-ally damped van der Pol equationrdquo Chaos Solitons and Fractalsvol 35 no 1 pp 188ndash198 2008

[14] M Shinozuka ldquoProbability modeling of concrete structuresrdquoASCE Journal of the EngineeringMechanics Division vol 98 no6 pp 1433ndash1451 1972

[15] R Ghamem and P Spans Stochastic Finite Element A SpectralApproach Springer Berlin Germany 1991

[16] C L Pettit and P S Beran ldquoSpectral and multiresolutionWiener expansions of oscillatory stochastic processesrdquo Journalof Sound and Vibration vol 294 no 4 pp 752ndash779 2006

[17] O P Le Maıtre O M Knio H N Najm and R G GhanemldquoUncertainty propagation using Wiener-Haar expansionsrdquoJournal of Computational Physics vol 197 no 1 pp 28ndash57 2004

[18] O P Le Maıtre H N Najm R G Ghanem and O M KnioldquoMulti-resolution analysis ofWiener-type uncertainty propaga-tion schemesrdquo Journal of Computational Physics vol 197 no 2pp 502ndash531 2004

[19] D Xiu and G E Karniadakis ldquoModeling uncertainty in steadystate diffusion problems via generalized polynomial chaosrdquoComputer Methods in Applied Mechanics and Engineering vol191 no 43 pp 4927ndash4948 2002

[20] X Wan and G E Karniadakis ldquoAn adaptive multi-elementgeneralized polynomial chaosmethod for stochastic differentialequationsrdquo Journal of Computational Physics vol 209 no 2 pp617ndash642 2005

[21] X L Leng C L Wu X P Ma G Meng and T Fang ldquoBifur-cation and chaos analysis of stochastic Duffing system underharmonic excitationsrdquo Nonlinear Dynamics vol 42 no 2 pp185ndash198 2005

[22] S-J MaW XuW Li and T Fang ldquoAnalysis of stochastic bifur-cation and chaos in stochastic Duffing-van der Pol system viaChebyshev polynomial approximationrdquo Chinese Physics vol 15no 6 article 017 pp 1231ndash1238 2006

[23] S J MaW Xu and T Fang ldquoAnalysis of period-doubling bifur-cation in double-well stochastic Duffing system via Laguerrepolynomial approximationrdquoNonlinear Dynamics vol 52 no 3pp 289ndash299 2008

[24] S-J Ma Q Shen and J Hou ldquoModified projective synchro-nization of stochastic fractional order chaotic systems withuncertain parametersrdquo Nonlinear Dynamics vol 73 no 1-2 pp93ndash100 2013

[25] X Liu L Hong and L Yang ldquoFractional-order complex Tsystem bifurcations chaos control and synchronizationrdquoNon-linear Dynamics vol 75 no 3 pp 589ndash602 2014

[26] K Diethelm N J Ford and A D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 3ndash22 2002

[27] A Charef H H Sun Y-Y Tsao and B Onaral ldquoFractal systemas represented by singularity functionrdquo IEEE Transactions onAutomatic Control vol 37 no 9 pp 1465ndash1470 1992

[28] M S Tavazoei andM Haeri ldquoLimitations of frequency domainapproximation for detecting chaos in fractional order systemsrdquoNonlinear Analysis Theory Methods amp Applications vol 69 no4 pp 1299ndash1320 2008

[29] P Borwein and T Erdelyi Polynomials and Polynomial Inequal-ities Springer New York NY USA 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Abstract and Applied Analysis

0 50 100 150 200 250 300

0

025

t

005

01

015

02

minus015

minus01

minus005

x0

(a) The time trajectory of EMR

0 005 01 015 02 025

0

005

01

015

02

y0

x0

minus015

minus01

minus005

minus02minus015 minus01 minus005

(b) The phase diagram of EMR in 1199090minus 1199100plane

0 50 100 150 200 250 300

0

025

t

x(t0)

005

01

015

02

minus015

minus01

minus005

(c) The time trajectory of SRM

0 005 01 015 02 025

0

005

01

015

minus015

minus01

minus005

minus02minus015 minus01 minus005

x(t 0)

y(t0)

(d) The phase diagram of SRM

Figure 2 The time trajectories and phase diagrams for 119906 = minus009

According to Theorem 1 the following parameter rela-tions which can make the equality Δ

9= 0 are derived

120583 = 0 120583 = plusmn120575

2 120583 = plusmn

120575

4 120583 = plusmn

radic3120575

2

120583 = plusmnradic3120575

4 120583 =

120575

4plusmn

radic3120575

4 120583 = minus

120575

4plusmn

radic3120575

4

(31)

Then (31) is substituted into the 8-dimensional and 7-dimensional Routh-Hurwitz determinants Δ

8 Δ7 and only

the following parameters can make Δ8

= 0 Δ7

= 0 satisfied

120583 = plusmn120575

2

120583 = plusmnradic3120575

2

(32)

Meanwhile the relations which can make all the coefficientssatisfied 119886

119894gt 0 (119894 = 0 1 10) are

120583 = plusmnradic3120575

2 (33)

We substitute the relations (33) intoΔ1015840

9(120583) andfind thatΔ1015840

9(120583)

is not equal to 0 therefore according to Theorem 1 the firsttwo existence conditions of Hopf bifurcation are satisfiedWhen (33) is substituted into the characteristic equation allthe eigenvalues of it can be obtained

12058212

= plusmnradic119887119894

12058234

=120575

4minus

radic3120575

4plusmn

1

4

radic41205752 minus 2radic31205752 minus 16119887

12058256

= minusradic3120575

4plusmn

1

4

radic31205752 minus 16119887

Abstract and Applied Analysis 7

0 50 100 150 200 250 300

0

025

t

005

01

015

02

minus015

minus01

minus005

x0

minus02

(a) The time trajectory of EMR

y0

x0

0 005 01

0

002

004

006

008

01

minus01minus005minus01

minus002

minus004

minus006

minus008

(b) The phase diagram of EMR in 1199090minus 1199100plane

0 50 100 150 200 250 300

025

t

005

01

015

02

minus015

minus01

minus005

0

x(t0)

(c) The time trajectory of SRM

0 001 002 003

0

001

002

003

minus003 minus002 minus001minus003

minus002

minus001

x(t 0)

y(t0)

(d) The phase diagram of SRM

Figure 3 The time trajectories and phase diagrams for 119906 = minus001

12058278

= minus120575

4minus

radic3120575

4plusmn

1

4

radic41205752 minus 2radic31205752 minus 16119887

120582910

= minusradic3120575

2plusmn

1

2

radic31205752 minus 4119887

(34)

Obviously when the intensity of random parameter 120575 = 0 thereal parts of all the other eigenvalues are less than 0 except1205821and 120582

2 The characteristic equation does not have zero

solution for 119887 = 0 Therefore all the existence conditions ofHopf bifurcation are satisfied When 120583

