symbolic computation of normal forms for resonant double hopf bifurcations using a perturbation...

Journal of Sound and <ibration (2001) 247(4), 615}632doi:10.1006/jsvi.2001.3732, available online at http://www.idealibrary.com on

SYMBOLIC COMPUTATION OF NORMAL FORMS FORRESONANT DOUBLE HOPF BIFURCATIONS USING

A PERTURBATION TECHNIQUE

P. YU

Department of Applied Mathematics, ;niversity of=estern Ontario, ¸ondon, Ont., Canada N6A 5B7.E-mail: [email protected]

(Received 28 September 2000, and in ,nal form 2 April 2001)

This paper presents a perturbation method for computing the normal forms of resonantdouble Hopf bifurcations with the aid of a computer algebraic system. This technique, basedon the method of multiple time scales, can be used to deal with general n-dimensionalsystems without the application of center manifold theory. Explicit iterative formulas havebeen derived for uniquely determining the coe$cients of normal forms and associatednon-linear transformations up to an arbitrary order. User-friendly symbolic computerprograms written in Maple have been developed, which can be easily applied for computingthe normal forms of a given system. Examples are presented to show the applicability of themethodology and the convenience of using the computer software.

( 2001 Academic Press

1. INTRODUCTION

Normal form theory has been widely used to simplify the analysis of dynamic behavior ofdi!erential equations near an equilibrium [1}3]. The basic idea of the method of normalforms is to employ successive co-ordinate transformations to systematically constructa simpli"ed form of the original di!erential equations without changing the fundamentaldynamical behavior of the system in the vicinity of the equilibrium. Normal form theory isusually applied along with center manifold theory [4]. Center manifold theory is appliedbefore normal form theory, to obtain a locally invariant small dimensional manifold in thevicinity of an equilibrium, called center manifold. Although it is easy to construct an&&abstract'' normal form for a given singularity, it is di$cult to compute the explicitexpressions of a normal form in terms of the coe$cients of the original di!erentialequations. Therefore, the crucial part in computing the normal form of a system is thecomputation e$ciency. Moreover, algebraic manipulations become very involved and timeconsuming as the order of the normal form increases. The introduction of computer algebrasystems such as Maple, Mathematica, Reduce, etc., becomes necessary at this point.Recently, many researchers have paid attention to the development of e$cientcomputational methods with the aid of computer algebra systems [5}11].

In this paper, a perturbation technique is used to compute the normal forms ofdi!erential equations whose Jacobian, evaluated at an equilibrium, has two pairs of purelyimaginary eigenvalues. The ratio of the two eigenvalues is a rational number, giving rise toresonant oscillations. The perturbation method, combined with multiple time scales, hasbeen successfully developed for computing normal forms of Hopf bifurcation and othersingularities [7, 12, 13]. It has been shown that the approach is very computationally

0022-460X/01/440615#18 $35.00/0 ( 2001 Academic Press

616 P. YU

e$cient and is simple enough to use by a novice to computer algebra. This technique issystematic and can be directly applied to the original n-dimensional di!erential equations,without the application of centre manifold theory. This approach reduces the steps involvedin the application of normal form theory to simultaneously obtain the normal form (on thecenter manifold) as well as the associated non-linear transformation.

This paper focuses on the development of the methodology and symbolic software forcomputing the normal forms of a double Hopf bifurcation associated with various resonantcases, including k

1: k

2(k

1Ok

2, k

1and k

2are positive integers) resonances and 1 : 1 primary

resonance. The separate treatment for the two cases is due to di!erent scaling which isnecessary for the application of perturbation techniques. Uniform scaling is usedfor the k

1: k

2(k

1Ok

2) resonant cases while non-uniform scaling has to be applied for

the 1 : 1 resonance to match the perturbation orders. All the formulas derived in this paperfor computing the coe$cients of the normal forms and the non-linear transformationsare given explicitly in terms of the original system coe$cients. This facilitatesdirect applications to special problems [8, 10, 14]. Based on the explicit recursiveformulas, user-friendly symbolic software using Maple has been developed. Examples arepresented to show that the perturbation technique is computationally e$cient andtherefore, is particularly useful for computing high-dimensional systems and high ordernormal forms.

The k1: k

2(k

1Ok

2) resonant cases are described in the next section, while section 3 deals

with 1 : 1 resonant cases. Section 4 outlines the symbolic computer programs, and examplesare presented in section 5. Conclusions are drawn in section 6.

2. PERTURBATION TECHNIQUE AND FORMULATIONS

2.1. k1: k

2RESONANT CASES

Consider a general n-dimensional system described by the following di!erential equation:

x5 "Jx#f(x), x3Rn, f : RnPRn, (1)

where Jx represents the linear term, and the non-linear function f is assumed to be analytic;and x"0 is an equilibrium of the system, i.e., f (0)"0. Further, assume that the Jacobian ofsystem (1) evaluated at the equilibrium 0 involves two pairs of purely imaginary eigenvalues$iu

1cand $iu

2c. In this section, we assume that

u1c

u2c

"

k1

k2

, where k1

and k2

are integers, (2)

which is called a k1: k

2resonance. Thus, for the convenience of analysis, we may let u

1c"

k1u and u

2c"k

2u, and further, without loss of generality, we may assume that u"1.