119888= minusradic31205752 the Hopf

bifurcation occurs around the equilibrium (0 0)The parameters are taken as 119886 = 1 119887 = 1 and 120575 =

01 and the derivative order is 119902 = 0995 And initialconditions are [119909

0(0) 119909

4(0)] = (02 02 02 02)

119879 and[1199100(0) 119910

4(0)] = (minus02 minus02 minus02 minus02)

119879 According tothe above analysis we can get that the critical parameter valueof the bifurcation is 120583

119888= minusradic31205752 asymp minus00866 which implies

that if 120583 lt 120583119888 the system will converge to the equilibrium

otherwise it will converge to a limit cycle The parametervalues are chosen as 120583 = minus009 and 119906 = minus001 respectivelyThen we obtain the time trajectories and phase diagrams forthe EMR and SRM of the stochastic system by simulationcomputations see Figures 2 and 3 It is clearly seen that thesystem converges to the fixed point (0 0) for 120583 = minus009 andto a limit cycle for 119906 = minus001 which implies that the Hopfbifurcation occurs in the stochastic system (18) Meanwhilethe SRM and EMR of phase plots coincide well enoughIt means that the Chebyshev polynomial approximationmethod is effective for the stochastic fractional-order system(18)

When the others parameters are fixed and 120583 = minus002 wealso found that Hopf bifurcation occurs with the variationof the derivative order The stochastic system converges tothe equilibrium when 119902 le 098 and to a limit cycle for 119902 ge

099 which means that the Hopf bifurcation occurs whenthe derivative order 119902 = 099 see Figure 4 from which it

8 Abstract and Applied Analysis

0 005 01 015 02 025

0

005

01

015

02

y0

x0

minus015

minus01

minus005

minus02minus015 minus01 minus005

(a) 119902 = 098

0 005 01

0

002

004

006

008

01

minus002

minus004

minus006

minus008

minus005

y0

x0

minus01minus01

(b) 119902 = 099

Figure 4 The phase diagrams for different values of order 119902

0 100 200 300 400 500 600 700 800

0

005

01

015

02

025

minus005

minus015

minus01

minus02

t

x0

(a)

0

0y0

x0

005

01

015

02

minus005

minus015

minus01

minus02005 01 015 02 025minus005minus015 minus01minus02

(b)

Figure 5 The time trajectory and phase diagram of EMR when 119905 isin [0 800] (a) The time trajectory (b) The phase diagram in 1199090minus 1199100plane

is obtained that the derivative order can also be taken as abifurcation parameter for fractional-order system

The simulation time is taken as 119905 = 800 one integrationstep of time is taken as Δ119905 = 0001 derivative order 119902 = 097and divide the time interval [0 800] into 8 equal parts Thetime trajectory and phase diagram for 119905 = 800 are shownin Figure 5 from which it appears that the 119909

0converges to a

fixed point as 119905 ge 300 However the time trajectory is zoomedin when 119905 belongs to the interval [300 400] see Figure 6(a)fromwhich we can see that the trajectory gradually decreasesbut does not converge to a point directly Secondly the timetrajectory is zoomed in when 119905 isin [400 500] see Figure 6(b)Meanwhile we also obtain the trajectories for 119905 isin [500 600]119905 isin [600 700] and 119905 isin [700 800] which are shown in Figures6(c)ndash6(e) respectively Comparing these figures it is seenthat the amplitude of 119909

0decreases as the time 119905 increases

Based on these observations we can suggest that the EMRsof the stochastic system (18) have an extremely long transientprocess while converging to the equilibrium as 119905 rarr infin

6 Conclusions

In this paper the Hopf bifurcation of a fractional-order VDPsystem with a random parameter is analyzed An equivalentdeterministic system is obtained with the approximationprinciple of Chebyshev orthogonal polynomials Based onbifurcation theory of nonlinear dynamic systems the criticalparameter value of the bifurcation is obtained by theoreticalanalysis Meanwhile numerical simulations are performedto verify the theoretical results Besides the derivative ordercan also be taken as a bifurcation parameter and the Hopfbifurcation is observed with the variation of the order

Abstract and Applied Analysis 9

300 320 340 360 380 400

012345times10minus4

minus1

minus2

minus3

minus4

minus5

t

x0

(a) 119905 isin [300 400]

400 420 440 460 480 500

0

05

1

15times10minus4

minus1

minus15

minus05

t

x0

(b) 119905 isin [400 500]

500 520 540 560 580 600

0

1

2

3

4times10minus5

minus1

minus2

minus3

minus4

t

x0

(c) 119905 isin [500 600]

600 620 640 660 680 700

0

2

4

6

8times10minus6

minus2

minus4

minus6

minus8

minus10

t

x0

(d) 119905 isin [600 700]

700 720 740 760 780 800t

0

2

05

1

15

times10minus6

minus2

minus25

minus15

minus1

minus05

x0

(e) 119905 isin [700 800]

Figure 6 The time trajectories of EMR 1199090for different time intervals

10 Abstract and Applied Analysis

It is worth mentioning that the orthogonal polyno-mial approximation is an effective method for integer-orderstochastic systems but the universality of the method forfractional-order ones still remains an open question Due tothe absences of analytical methods the threshold value ofbifurcation when the derivative order is taken as a bifurcationparameter cannot be obtained by theoretical analysis but onlythe numerical experiments

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the National Natural ScienceFoundation of China (NSFC) under Grant nos 1133200811172223 and 11172224

References

[1] C P Li W H Deng and D Xu ldquoChaos synchronization ofthe Chua system with a fractional orderrdquo Physica A StatisticalMechanics and Its Applications vol 360 no 2 pp 171ndash185 2006

[2] Z M Ge and C Y Ou ldquoChaos in a fractional order modifiedDuffing systemrdquo Chaos Solitons amp Fractals vol 34 no 2 pp262ndash291 2007

[3] J G Lu and G Chen ldquoA note on the fractional-order Chensystemrdquo Chaos Solitons and Fractals vol 27 no 3 pp 685ndash6882006

[4] H Sun W Chen and Y Chen ldquoVariable-order fractional dif-ferential operators in anomalous diffusion modelingrdquo PhysicaA Statistical Mechanics and Its Applications vol 388 no 21 pp4586ndash4592 2009

[5] Y Liang and W Chen ldquoA survey on computing Levy stabledistributions and a new MATLAB toolboxrdquo Signal Processingvol 93 no 1 pp 242ndash251 2013

[6] Y J Liang and W Chen ldquoReliability analysis for sluice gateanti-sliding stability using Levy stable distributionsrdquo SignalProcessing 2014

[7] V Venkatasubramanian ldquoSingularity induced bifurcation andthe van der Pol oscillatorrdquo IEEE Transactions on Circuits andSystems I Fundamental Theory and Applications vol 41 no 11pp 765ndash769 1994

[8] R Mettin U Parlitz andW Lauterborn ldquoBifurcation structureof the driven van der Pol oscillatorrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol3 no 6 pp 1529ndash1555 1993

[9] U Parlitz and W Lauterborn ldquoPeriod-doubling cascades anddevils staircases of the driven van der Pol oscillatorrdquo PhysicalReview A vol 36 no 3 pp 1428ndash1434 1987

[10] A Buonomo ldquoThe periodic solution of van der Polrsquos equationrdquoSIAM Journal on AppliedMathematics vol 59 no 1 pp 156ndash1711998

[11] J-X Xu and J Jiang ldquoThe global bifurcation characteristics ofthe forced van der Pol oscillatorrdquo Chaos Solitons and Fractalsvol 7 no 1 pp 3ndash19 1996

[12] R S Barbosa J A T MacHado B M Vinagre and A JCaldern ldquoAnalysis of the Van der Pol oscillator containing

derivatives of fractional orderrdquo Journal of Vibration and Controlvol 13 no 9-10 pp 1291ndash1301 2007

[13] J-H Chen andW-C Chen ldquoChaotic dynamics of the fraction-ally damped van der Pol equationrdquo Chaos Solitons and Fractalsvol 35 no 1 pp 188ndash198 2008

[14] M Shinozuka ldquoProbability modeling of concrete structuresrdquoASCE Journal of the EngineeringMechanics Division vol 98 no6 pp 1433ndash1451 1972

[15] R Ghamem and P Spans Stochastic Finite Element A SpectralApproach Springer Berlin Germany 1991

[16] C L Pettit and P S Beran ldquoSpectral and multiresolutionWiener expansions of oscillatory stochastic processesrdquo Journalof Sound and Vibration vol 294 no 4 pp 752ndash779 2006

[17] O P Le Maıtre O M Knio H N Najm and R G GhanemldquoUncertainty propagation using Wiener-Haar expansionsrdquoJournal of Computational Physics vol 197 no 1 pp 28ndash57 2004

[18] O P Le Maıtre H N Najm R G Ghanem and O M KnioldquoMulti-resolution analysis ofWiener-type uncertainty propaga-tion schemesrdquo Journal of Computational Physics vol 197 no 2pp 502ndash531 2004

[19] D Xiu and G E Karniadakis ldquoModeling uncertainty in steadystate diffusion problems via generalized polynomial chaosrdquoComputer Methods in Applied Mechanics and Engineering vol191 no 43 pp 4927ndash4948 2002

[20] X Wan and G E Karniadakis ldquoAn adaptive multi-elementgeneralized polynomial chaosmethod for stochastic differentialequationsrdquo Journal of Computational Physics vol 209 no 2 pp617ndash642 2005

[21] X L Leng C L Wu X P Ma G Meng and T Fang ldquoBifur-cation and chaos analysis of stochastic Duffing system underharmonic excitationsrdquo Nonlinear Dynamics vol 42 no 2 pp185ndash198 2005

[22] S-J MaW XuW Li and T Fang ldquoAnalysis of stochastic bifur-cation and chaos in stochastic Duffing-van der Pol system viaChebyshev polynomial approximationrdquo Chinese Physics vol 15no 6 article 017 pp 1231ndash1238 2006

[23] S J MaW Xu and T Fang ldquoAnalysis of period-doubling bifur-cation in double-well stochastic Duffing system via Laguerrepolynomial approximationrdquoNonlinear Dynamics vol 52 no 3pp 289ndash299 2008

[24] S-J Ma Q Shen and J Hou ldquoModified projective synchro-nization of stochastic fractional order chaotic systems withuncertain parametersrdquo Nonlinear Dynamics vol 73 no 1-2 pp93ndash100 2013

[25] X Liu L Hong and L Yang ldquoFractional-order complex Tsystem bifurcations chaos control and synchronizationrdquoNon-linear Dynamics vol 75 no 3 pp 589ndash602 2014

[26] K Diethelm N J Ford and A D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 3ndash22 2002

[27] A Charef H H Sun Y-Y Tsao and B Onaral ldquoFractal systemas represented by singularity functionrdquo IEEE Transactions onAutomatic Control vol 37 no 9 pp 1465ndash1470 1992

[28] M S Tavazoei andM Haeri ldquoLimitations of frequency domainapproximation for detecting chaos in fractional order systemsrdquoNonlinear Analysis Theory Methods amp Applications vol 69 no4 pp 1299ndash1320 2008

[29] P Borwein and T Erdelyi Polynomials and Polynomial Inequal-ities Springer New York NY USA 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

0 50 100 150 200 250 300

0

025

t

005

01

015

02

minus015

minus01

minus005

x0

minus02

(a) The time trajectory of EMR

y0

x0

0 005 01

0

002

004

006

008

01

minus01minus005minus01

minus002

minus004

minus006

minus008

(b) The phase diagram of EMR in 1199090minus 1199100plane

0 50 100 150 200 250 300

025

t

005

01

015

02

minus015

minus01

minus005

0

x(t0)

(c) The time trajectory of SRM

0 001 002 003

0

001

002

003

minus003 minus002 minus001minus003

minus002

minus001

x(t 0)

y(t0)

(d) The phase diagram of SRM

Figure 3 The time trajectories and phase diagrams for 119906 = minus001

12058278

= minus120575

4minus

radic3120575

4plusmn

1

4

radic41205752 minus 2radic31205752 minus 16119887

120582910

= minusradic3120575

2plusmn

1

2

radic31205752 minus 4119887

(34)

Obviously when the intensity of random parameter 120575 = 0 thereal parts of all the other eigenvalues are less than 0 except1205821and 120582

2 The characteristic equation does not have zero

solution for 119887 = 0 Therefore all the existence conditions ofHopf bifurcation are satisfied When 120583

119888= minusradic31205752 the Hopf

bifurcation occurs around the equilibrium (0 0)The parameters are taken as 119886 = 1 119887 = 1 and 120575 =

01 and the derivative order is 119902 = 0995 And initialconditions are [119909

0(0) 119909

4(0)] = (02 02 02 02)

119879 and[1199100(0) 119910

4(0)] = (minus02 minus02 minus02 minus02)

119879 According tothe above analysis we can get that the critical parameter valueof the bifurcation is 120583

119888= minusradic31205752 asymp minus00866 which implies

that if 120583 lt 120583119888 the system will converge to the equilibrium

otherwise it will converge to a limit cycle The parametervalues are chosen as 120583 = minus009 and 119906 = minus001 respectivelyThen we obtain the time trajectories and phase diagrams forthe EMR and SRM of the stochastic system by simulationcomputations see Figures 2 and 3 It is clearly seen that thesystem converges to the fixed point (0 0) for 120583 = minus009 andto a limit cycle for 119906 = minus001 which implies that the Hopfbifurcation occurs in the stochastic system (18) Meanwhilethe SRM and EMR of phase plots coincide well enoughIt means that the Chebyshev polynomial approximationmethod is effective for the stochastic fractional-order system(18)