(Otherwise, one may use a time scaling t@"ut to scale the frequency u to 1.) Consequently,the Jacobian matrix of system (1) evaluated at the equilibrium, x"0, can be assumed(with the aid of linear transformation if necessary) in the Jordan canonical form, to be

J"

0 k1

0 0 0

!k1

0 0 0 0

0 0 0 k2

0

0 0 !k2

0 0

0 0 0 0 A

, A3R(n~4)](n~4), (3)

NORMAL FORMS FOR DOUBLE HOPF BIFURCATION 617

such that k1(k

2. Here A is hyperbolic, i.e., all of its eigenvalues have non-zero real parts,

with either a real or complex conjugate. For physically interesting cases, the unstablemanifold is assumed to be empty, i.e., all the eigenvalues of A have negative real parts. Let usrewrite system (1) in component form as

xR1"k

1x2#f

1(x), xR

2"!k

1x1#f

2(x), (4)

xR3"k

2x4#f

3(x), xR

4"!k

2x3#f

4(x), (5)

xRp"!a

pxp#f

p(x) (p"5, 6,2 , m

1#4), (6)

xRq"!a

qxq#u

qxq`1

#fq(x),

xRq`1

"!uqxq!a

qxq`1

#fq`1

(x) (q"m1#5, m

1#7,2, n!1), (7)

where ap'0, a

q'0, u

q'0, and 4#m

1#2m

2"n. Equations (4)}(7) show that the matrix

A has m1

real eigenvalues, !ap, p"1, 2,2, m

1, and m

2pairs of complex conjugate

eigenvalues, !aq$iu

q, q"1, 2,2, m

2. The functions f

i(x) satisfy f

i(0)"0 and

L fi(0)/Lx

j"0, i, j"1, 2,2, n. That is, the Taylor expansion of f

i(x) about x"0 starts with

quadratic terms.The underlying idea of the method of multiple scales is to consider the expansion of

representing the response as a function of multiple independent variables, or scales, insteadof a single time variable. This can be achieved by introducing new independent variablesaccording to

¹k"ekt, k"0, 1, 2,2 . (8)

It follows that the derivatives with respect to t now become expansions in terms of thepartial derivatives with respect to ¹

k, given by

d

dt"

d¹0

dt

LL¹

0

#

d¹1

dt

LL¹

1

#

d¹2

dt

LL¹

2

#2"D0#eD

1#e2D

2#2, (9)

where the di!erentiation operator Dk"L/L¹

k.

Now suppose that the solution of equation (1) (or equivalently, equations (4)}(7)) in theneighborhood of x"0 is represented by an expansion of the form

xi(t; e)"ex

i1(¹

0, ¹

1,2)#e2x

i2(¹

0, ¹

1,2)#2 (i"1, 2,2, n). (10)

Note that the number of independent time scales needed depends on the order to which theexpansion is carried out. In general, to "nd a normal form up to an order n, ¹

0, ¹

1,2, ¹

nshould be used in solution (10). However, one can employ the idea of the intrinsic harmonicbalance (IHB) technique [15] and let n be open so that the process of the method will takeinto account all essential terms up to the required order, as seen in the following procedure.

Substituting solution (10) into equations (4)} (7) with the aid of equation (9) and balancingthe like powers of e results in the perturbation equations in the order given:

e1 : D0x11"k

1x21

, D0x21"!k

1x11

, (11)

D0x31"k

2x41

, D0x41"!k

2x31

, (12)

D0xp1"!a

pxp1

(p"3, 4,2 , m1#2), (13)

618 P. YU

D0xq1"!a

qxq1#u

qx(q`1)1

,

D0x(q`1)1

"!uqxq1!a

qx(q`1)1

(q"m1#3, m

1#5,2 , n!1). (14)

e2 : D0x12"k

1x22!D

1x11#f

12(x

1) , D

0x22

"!k1x12!D

1x21#f

22(x

1) , (15)

D0x32"k

2x42!D

1x31#f

32(x

1), D

0x42"!k

2x32!D

1x41#f

42(x

1), (16)

D0xp2"!a

pxp2#f

p2(x

1) (p"3, 4,2, m

1#2), (17)

D0xq2"!a

qxq2#u

qx(q`1)2

#fq2

(x1),

D0x(q`1)2

"!uqxq2!a

qx(q`1)2

#f(q`1)2

(x1) (q"m

1#3, m

1#5,2 , n!1), (18)

etc., where fi2"(d2/de2) [ f

i(x

1)]e/0

are functions of xi1

(i"1, 2,2 , n) which have beenobtained from the e1 order perturbation equations (11)}(14). In general, functions f

ikonly

involve the variables which have been solved from the previous 1, 2,2, (k!1) orderperturbation equations.