When the others parameters are fixed and 120583 = minus002 wealso found that Hopf bifurcation occurs with the variationof the derivative order The stochastic system converges tothe equilibrium when 119902 le 098 and to a limit cycle for 119902 ge

099 which means that the Hopf bifurcation occurs whenthe derivative order 119902 = 099 see Figure 4 from which it

8 Abstract and Applied Analysis

0 005 01 015 02 025

0

005

01

015

02

y0

x0

minus015

minus01

minus005

minus02minus015 minus01 minus005

(a) 119902 = 098

0 005 01

0

002

004

006

008

01

minus002

minus004

minus006

minus008

minus005

y0

x0

minus01minus01

(b) 119902 = 099

Figure 4 The phase diagrams for different values of order 119902

0 100 200 300 400 500 600 700 800

0

005

01

015

02

025

minus005

minus015

minus01

minus02

t

x0

(a)

0

0y0

x0

005

01

015

02

minus005

minus015

minus01

minus02005 01 015 02 025minus005minus015 minus01minus02

(b)

Figure 5 The time trajectory and phase diagram of EMR when 119905 isin [0 800] (a) The time trajectory (b) The phase diagram in 1199090minus 1199100plane

is obtained that the derivative order can also be taken as abifurcation parameter for fractional-order system

The simulation time is taken as 119905 = 800 one integrationstep of time is taken as Δ119905 = 0001 derivative order 119902 = 097and divide the time interval [0 800] into 8 equal parts Thetime trajectory and phase diagram for 119905 = 800 are shownin Figure 5 from which it appears that the 119909

0converges to a

fixed point as 119905 ge 300 However the time trajectory is zoomedin when 119905 belongs to the interval [300 400] see Figure 6(a)fromwhich we can see that the trajectory gradually decreasesbut does not converge to a point directly Secondly the timetrajectory is zoomed in when 119905 isin [400 500] see Figure 6(b)Meanwhile we also obtain the trajectories for 119905 isin [500 600]119905 isin [600 700] and 119905 isin [700 800] which are shown in Figures6(c)ndash6(e) respectively Comparing these figures it is seenthat the amplitude of 119909

0decreases as the time 119905 increases

Based on these observations we can suggest that the EMRsof the stochastic system (18) have an extremely long transientprocess while converging to the equilibrium as 119905 rarr infin

6 Conclusions

In this paper the Hopf bifurcation of a fractional-order VDPsystem with a random parameter is analyzed An equivalentdeterministic system is obtained with the approximationprinciple of Chebyshev orthogonal polynomials Based onbifurcation theory of nonlinear dynamic systems the criticalparameter value of the bifurcation is obtained by theoreticalanalysis Meanwhile numerical simulations are performedto verify the theoretical results Besides the derivative ordercan also be taken as a bifurcation parameter and the Hopfbifurcation is observed with the variation of the order

Abstract and Applied Analysis 9

300 320 340 360 380 400

012345times10minus4

minus1

minus2

minus3

minus4

minus5

t

x0

(a) 119905 isin [300 400]

400 420 440 460 480 500

0

05

1

15times10minus4

minus1

minus15

minus05

t

x0

(b) 119905 isin [400 500]

500 520 540 560 580 600

0

1

2

3

4times10minus5

minus1

minus2

minus3

minus4

t

x0

(c) 119905 isin [500 600]

600 620 640 660 680 700

0

2

4

6

8times10minus6

minus2

minus4

minus6

minus8

minus10

t

x0

(d) 119905 isin [600 700]

700 720 740 760 780 800t

0

2

05

1

15

times10minus6

minus2

minus25

minus15

minus1

minus05

x0

(e) 119905 isin [700 800]

Figure 6 The time trajectories of EMR 1199090for different time intervals

10 Abstract and Applied Analysis

It is worth mentioning that the orthogonal polyno-mial approximation is an effective method for integer-orderstochastic systems but the universality of the method forfractional-order ones still remains an open question Due tothe absences of analytical methods the threshold value ofbifurcation when the derivative order is taken as a bifurcationparameter cannot be obtained by theoretical analysis but onlythe numerical experiments

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the National Natural ScienceFoundation of China (NSFC) under Grant nos 1133200811172223 and 11172224

References

[1] C P Li W H Deng and D Xu ldquoChaos synchronization ofthe Chua system with a fractional orderrdquo Physica A StatisticalMechanics and Its Applications vol 360 no 2 pp 171ndash185 2006

[2] Z M Ge and C Y Ou ldquoChaos in a fractional order modifiedDuffing systemrdquo Chaos Solitons amp Fractals vol 34 no 2 pp262ndash291 2007

[3] J G Lu and G Chen ldquoA note on the fractional-order Chensystemrdquo Chaos Solitons and Fractals vol 27 no 3 pp 685ndash6882006

[4] H Sun W Chen and Y Chen ldquoVariable-order fractional dif-ferential operators in anomalous diffusion modelingrdquo PhysicaA Statistical Mechanics and Its Applications vol 388 no 21 pp4586ndash4592 2009

[5] Y Liang and W Chen ldquoA survey on computing Levy stabledistributions and a new MATLAB toolboxrdquo Signal Processingvol 93 no 1 pp 242ndash251 2013

[6] Y J Liang and W Chen ldquoReliability analysis for sluice gateanti-sliding stability using Levy stable distributionsrdquo SignalProcessing 2014

[7] V Venkatasubramanian ldquoSingularity induced bifurcation andthe van der Pol oscillatorrdquo IEEE Transactions on Circuits andSystems I Fundamental Theory and Applications vol 41 no 11pp 765ndash769 1994

[8] R Mettin U Parlitz andW Lauterborn ldquoBifurcation structureof the driven van der Pol oscillatorrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol3 no 6 pp 1529ndash1555 1993

[9] U Parlitz and W Lauterborn ldquoPeriod-doubling cascades anddevils staircases of the driven van der Pol oscillatorrdquo PhysicalReview A vol 36 no 3 pp 1428ndash1434 1987

[10] A Buonomo ldquoThe periodic solution of van der Polrsquos equationrdquoSIAM Journal on AppliedMathematics vol 59 no 1 pp 156ndash1711998

[11] J-X Xu and J Jiang ldquoThe global bifurcation characteristics ofthe forced van der Pol oscillatorrdquo Chaos Solitons and Fractalsvol 7 no 1 pp 3ndash19 1996

[12] R S Barbosa J A T MacHado B M Vinagre and A JCaldern ldquoAnalysis of the Van der Pol oscillator containing

derivatives of fractional orderrdquo Journal of Vibration and Controlvol 13 no 9-10 pp 1291ndash1301 2007