To "nd the solutions of the e1 order equations (11)} (14), "rst note that these equationscan be divided into two groups, one of which consists of the "rst four equations given inequations (11) and (12), and the other one includes the remaining equations. The formergroup is associated with the critical eigenvalues (having zero real parts), and the lattercorresponds to the non-critical eigenvalues (having non-zero real parts). Second, the "rstorder solutions for the variables of the second group are obtained only from the "rst fourvariables x

1, x

2, x

3and x

4, since the perturbation technique is based on the assumed

asymptotic solution (10). However, it is interesting to note that the normal form obtainedbased on equation (10) not only represents the asymptotic behavior, but also the transientproperty of the system.

Now consider equation set (11). Di!erentiating the "rst equation of equation (11) andthen substituting the second equation of equation (11) into the resulting equation yieldsa simple second order ordinary di!erential equation,

D20x11#k2

1x11"0. (19)

The solution of equation (19) can be written in a general form as

x11"r

1(¹

1, ¹

2,2) cos [k

1¹0#/

1(¹

1, ¹

2,2)],r

1cos (k

1¹0#/

1),r

1cos h

1, (20)

where r1

and /1

represent, respectively, the amplitude and phase of motion, andh1"k

1¹0#/

1. Once x

11is determined, x

21can be directly solved from the "rst equation

of equation (11). Note that solution (20) implies that

D0r1"0 and D

0/

1"0, (21)

since r1

and /1

do not contain the variable ¹0. (Remember that D

0denotes di!erentiation

with respect to ¹0.) Similarly, we can obtain solutions from equation (14) as follows:

x31"r

2(¹

1, ¹

2,2) cos [k

2¹0#/

2(¹

1, ¹

2,2)],r

2cos (k

2¹0#/

1),r

2cos h

2(22)

and

D0r2"0 and D

0/

2"0. (23)


x41

can be determined from the "rst equation of equation (12) after x31

is solved. Theasymptotic e1 order solutions of the second group, described by equations (13) and (14), areobviously given by

xi1"0, i"5, 6,2 , n, (24)

which actually represent the "rst order steady state solutions of the second group equations.Next, to solve the e2 order perturbation equations (15)}(18), the procedure described

above can be applied. Thus, di!erentiating the "rst equation of equation (15) and thensubstituting the second equation of equation (15) into the resulting equation results ina non-linear homogeneous ordinary di!erential equation:

D20x12#k2

1x12"!D

1D

0x11!D

1x21

#D0f12#f

22. (25)

Then, substituting the solutions x11

and x21

into the right-hand side of equation (25)gives an expression in terms of the trigonometric functions cos k¹

0and sin k¹

0. To

eliminate possible secular terms which may appear in the solution of x12

, it is requiredthat the coe$cients of the two terms cos k

1¹0

and sin k1¹0

be equal to zero, whichin turn yields the explicit solutions for the derivatives D

1r1

and D1/1. Then, the solution

of x12

can be found from the remaining terms of equation (23), and thus, x12

containsonly a particular solution. Having found x

12, x

22is determined from the "rst

equation of equation (15) as x22"(1/k

1) (D

0x12#D

1x11!f

12(x

1)). This procedure

can be carried out to higher order perturbation equations, and thus we can "nd D1r2,

D1/2, x

32and x

42, etc. Finally, the normal forms, given in polar co-ordinates, can be

written as

dr1

dt"

Lr1

L¹0

L¹0

Lt#

Lr1

L¹1

L¹1

Lt#

Lr1

L¹2

L¹2

Lt#2"D

0r1#D

1r1#D

2r1#2, (26)

dh1

dt"k

1#

L/1

L¹0

L¹0

Lt#

L/1

L¹1

L¹1

Lt#

L/1

L¹2

L¹2

Lt#2"k

1#D

0/

1#D

1/1#D

2/1#2,

(27)

dr2

dt"

Lr2

L¹0

L¹0

Lt#

Lr2

L¹1

L¹1

Lt#

Lr2

L¹2

L¹2

Lt#2"D

0r2#D

1r2#D

2r2#2, (28)

dh2

dt"k

2#

L/2

L¹0

L¹0

Lt#

L/2

L¹1

L¹1

Lt#

L/2

L¹2

L¹2

Lt#2"k

2#D

0/

2#D

1/2#D

2/2#2,

(29)

where the back scaling eriPr

i(i.e., ex

iPx

i) has been used (or simply set e"1). It should be

pointed out here that the particular solutions of equation (25), etc., can be found by usingthe IHB method [15] so that the solution is uniquely determined. Thus, D

irj

and Di/j

( j"1, 2) are also uniquely de"ned, which implies that the normal form given in equations(26)}(29) are actually uniquely determined. It is also noted that, unlike the non-resonantcase, in which the derivatives D

irjand D

i/jare functions of r

1and r

2only, here they are

given explicitly in terms of amplitudes r1, r

2and phase k

2h1!k

1h2.

620 P. YU

2.2. 1 : 1 RESONANT CASE

Again, consider system (1), but now the Jacobian evaluated on the equilibrium x"0 isgiven by

J"

0 u 0 1 0

!u 0 0 0 1

0 0 0 u 0

0 0 !u 0 0

0 0 0 0 A

, A3R(n~4)](n~4), (30)

where u can be assumed as u"1 if a time scaling t@"ut is applied. Note that this is calledthe non-semisimple case since two 1's are present in J.

This resonant case is quite similar to the resonant cases discussed in section 2.1. The onlydi!erence for this particular case is that unlike a general k

1: k

2(k

1Ok

2) resonance, we

cannot use a uniform scaling xiPex

ito obtain the ordered perturbation equations. It is

straightforward to show that a uniform scaling leads to the solution of non-lineardi!erential equations at the "rst perturbation order. This di$culty is due to thenon-semisimple resonance, that is, linear terms x

3and x

4are present in the "rst and second

equations (see the Jacobian given by equation (30)). In order for the perturbation techniqueto work on the 1 : 1 non-semisimple resonant case, we must use di!erent scalings betweenx1, x

2and other variables, x

i, i"3, 4, 5,2 , n. A careful consideration shows that the ratio

of the perturbation order for x1, x

2and the remaining variables should be 2 : 3. Thus, let

x1Pe2x

1, x

2Pe2x

2, x

iPe3x

i, i"3, 4,2 , n. (31)

Then, unlike the k1: k

2(k

1Ok

2) resonances, where the highest degree of harmonics at an

order k is consistently equal to k#1, now the kth order terms in the original di!erentialequations may spread into di!erent order ('k) perturbation equations due to the di!erentscalings. However, it can be shown that the highest order of the perturbation equationwhich should be included for the original kth order terms is 3k!2. Therefore, the highestdegree of the harmonics for each kth order must be 3k!2. In other words, if an nth ordernormal form is needed, then the perturbation equations up to 3n!2 order must beconsidered. Once this scaling is applied and the solution form is appropriately set, thena solution procedure similar to that presented in section 2.1 can be obtained and theformulas similar to that given in section 2.1 can be found.

It should be noted, unlike for the k1: k

2(k

1Ok

2) resonant cases, a kth order normal form

and the associated non-linear transformations (solutions) obtained for this 1 : 1 case mayinvolve higher order terms which actually do not belong to the kth order expression. This isdue to the di!erence in scaling. This suggests that the di!erence in scaling has caused thelower order terms to spread into higher order perturbation equations, and has causedhigher order terms to appear in lower order ((3n!2) perturbation equations. This is thefundamental di!erence between the 1 : 1 primary resonance and the k

1: k

2(k

1Ok

2)

resonance. In order to remove the terms which do not belong to the nth order normal form,one may just easily truncate the normal form and solutions up to nth order. (Note: here theorder means the order of the original di!erential equations.) This redundant calculationincreases the complexity and the time required for the computation.

It is interesting to note that the normal form obtained using the perturbation techniquevia di!erent scalings actually agree with that found by using other theories and methods


such as PoincareH normal form theory and the IHB technique. This will be demonstrated bythe examples presented in section 4.

3. OUTLINE OF THE SYMBOLIC COMPUTATION

All the formulas presented in the previous section are given explicitly in terms of thecoe$cients of the original di!erential equations, and thus can be easily implemented usinga computer algebraic system. Maple has been used to code these explicit formulas. In thissection, we shall outline the Maple programs.s

3.1. MAPLE SOURCE CODE

(1) Read a prepared input "le. The input "le provides the original di!erent equations,with the number of non-zero real eigenvalues, N

1, and the number of the pairs of

complex conjugate eigenvalues, N2, as well as the order for computing the normal

form, norder.(2) Create procedures.* The procedure Findcoef for computing all the coe$cients of harmonics in terms of

cos (mh1#nh

2) and sin (mh

1#nh

2) for a given trigonometric function F.

* The procedure solu1A}B for computing the coe$cients of harmonics in solutions

xi, i"1, 2, 3, 4 (corresponding to the center manifold) using the method of

harmonic balancing.* The procedure solu2A

}B for computing the coe$cients of harmonics in solutions

xi, i"5, 6,2, N

1#4 (corresponding to the non-zero real eigenvalues) using the

method of harmonic balancing.* The procedure solu3A

}B for computing the coe$cients of harmonics in solutions

xi, i"N

i#5, N

1#6,2, N

1#2]N

2(corresponding to the complex conjugate

eigenvalues) using the method of harmonic balancing.(3) Create the main procedure solution for computing the normal form and non-linear

transformation (i.e., periodic solution).* Establish ordered perturbation equations f

iby separating the original di!erential

equations Dxiaccording to the powers of e.

* Create the formal solutions for xi, i"1, 2, 3, 4, given in terms of basic

trigonometric functions.* Call the procedure Findcoef to "nd the harmonic coe$cients of the original

di!erential equations after the formal solutions are used.* Find the normal form terms D

nr1, D

np1, D

nr2, D

np2

by eliminating the secularterms.

* Call the procedure solu1A}B to "nd the solutions for x

i, i"1, 2, 3, 4.

* For the part associated with the non-zero real eigenvalues, call the procedureFindcoef "rst and then the procedure solu2A

}B to obtain the solutions for x

i,

i"5, 6,2 , N1#4.