[13] J-H Chen andW-C Chen ldquoChaotic dynamics of the fraction-ally damped van der Pol equationrdquo Chaos Solitons and Fractalsvol 35 no 1 pp 188ndash198 2008

[14] M Shinozuka ldquoProbability modeling of concrete structuresrdquoASCE Journal of the EngineeringMechanics Division vol 98 no6 pp 1433ndash1451 1972

[15] R Ghamem and P Spans Stochastic Finite Element A SpectralApproach Springer Berlin Germany 1991

[16] C L Pettit and P S Beran ldquoSpectral and multiresolutionWiener expansions of oscillatory stochastic processesrdquo Journalof Sound and Vibration vol 294 no 4 pp 752ndash779 2006

[17] O P Le Maıtre O M Knio H N Najm and R G GhanemldquoUncertainty propagation using Wiener-Haar expansionsrdquoJournal of Computational Physics vol 197 no 1 pp 28ndash57 2004

[18] O P Le Maıtre H N Najm R G Ghanem and O M KnioldquoMulti-resolution analysis ofWiener-type uncertainty propaga-tion schemesrdquo Journal of Computational Physics vol 197 no 2pp 502ndash531 2004

[19] D Xiu and G E Karniadakis ldquoModeling uncertainty in steadystate diffusion problems via generalized polynomial chaosrdquoComputer Methods in Applied Mechanics and Engineering vol191 no 43 pp 4927ndash4948 2002

[20] X Wan and G E Karniadakis ldquoAn adaptive multi-elementgeneralized polynomial chaosmethod for stochastic differentialequationsrdquo Journal of Computational Physics vol 209 no 2 pp617ndash642 2005

[21] X L Leng C L Wu X P Ma G Meng and T Fang ldquoBifur-cation and chaos analysis of stochastic Duffing system underharmonic excitationsrdquo Nonlinear Dynamics vol 42 no 2 pp185ndash198 2005

[22] S-J MaW XuW Li and T Fang ldquoAnalysis of stochastic bifur-cation and chaos in stochastic Duffing-van der Pol system viaChebyshev polynomial approximationrdquo Chinese Physics vol 15no 6 article 017 pp 1231ndash1238 2006

[23] S J MaW Xu and T Fang ldquoAnalysis of period-doubling bifur-cation in double-well stochastic Duffing system via Laguerrepolynomial approximationrdquoNonlinear Dynamics vol 52 no 3pp 289ndash299 2008

[24] S-J Ma Q Shen and J Hou ldquoModified projective synchro-nization of stochastic fractional order chaotic systems withuncertain parametersrdquo Nonlinear Dynamics vol 73 no 1-2 pp93ndash100 2013

[25] X Liu L Hong and L Yang ldquoFractional-order complex Tsystem bifurcations chaos control and synchronizationrdquoNon-linear Dynamics vol 75 no 3 pp 589ndash602 2014

[26] K Diethelm N J Ford and A D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 3ndash22 2002

[27] A Charef H H Sun Y-Y Tsao and B Onaral ldquoFractal systemas represented by singularity functionrdquo IEEE Transactions onAutomatic Control vol 37 no 9 pp 1465ndash1470 1992

[28] M S Tavazoei andM Haeri ldquoLimitations of frequency domainapproximation for detecting chaos in fractional order systemsrdquoNonlinear Analysis Theory Methods amp Applications vol 69 no4 pp 1299ndash1320 2008

[29] P Borwein and T Erdelyi Polynomials and Polynomial Inequal-ities Springer New York NY USA 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

0 005 01 015 02 025

0

005

01

015

02

y0

x0

minus015

minus01

minus005

minus02minus015 minus01 minus005

(a) 119902 = 098

0 005 01

0

002

004

006

008

01

minus002

minus004

minus006

minus008

minus005

y0

x0

minus01minus01

(b) 119902 = 099

Figure 4 The phase diagrams for different values of order 119902

0 100 200 300 400 500 600 700 800

0

005

01

015

02

025

minus005

minus015

minus01

minus02

t

x0

(a)

0

0y0

x0

005

01

015

02

minus005

minus015

minus01

minus02005 01 015 02 025minus005minus015 minus01minus02

(b)

Figure 5 The time trajectory and phase diagram of EMR when 119905 isin [0 800] (a) The time trajectory (b) The phase diagram in 1199090minus 1199100plane

is obtained that the derivative order can also be taken as abifurcation parameter for fractional-order system

The simulation time is taken as 119905 = 800 one integrationstep of time is taken as Δ119905 = 0001 derivative order 119902 = 097and divide the time interval [0 800] into 8 equal parts Thetime trajectory and phase diagram for 119905 = 800 are shownin Figure 5 from which it appears that the 119909

0converges to a

fixed point as 119905 ge 300 However the time trajectory is zoomedin when 119905 belongs to the interval [300 400] see Figure 6(a)fromwhich we can see that the trajectory gradually decreasesbut does not converge to a point directly Secondly the timetrajectory is zoomed in when 119905 isin [400 500] see Figure 6(b)Meanwhile we also obtain the trajectories for 119905 isin [500 600]119905 isin [600 700] and 119905 isin [700 800] which are shown in Figures6(c)ndash6(e) respectively Comparing these figures it is seenthat the amplitude of 119909

0decreases as the time 119905 increases

Based on these observations we can suggest that the EMRsof the stochastic system (18) have an extremely long transientprocess while converging to the equilibrium as 119905 rarr infin

6 Conclusions

In this paper the Hopf bifurcation of a fractional-order VDPsystem with a random parameter is analyzed An equivalentdeterministic system is obtained with the approximationprinciple of Chebyshev orthogonal polynomials Based onbifurcation theory of nonlinear dynamic systems the criticalparameter value of the bifurcation is obtained by theoreticalanalysis Meanwhile numerical simulations are performedto verify the theoretical results Besides the derivative ordercan also be taken as a bifurcation parameter and the Hopfbifurcation is observed with the variation of the order

Abstract and Applied Analysis 9

300 320 340 360 380 400

012345times10minus4

minus1

minus2

minus3

minus4

minus5

t

x0

(a) 119905 isin [300 400]

400 420 440 460 480 500

0

05

1

15times10minus4

minus1

minus15

minus05

t

x0

(b) 119905 isin [400 500]

500 520 540 560 580 600

0

1

2

3

4times10minus5

minus1

minus2

minus3

minus4

t

x0

(c) 119905 isin [500 600]

600 620 640 660 680 700

0

2

4

6

8times10minus6

minus2

minus4

minus6

minus8

minus10

t

x0

(d) 119905 isin [600 700]

700 720 740 760 780 800t

0

2

05

1

15

times10minus6

minus2

minus25

minus15

minus1

minus05

x0

(e) 119905 isin [700 800]