* For the part associated with the complex conjugate eigenvalues, call theprocedure Findcoef "rst and then the procedure solu3A

}B to solve the solutions

for x , i"N #5, N #6,2 , N #2]N .

sThe Maple source codes and sample computer input "les can be found from the author's website:http://pyu1.apmaths.uwo.ca/pyu/software. The source code name is program5 and the name of the sample input"le is input5.

i 1 1 1 2

TABLE 1

Input ,le to Maple source code

N1 :" 1:N2 :" 1:N :" 4#N1#N2*2:norder :" 4:case :" 10:Dx[1] :" omg1*x[2]#x[1] L 2#x[1]*x[3]#2/3*x[3]*x[4]#x[4]*x[5]:Dx[2] :"!omg1*x[1]#2*x[4] L 2#x[5] ( 2#1/2*x[1]*x[2]#x[2]*x[4]:Dx[3] :" omg2*x[4]#4*x[3] L 2#x[5] ( 2#x[1]*x[2]:Dx[4] :"!omg2*x[3]#2*x[2] L 2#x[3]*x[4]#x[3]*x[5]:Dx[5] :"!x[5]#x[2] ( 2#1/3*x [2] *x[4]:Dx[6] :"!x[6]#x[7]#x[2] ( 2#5*x[1]*x[3]:Dx[7] :"!x[6]!x[7]#2/5*x[1] ( 2:if case"11 then

omg1 :"1:omg2 :"2:Dx[1] :"Dx[1]#x[3]:Dx[2] :"Dx[2]#x[4]:

elseomg1 :"2:omg2 :"3:

":

622 P. YU

(4) Transform the original di!erential equations into the summation of ordered terms byintroducing perturbation parameter e.

(5) Transform the original variables xi

into the summation of ordered terms byintroducing perturbation parameter e.

(6) Set the zero order solutions xi0

and zero order normal form terms D0r1, D

0p1, D

0r2,

D0p2.

(7) Call the main procedure solution to obtain the normal form and solutions for anarbitrary perturbation order n.

(8) Write the normal form dr1, dp1, dr2, dp2 into the output "le &&Nform''.

3.2. CREATE THE INPUT FILE

(1) Set the variables:N1*the number of the non-zero real eigenvalues of the Jacobian.N2*the number of pairs of complex conjugate eigenvalues of the Jacobian.N2the dimension of the system.norder2the order of normal forms to be computed.

(2) Create the vector "eld, i.e., the functions Dxi, i"1, 2,2 , n; n"4#N

1#2N

2, the

dimension of the system.

As an example, the input "le for the third example (2 : 3 resonant case) given in the nextsection is shown in Table 1.

4. EXAMPLES

In this section, three examples are presented to show the applicability of the method andthe e$ciency of the Maple programs. The "rst two examples are chosen from the problems


which have been studied before, in order to verify the results obtained using the programsdeveloped in this paper. The third example is used to demonstrate the computationale$ciency of the Maple programs for computing high order normal forms ofa higher-dimensional system. All the results including the normal forms and correspondingnon-linear transformations are obtained by executing the Maple programs on a PC(PENTIUM III-700MMX 256K system).

4.1. EXAMPLE 1 FOR 1 : 2 RESONANCE

Consider the following four-dimensional general system, described by

x5 "Jx#f (x, l), x3R4, l3R2, (32)

where l is a two-dimensional vector parameter called perturbation parameter (unfolding),and the Jacobian is given by

J"

0 u1c

0 0

!u1c

0 0 0

0 0 0 u2c

0 0 !u2c

0 (x"0, l"0),

(33)

satisfying u1c

/u2c"1 : 2. Thus, we may assume that u

1c"1 and u

2c"2. This example was

studied before using the IHB technique [16]. Since our interest here is focused on thecomputation of the normal form, we may ignore the perturbation parameter l (or simply setl"0). The general normal form up to second order was obtained using the IHB method inreference [16] as

dr1

dq1

"r1r2(A cosU#B sinU), r

1

d/1

dq1

"r1r2(B cosU!A sinU),

dr2

dq2

"r21(C cosU#D sinU), r

2

d/2

dq2

"r21(!D cosU#C sinU), (34)

where U"2/1!/

2, and

A"14

[( f113

!f223

)#( f124

#f214

)], C"14

[( f114

!f224

)!( f123

#f213

)],

C"14

[( f311

!f322

)#( f412

#f421

)], D"14

[( f411

!f422

)!( f312

#f321

)], (35)

in which fijk"L f

i(0, 0)/Lx

jLx

kfor f"( f

1, f

2, f

3, f

4)T. Note that the scaled times

q2"2q

1,2q, and q

k"u

kt.