Figure 6 The time trajectories of EMR 1199090for different time intervals

10 Abstract and Applied Analysis

It is worth mentioning that the orthogonal polyno-mial approximation is an effective method for integer-orderstochastic systems but the universality of the method forfractional-order ones still remains an open question Due tothe absences of analytical methods the threshold value ofbifurcation when the derivative order is taken as a bifurcationparameter cannot be obtained by theoretical analysis but onlythe numerical experiments

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the National Natural ScienceFoundation of China (NSFC) under Grant nos 1133200811172223 and 11172224

References

[1] C P Li W H Deng and D Xu ldquoChaos synchronization ofthe Chua system with a fractional orderrdquo Physica A StatisticalMechanics and Its Applications vol 360 no 2 pp 171ndash185 2006

[2] Z M Ge and C Y Ou ldquoChaos in a fractional order modifiedDuffing systemrdquo Chaos Solitons amp Fractals vol 34 no 2 pp262ndash291 2007

[3] J G Lu and G Chen ldquoA note on the fractional-order Chensystemrdquo Chaos Solitons and Fractals vol 27 no 3 pp 685ndash6882006

[4] H Sun W Chen and Y Chen ldquoVariable-order fractional dif-ferential operators in anomalous diffusion modelingrdquo PhysicaA Statistical Mechanics and Its Applications vol 388 no 21 pp4586ndash4592 2009

[5] Y Liang and W Chen ldquoA survey on computing Levy stabledistributions and a new MATLAB toolboxrdquo Signal Processingvol 93 no 1 pp 242ndash251 2013

[6] Y J Liang and W Chen ldquoReliability analysis for sluice gateanti-sliding stability using Levy stable distributionsrdquo SignalProcessing 2014

[7] V Venkatasubramanian ldquoSingularity induced bifurcation andthe van der Pol oscillatorrdquo IEEE Transactions on Circuits andSystems I Fundamental Theory and Applications vol 41 no 11pp 765ndash769 1994

[8] R Mettin U Parlitz andW Lauterborn ldquoBifurcation structureof the driven van der Pol oscillatorrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol3 no 6 pp 1529ndash1555 1993

[9] U Parlitz and W Lauterborn ldquoPeriod-doubling cascades anddevils staircases of the driven van der Pol oscillatorrdquo PhysicalReview A vol 36 no 3 pp 1428ndash1434 1987

[10] A Buonomo ldquoThe periodic solution of van der Polrsquos equationrdquoSIAM Journal on AppliedMathematics vol 59 no 1 pp 156ndash1711998

[11] J-X Xu and J Jiang ldquoThe global bifurcation characteristics ofthe forced van der Pol oscillatorrdquo Chaos Solitons and Fractalsvol 7 no 1 pp 3ndash19 1996

[12] R S Barbosa J A T MacHado B M Vinagre and A JCaldern ldquoAnalysis of the Van der Pol oscillator containing

derivatives of fractional orderrdquo Journal of Vibration and Controlvol 13 no 9-10 pp 1291ndash1301 2007

[13] J-H Chen andW-C Chen ldquoChaotic dynamics of the fraction-ally damped van der Pol equationrdquo Chaos Solitons and Fractalsvol 35 no 1 pp 188ndash198 2008

[14] M Shinozuka ldquoProbability modeling of concrete structuresrdquoASCE Journal of the EngineeringMechanics Division vol 98 no6 pp 1433ndash1451 1972

[15] R Ghamem and P Spans Stochastic Finite Element A SpectralApproach Springer Berlin Germany 1991

[16] C L Pettit and P S Beran ldquoSpectral and multiresolutionWiener expansions of oscillatory stochastic processesrdquo Journalof Sound and Vibration vol 294 no 4 pp 752ndash779 2006

[17] O P Le Maıtre O M Knio H N Najm and R G GhanemldquoUncertainty propagation using Wiener-Haar expansionsrdquoJournal of Computational Physics vol 197 no 1 pp 28ndash57 2004

[18] O P Le Maıtre H N Najm R G Ghanem and O M KnioldquoMulti-resolution analysis ofWiener-type uncertainty propaga-tion schemesrdquo Journal of Computational Physics vol 197 no 2pp 502ndash531 2004

[19] D Xiu and G E Karniadakis ldquoModeling uncertainty in steadystate diffusion problems via generalized polynomial chaosrdquoComputer Methods in Applied Mechanics and Engineering vol191 no 43 pp 4927ndash4948 2002

[20] X Wan and G E Karniadakis ldquoAn adaptive multi-elementgeneralized polynomial chaosmethod for stochastic differentialequationsrdquo Journal of Computational Physics vol 209 no 2 pp617ndash642 2005

[21] X L Leng C L Wu X P Ma G Meng and T Fang ldquoBifur-cation and chaos analysis of stochastic Duffing system underharmonic excitationsrdquo Nonlinear Dynamics vol 42 no 2 pp185ndash198 2005

[22] S-J MaW XuW Li and T Fang ldquoAnalysis of stochastic bifur-cation and chaos in stochastic Duffing-van der Pol system viaChebyshev polynomial approximationrdquo Chinese Physics vol 15no 6 article 017 pp 1231ndash1238 2006

[23] S J MaW Xu and T Fang ldquoAnalysis of period-doubling bifur-cation in double-well stochastic Duffing system via Laguerrepolynomial approximationrdquoNonlinear Dynamics vol 52 no 3pp 289ndash299 2008

[24] S-J Ma Q Shen and J Hou ldquoModified projective synchro-nization of stochastic fractional order chaotic systems withuncertain parametersrdquo Nonlinear Dynamics vol 73 no 1-2 pp93ndash100 2013

[25] X Liu L Hong and L Yang ldquoFractional-order complex Tsystem bifurcations chaos control and synchronizationrdquoNon-linear Dynamics vol 75 no 3 pp 589ndash602 2014

[26] K Diethelm N J Ford and A D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 3ndash22 2002

[27] A Charef H H Sun Y-Y Tsao and B Onaral ldquoFractal systemas represented by singularity functionrdquo IEEE Transactions onAutomatic Control vol 37 no 9 pp 1465ndash1470 1992

[28] M S Tavazoei andM Haeri ldquoLimitations of frequency domainapproximation for detecting chaos in fractional order systemsrdquoNonlinear Analysis Theory Methods amp Applications vol 69 no4 pp 1299ndash1320 2008

[29] P Borwein and T Erdelyi Polynomials and Polynomial Inequal-ities Springer New York NY USA 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

300 320 340 360 380 400

012345times10minus4

minus1

minus2

minus3

minus4

minus5

t

x0

(a) 119905 isin [300 400]

400 420 440 460 480 500

0

05

1

15times10minus4

minus1

minus15

minus05

t

x0

(b) 119905 isin [400 500]