Now rewrite equation (32) in the expanded form

x5 "Jx#Q(x)#2, (36)

where Q(x)"(q1, q

2, q

3, q

4)T represents the quadratic terms with

qi"1

2f111

x21#1

2f122

x21#1

2f133

x23#1

2f144

x24#f

112x1x2#f

113x1x3

#f114

x1x4#f

123x2x3#f

124x2x4#f

134x3x4

(i"1, 2, 3, 4). (37)

624 P. YU

Then executing the Maple program developed in this paper yields the following normalform:

dr1

dt"!r

1r2(A cosW#B sinW ), r

1

dh1

dt"r

1!r

1r2(B cosW!A sin W),

dr2

dt"!

1

2r21(C cosW#D sinW ), r

2

dh2

dt"2r

2!

1

2r21(!D cosW#C sinW), (38)

where A, B, C and D are given in equation (35), and W"h2!2h

1. One can show that

equation (38) is actually identical to equation (34) by noting the di!erent time scales, and thephase di!erence between system (38) and (34) (due to the di!erent notations used in theMaple program), which is !n/2. This indicates that the results obtained in this paper usingthe multiple scales are the same as those given by a di!erent method such as the IHBtechnique [16].

Here, it should be pointed out that the Maple program developed in this paper can beemployed to "nd normal forms up to an arbitrary order as long as the machine memory isallowed. However, for the IHB method, only the form up to the leading order terms iscorrect. For example, for the 1 : 2 resonant case, it can be shown that, in general, the normalform obtained using the IHB approach is correct only up to the second order.


The double pendulum system shown in Figure 1 consists of two rigid weigthless links ofequal length l which carry two concentrated masses 2m and m respectively. A follower forceP1

and a constant directional force (vertical) P2

are applied to this system. This system hasbeen studied by many authors for di!erent types of bifurcations (e.g., see references[14, 17]).

Figure 1. A double pendulum system.


The system energy for the three linear springs h1, h

2and h

3is assumed to be given by [17]

<"12[(h

1#h

2#h

3l2 )h2

1#2(h

3l2!h

2)h

1h2#(h

2#h

3l2 )h2

2]!1

6h3l2(h

1#h

2) (h3

1#h3

2),

(39)

where h1

and h2

are the generalized co-ordinates which specify the con"guration of thesystem completely.

The kinetic energy ¹ of the system is expressed by

¹"

ml2

2X2[3hQ 2

1#hQ 2

2#2hQ

1hQ2cos (h

1!h

2)], (40)

where X is an arbitrary value rendering the time variable non-dimensional, and the overdotdenotes di!erentiation with respect to the non-dimensional time variable q and q"Xt.

The components of the generalized forces corresponding to the generalized co-ordinatesh1

and h2

may be written as

Q1"P

1l sin (h

1!h

2)#2P

2l sin h

2, Q

2"P

2l sin h

2(41)

and the damping can be expressed by

D"12[d

1hQ 21#d

2(hQ

1!hQ

2)2]#1

4[d

3hQ 41#d

4(hQ

1!hQ

2)4], (42)

where d1

and d2

represent the linear parts, while d3

and d4

describe the non-linear parts.With the aid of the ¸agrangian equations, in addition, choosing the state variables

z1"h

1, z

2"hQ

1, z

3"h

2and z

4"hQ

2(43)

and by rescaling the coe$cients to be dimensionless coe$cients as

f1"

h1X2

ml2, f

2"

h2X2

ml2, f

3"

h3X2

m, f

4"

P1X2

ml, f

5"

P2X2

ml,

f6"

d3X4

ml2, f

7"

d4X4

ml2, g

1"

d1X2

ml2, g

2"

d2X2

ml2, (44)

one can derive a set of "rst order di!erential equations up to third order terms as follows:

dz1

dq"z

2,

dz2

dq"!(1

2f1#f

2!1

2f4!f

5) z

1!(1

2g1#g

2) z

2#( f

2!1

2f4!1

2f5) z

3#g

2z4

#(14

f1#3

4f2!1

3f4!2

3f5) z3

1!(3

4f2#1

2f3!1

3f4! 7

12f5) z3

3#f

7z34

!(12

f6#f

7) z3

2#(1

4g1#3

4g2) z2

1z2!(1

2f1#9

4f2!1

2f3!f

4!3

2f5) z2

1z3

!34

g2z21z4!1

2z1z22#1

2z22z3#3 f

7z22z4#(1

4f1#9

4f2!f

4!3

2f5)z

1z23

#(14

g1#3

4n2) z

2z23!3

4g2z23z4!1

2z1z24!3 f

7z2z24#1

2z3z24

!( 12

g1#3

2g2) z

1z2z3#3

2g2z1z3z4,

626 P. YU

dz3

dq"z

4,

dz4

dq"(1

2f1#2 f

2!f

3!1

2f4!f

5) z

1#(1

2g1#2g

2)z

2!( 2 f

2#f

3!1

2f4!3

2f5) z

3

!2 g2z4!(1

2f1#5

4f2!1

6f3! 7

12f4!7

6f5) z3

1#(5

4f2#7

6f3! 7

12f4!f

5) z3

3

#(12

f6#2 f

7) z3

2!2f

7z34!(1

2g1#5

4g2) z2

1z2#5

4g2z21z4#3

2z1z22

#( f1#15

4f2!1

2f3!7

4f4!11

4f5)z2

1z3!(1

2f1#15

4f2!1

2f3!7

4f4!5

2f5)z

1z23

!32z22z3!6 f

7z22z4!(1

2g1#5

4g2) z

2z23#5

4g2z23z4#1

2z1z24

#6 f7z2z24!1

2z3z24#(g

1#5

2g2) z

1z2z3!5

2g2z1z3z4, (45)

where fi*0 (for i"1, 2, 3) due to physical conditions, and g

1and g

2are used to indicate

the system parameters. Note that g1

and g2

represent the linear parts of the damping,assumed to be non-negative.

We choose a critical point de"ned by

g1c"g

2c"0 (46)

and the parameter values

f1"10, f

2"2, f

3"5, f

4"!6, f

5"8, f

6"!14, f

7"30, (47)

at which the Jacobian of equation (45) has non-semisimple double imaginary eigenvalues:

j1"I, j

2"!I, j

3"I, j

4"!I. (48)

This 1 : 1 resonant case has been studied in reference [14], where the normal form wasobtained using PoincareH normal form theory. In the following, we list the two normal formsfor a comparison. The Maple program developed in reference [9] based on PoincareHnormal form theory can be applied to "nd the following normal form for system (45) up tothird order, given in polar co-ordinates:

rR1"r

2cos (t), r

1hQ1"r

1#r

2sin (t),

rR2"21 r2

1r2!5r3

1cos (t)#

21

2r31sin (t)!58 r

1r22cos (t)#63 r

1r22sin (t)

!21 r21r2cos (2t)#r2

1r2sin (2t),

r2hQ2"r

2!22r2

1r2#

21

2r31cos (t)#5 r3

1sin (t)

#63 r1r22cos (t)#58 r

1r22sin (t)#r2

1r2cos (2t)#

21

2r21r2sin (2t), (49)

where the phase di!erence t"h2!h

1. Note that these four di!erential equations are not

independent, and can be reduced to three independent equations by combining the second


and fourth equations to form an equation for tQ . Note that since normal forms are generallynot unique, equation (49) only represents one of the normal forms.

Similarly, we can execute the Maple program developed in this paper based on themethod of multiple scales to obtain the normal form:

rR1"r

2cos (t)!

21

2r31!

105

4r1r22#

83

4r21r2cos (t)!

315

8r21r2sin (t)

#

131

16r32cos (t)#

22347

32r32sin (t)#

21

2r1r22cos (2t)!31 r

1r22sin (2t),

r1hQ1"r

1#r

2sin (t)!9r3

1!12 r

1r22!

63

8r21r2cos (t)#

129

4r21r2sin (t)

!

22347

32r32cos (t)#

131

16r32sin (t)#31 r

1r22cos (2t)#

21

2r1r22sin (2t),

rR2"41 r2

1r2!

1971

4r32!5r3

1cos (t)#

21

2r31sin(t)

#

21

2r1r22cos (t)!

53

4r1r22sin (t)!

21

2r21r2cos (2t)#10 r2

1r2sin (2t),

r2hQ2"r

2!4r2

1r2!

63

4r32#

21

2r31cos (t)#5 r3

1sin (t)

#

189

8r1r22cos (t)#

71

4r1r22sin (t)#10r2

1r2cos (2t)#

21

2r21r2sin (2t). (50)

It is obvious from equations (49) and (50) that the normal form obtained using the multiplescales has more terms than that obtained using PoincareH normal form theory. However, thenormal forms obtained using the two di!erent methods for the non-resonance and allresonant cases (except for the 1 : 1 resonance) have been found to actually have the samenumber of terms up to a "xed order. The di!erence that appeared in the 1 : 1 resonance isdue to the fact that the approach of multiple scales chooses a "xed basis and thus the form isuniquely determined at each order, while in applying PoincareH normal form theory, theform is not uniquely determined and one may select a di!erent basis which may result indi!erent number of terms. However, we can show that normal form (50) is equivalent tonormal form (49). In other words, we can "nd a third order near identity non-lineartransform which transforms equation (50) into equation (49). The proof is given in AppendixA. Further, it can be shown that we may use additional undetermined coe$cients, theso-called relaxing parameter approach which has been applied to compute the simplestnormal forms [18], in the process of calculating the normal form to directly "nd the normalform for the 1:1 resonance given in the form of equation (49). This will be discussed in moredetail in a di!erent paper.

Since the bifurcation analysis results obtained from equation (49) have been veri"ed bya numerical approach performed based on the original system (45) [14], we will not repeatthe numerical simulation here to verify equation (50).