500 520 540 560 580 600

0

1

2

3

4times10minus5

minus1

minus2

minus3

minus4

t

x0

(c) 119905 isin [500 600]

600 620 640 660 680 700

0

2

4

6

8times10minus6

minus2

minus4

minus6

minus8

minus10

t

x0

(d) 119905 isin [600 700]

700 720 740 760 780 800t

0

2

05

1

15

times10minus6

minus2

minus25

minus15

minus1

minus05

x0

(e) 119905 isin [700 800]

Figure 6 The time trajectories of EMR 1199090for different time intervals

10 Abstract and Applied Analysis

It is worth mentioning that the orthogonal polyno-mial approximation is an effective method for integer-orderstochastic systems but the universality of the method forfractional-order ones still remains an open question Due tothe absences of analytical methods the threshold value ofbifurcation when the derivative order is taken as a bifurcationparameter cannot be obtained by theoretical analysis but onlythe numerical experiments

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the National Natural ScienceFoundation of China (NSFC) under Grant nos 1133200811172223 and 11172224

References

[1] C P Li W H Deng and D Xu ldquoChaos synchronization ofthe Chua system with a fractional orderrdquo Physica A StatisticalMechanics and Its Applications vol 360 no 2 pp 171ndash185 2006

[2] Z M Ge and C Y Ou ldquoChaos in a fractional order modifiedDuffing systemrdquo Chaos Solitons amp Fractals vol 34 no 2 pp262ndash291 2007

[3] J G Lu and G Chen ldquoA note on the fractional-order Chensystemrdquo Chaos Solitons and Fractals vol 27 no 3 pp 685ndash6882006

[4] H Sun W Chen and Y Chen ldquoVariable-order fractional dif-ferential operators in anomalous diffusion modelingrdquo PhysicaA Statistical Mechanics and Its Applications vol 388 no 21 pp4586ndash4592 2009

[5] Y Liang and W Chen ldquoA survey on computing Levy stabledistributions and a new MATLAB toolboxrdquo Signal Processingvol 93 no 1 pp 242ndash251 2013

[6] Y J Liang and W Chen ldquoReliability analysis for sluice gateanti-sliding stability using Levy stable distributionsrdquo SignalProcessing 2014

[7] V Venkatasubramanian ldquoSingularity induced bifurcation andthe van der Pol oscillatorrdquo IEEE Transactions on Circuits andSystems I Fundamental Theory and Applications vol 41 no 11pp 765ndash769 1994

[8] R Mettin U Parlitz andW Lauterborn ldquoBifurcation structureof the driven van der Pol oscillatorrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol3 no 6 pp 1529ndash1555 1993

[9] U Parlitz and W Lauterborn ldquoPeriod-doubling cascades anddevils staircases of the driven van der Pol oscillatorrdquo PhysicalReview A vol 36 no 3 pp 1428ndash1434 1987

[10] A Buonomo ldquoThe periodic solution of van der Polrsquos equationrdquoSIAM Journal on AppliedMathematics vol 59 no 1 pp 156ndash1711998

[11] J-X Xu and J Jiang ldquoThe global bifurcation characteristics ofthe forced van der Pol oscillatorrdquo Chaos Solitons and Fractalsvol 7 no 1 pp 3ndash19 1996

[12] R S Barbosa J A T MacHado B M Vinagre and A JCaldern ldquoAnalysis of the Van der Pol oscillator containing

derivatives of fractional orderrdquo Journal of Vibration and Controlvol 13 no 9-10 pp 1291ndash1301 2007

[13] J-H Chen andW-C Chen ldquoChaotic dynamics of the fraction-ally damped van der Pol equationrdquo Chaos Solitons and Fractalsvol 35 no 1 pp 188ndash198 2008

[14] M Shinozuka ldquoProbability modeling of concrete structuresrdquoASCE Journal of the EngineeringMechanics Division vol 98 no6 pp 1433ndash1451 1972

[15] R Ghamem and P Spans Stochastic Finite Element A SpectralApproach Springer Berlin Germany 1991

[16] C L Pettit and P S Beran ldquoSpectral and multiresolutionWiener expansions of oscillatory stochastic processesrdquo Journalof Sound and Vibration vol 294 no 4 pp 752ndash779 2006

[17] O P Le Maıtre O M Knio H N Najm and R G GhanemldquoUncertainty propagation using Wiener-Haar expansionsrdquoJournal of Computational Physics vol 197 no 1 pp 28ndash57 2004

[18] O P Le Maıtre H N Najm R G Ghanem and O M KnioldquoMulti-resolution analysis ofWiener-type uncertainty propaga-tion schemesrdquo Journal of Computational Physics vol 197 no 2pp 502ndash531 2004

[19] D Xiu and G E Karniadakis ldquoModeling uncertainty in steadystate diffusion problems via generalized polynomial chaosrdquoComputer Methods in Applied Mechanics and Engineering vol191 no 43 pp 4927ndash4948 2002

[20] X Wan and G E Karniadakis ldquoAn adaptive multi-elementgeneralized polynomial chaosmethod for stochastic differentialequationsrdquo Journal of Computational Physics vol 209 no 2 pp617ndash642 2005

[21] X L Leng C L Wu X P Ma G Meng and T Fang ldquoBifur-cation and chaos analysis of stochastic Duffing system underharmonic excitationsrdquo Nonlinear Dynamics vol 42 no 2 pp185ndash198 2005

[22] S-J MaW XuW Li and T Fang ldquoAnalysis of stochastic bifur-cation and chaos in stochastic Duffing-van der Pol system viaChebyshev polynomial approximationrdquo Chinese Physics vol 15no 6 article 017 pp 1231ndash1238 2006

[23] S J MaW Xu and T Fang ldquoAnalysis of period-doubling bifur-cation in double-well stochastic Duffing system via Laguerrepolynomial approximationrdquoNonlinear Dynamics vol 52 no 3pp 289ndash299 2008

[24] S-J Ma Q Shen and J Hou ldquoModified projective synchro-nization of stochastic fractional order chaotic systems withuncertain parametersrdquo Nonlinear Dynamics vol 73 no 1-2 pp93ndash100 2013

[25] X Liu L Hong and L Yang ldquoFractional-order complex Tsystem bifurcations chaos control and synchronizationrdquoNon-linear Dynamics vol 75 no 3 pp 589ndash602 2014

[26] K Diethelm N J Ford and A D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 3ndash22 2002

[27] A Charef H H Sun Y-Y Tsao and B Onaral ldquoFractal systemas represented by singularity functionrdquo IEEE Transactions onAutomatic Control vol 37 no 9 pp 1465ndash1470 1992

[28] M S Tavazoei andM Haeri ldquoLimitations of frequency domainapproximation for detecting chaos in fractional order systemsrdquoNonlinear Analysis Theory Methods amp Applications vol 69 no4 pp 1299ndash1320 2008

[29] P Borwein and T Erdelyi Polynomials and Polynomial Inequal-ities Springer New York NY USA 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Abstract and Applied Analysis

It is worth mentioning that the orthogonal polyno-mial approximation is an effective method for integer-orderstochastic systems but the universality of the method forfractional-order ones still remains an open question Due tothe absences of analytical methods the threshold value ofbifurcation when the derivative order is taken as a bifurcationparameter cannot be obtained by theoretical analysis but onlythe numerical experiments

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

This work is supported by the National Natural ScienceFoundation of China (NSFC) under Grant nos 1133200811172223 and 11172224

References

[1] C P Li W H Deng and D Xu ldquoChaos synchronization ofthe Chua system with a fractional orderrdquo Physica A StatisticalMechanics and Its Applications vol 360 no 2 pp 171ndash185 2006

[2] Z M Ge and C Y Ou ldquoChaos in a fractional order modifiedDuffing systemrdquo Chaos Solitons amp Fractals vol 34 no 2 pp262ndash291 2007

[3] J G Lu and G Chen ldquoA note on the fractional-order Chensystemrdquo Chaos Solitons and Fractals vol 27 no 3 pp 685ndash6882006

[4] H Sun W Chen and Y Chen ldquoVariable-order fractional dif-ferential operators in anomalous diffusion modelingrdquo PhysicaA Statistical Mechanics and Its Applications vol 388 no 21 pp4586ndash4592 2009

[5] Y Liang and W Chen ldquoA survey on computing Levy stabledistributions and a new MATLAB toolboxrdquo Signal Processingvol 93 no 1 pp 242ndash251 2013

[6] Y J Liang and W Chen ldquoReliability analysis for sluice gateanti-sliding stability using Levy stable distributionsrdquo SignalProcessing 2014

[7] V Venkatasubramanian ldquoSingularity induced bifurcation andthe van der Pol oscillatorrdquo IEEE Transactions on Circuits andSystems I Fundamental Theory and Applications vol 41 no 11pp 765ndash769 1994

[8] R Mettin U Parlitz andW Lauterborn ldquoBifurcation structureof the driven van der Pol oscillatorrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol3 no 6 pp 1529ndash1555 1993

[9] U Parlitz and W Lauterborn ldquoPeriod-doubling cascades anddevils staircases of the driven van der Pol oscillatorrdquo PhysicalReview A vol 36 no 3 pp 1428ndash1434 1987

[10] A Buonomo ldquoThe periodic solution of van der Polrsquos equationrdquoSIAM Journal on AppliedMathematics vol 59 no 1 pp 156ndash1711998

[11] J-X Xu and J Jiang ldquoThe global bifurcation characteristics ofthe forced van der Pol oscillatorrdquo Chaos Solitons and Fractalsvol 7 no 1 pp 3ndash19 1996

[12] R S Barbosa J A T MacHado B M Vinagre and A JCaldern ldquoAnalysis of the Van der Pol oscillator containing

derivatives of fractional orderrdquo Journal of Vibration and Controlvol 13 no 9-10 pp 1291ndash1301 2007

[13] J-H Chen andW-C Chen ldquoChaotic dynamics of the fraction-ally damped van der Pol equationrdquo Chaos Solitons and Fractalsvol 35 no 1 pp 188ndash198 2008

[14] M Shinozuka ldquoProbability modeling of concrete structuresrdquoASCE Journal of the EngineeringMechanics Division vol 98 no6 pp 1433ndash1451 1972

[15] R Ghamem and P Spans Stochastic Finite Element A SpectralApproach Springer Berlin Germany 1991

[16] C L Pettit and P S Beran ldquoSpectral and multiresolutionWiener expansions of oscillatory stochastic processesrdquo Journalof Sound and Vibration vol 294 no 4 pp 752ndash779 2006

[17] O P Le Maıtre O M Knio H N Najm and R G GhanemldquoUncertainty propagation using Wiener-Haar expansionsrdquoJournal of Computational Physics vol 197 no 1 pp 28ndash57 2004

[18] O P Le Maıtre H N Najm R G Ghanem and O M KnioldquoMulti-resolution analysis ofWiener-type uncertainty propaga-tion schemesrdquo Journal of Computational Physics vol 197 no 2pp 502ndash531 2004

[19] D Xiu and G E Karniadakis ldquoModeling uncertainty in steadystate diffusion problems via generalized polynomial chaosrdquoComputer Methods in Applied Mechanics and Engineering vol191 no 43 pp 4927ndash4948 2002

[20] X Wan and G E Karniadakis ldquoAn adaptive multi-elementgeneralized polynomial chaosmethod for stochastic differentialequationsrdquo Journal of Computational Physics vol 209 no 2 pp617ndash642 2005

[21] X L Leng C L Wu X P Ma G Meng and T Fang ldquoBifur-cation and chaos analysis of stochastic Duffing system underharmonic excitationsrdquo Nonlinear Dynamics vol 42 no 2 pp185ndash198 2005

[22] S-J MaW XuW Li and T Fang ldquoAnalysis of stochastic bifur-cation and chaos in stochastic Duffing-van der Pol system viaChebyshev polynomial approximationrdquo Chinese Physics vol 15no 6 article 017 pp 1231ndash1238 2006

[23] S J MaW Xu and T Fang ldquoAnalysis of period-doubling bifur-cation in double-well stochastic Duffing system via Laguerrepolynomial approximationrdquoNonlinear Dynamics vol 52 no 3pp 289ndash299 2008

[24] S-J Ma Q Shen and J Hou ldquoModified projective synchro-nization of stochastic fractional order chaotic systems withuncertain parametersrdquo Nonlinear Dynamics vol 73 no 1-2 pp93ndash100 2013

[25] X Liu L Hong and L Yang ldquoFractional-order complex Tsystem bifurcations chaos control and synchronizationrdquoNon-linear Dynamics vol 75 no 3 pp 589ndash602 2014

[26] K Diethelm N J Ford and A D Freed ldquoA predictor-correctorapproach for the numerical solution of fractional differentialequationsrdquoNonlinear Dynamics vol 29 no 1ndash4 pp 3ndash22 2002

[27] A Charef H H Sun Y-Y Tsao and B Onaral ldquoFractal systemas represented by singularity functionrdquo IEEE Transactions onAutomatic Control vol 37 no 9 pp 1465ndash1470 1992

[28] M S Tavazoei andM Haeri ldquoLimitations of frequency domainapproximation for detecting chaos in fractional order systemsrdquoNonlinear Analysis Theory Methods amp Applications vol 69 no4 pp 1299ndash1320 2008

[29] P Borwein and T Erdelyi Polynomials and Polynomial Inequal-ities Springer New York NY USA 1995

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of