628 P. YU


In order to further demonstrate the e$ciency of the Maple program for computing highorder normal form of higher-dimensional systems without applying center manifold theory,consider the following seven-dimensional system:

xR1"u

1x2#x2

1#x

1x3#2

3x3x4#x

4x5,

xR2"u

1x1#2x2

4#x2

5#1

2x1x2#x

2x4,

xR3"u

2x4#4x2

3#x2

5#x

1x2,

xR4"!u

2x3#2x2

2#x

3x4#x

3x5,

xR5"!x

5#x2

2#1

3x2x4,

xR6"!x

6#x

7#x2

2#5x

1x3,

xR7"!x

6!x

7#2

5x21. (51)

Executing the Maple program on a PC (PENTIUM III-700MMX 256K system) takesabout 27 min to obtain the normal form up to 10th order. The normal form up to sixthorder is given below

rR1"

29

84r31#

29

104r1r22#

12713

78336r21r22cos (t)#

24143

19584r21r22sin (t)

#

23630620163

22368648960r51#

138177606761

681135436800r31r22#

177068030053

49695085440r1r42

!

3174158149507271

10440611639377920r41r22cos(t)#

2404182693451369

1305076454922240r41r22sin (t)

#

79586782468307866859

55648460037884313600r41r21cos (t)!

8438641581728237917

55648460037884313600r21r42sin (t),

r1hQ1"2r

1!

313

1344r31!

263

2184r1r22!

24143

19584r21r22cos(t)#

12713

78336r21r22sin(t)

!

4053892009

58432389120r51#

7248108167809

2043406310400r31r22#

12602271904219

12986982328320r1r42

!

373702778230261

522030581968896r41r22

cos (t)#23211401849611067

10440611639377920r41r22sin (t)

#

8438641581728237917

55648460037884313600r21r42cos (t)#

79586782468307866859

55648460037884313600r21r42sin (t),


rR2"

41

42r21r2#

1258879

1268064r31r2cos (t)#

868757

2536128r31r2sin (t)

!

2000235383

1091563200r41r2#

2427640143403

1260100558080r21r32!

39909823

49230720r52

#

629839075170792271

278996344363376640r51r2

cos (t)!3982767058488934319

1673978066180259840r51r2sin (t)

#

1000025370085244051

176767944831878400r31r32cos (t)!

221984821019707783

282828711731005440r31r32sin (t),

r2hQ2"3r

2!

5

84r21r2!

5

8r32#

868757

2536128r31r2cos (t)!

1258879

1268064r31r2sin (t)

#

19758721

30321200r41r2!

23095337868209

6930553069440r21r32#

8592833

16410240r52

!

3982767058488934319

1673978066180259840r51r2cos (t)!

629839075170792271

278996344363376640r51r2sin (t)

!

29817963307132463

68743089656841600r31r32cos (t)!

2692641827024565277

1583840785693630464r31r32sin (t), (52)

where t"2h2!3h

1.

5. CONCLUSIONS

A perturbation technique and Maple computer programs have been developed forcomputing the explicit normal forms of resonant double Hopf bifurcations. For a givenarbitrary n-dimensional system, this approach does not require the application of centermanifold theory, which has been included in the perturbation technique so that the normalform on the center manifold and the associated non-linear transformations can be obtainedsimultaneously. The formulas are given in an explicit iterative procedure, and thus areimplemented using computer algebra. Maple programs have been developed forautomating the computations. Examples are presented to verify the method and to showthat the method is computationally e$cient, which is particularly useful forhigh-dimensional systems and higher order normal forms.

ACKNOWLEDGMENT

This work was supported by the Natural Sciences and Engineering Research Council of Canada.

REFERENCES

1. J. GUCKENHEIMER and P. HOLMES 1983 Nonlinear Oscillations, Dynamical Systems, andBifurcations of <ector Fields. New York: Springer-Verlag.

2. A. H. NAYFEH 1993 Methods of Normal Forms. New York: Springer-Verlag.

630 P. YU

3. S. N. CHOW, C. LI and D. WANG 1994 Normal Forms and Bifurcation of Planar <ector Fields.Cambridge: Cambridge University Press.

4. J. CARR 1981 Applications of Center Manifold ¹heory. New York: Springer-Verlag.5. R. H. RAND and W. L. KEITH 1986 in Applications of Computer Algebra (R. Pavelle, editor)

309}328. Boston: Kluwer Academic Press. Normal forms and center manifold calculations onMACSYMA.

6. S. N. CHOW, B. DRACHMAN and D. WANG 1991 Journal of Computation and Applied Mathematics29, 129}143. Computation of normal forms.

7. P. YU 1988 Journal of Sound and <ibration 211, 19}38. Computation of normal forms viaa perturbation technique.

8. Q. BI and P. YU 1998 International Journal of Bifurcation and Chaos 8, 2279}2319. Computationof normal forms of di!erential equations associated with non-semi-simple zero eigenvalues.

9. Q. BI and P. YU 1999 Mathematical and Computer Modelling 29, 49}70. Symbolic softwaredevelopment for computing the normal forms of double Hopf bifurcation.

10. P. YU, W. ZHANG and Q. BI 2001 International Journal of Non-¸inear Mechanics 36, 597}627.Vibration analysis on a thin plate with the aid of computation of normal forms.

11. W. ZHANG and K. HUSEYIN 2000 Applied Mathematical Modelling 24, 279}295. On the relationbetween the methods of averaging and normal forms.

12. S. ZHU and P. YU 1999 Proceedings of the 17th CANCAM, Hamilton, Canada, 69}70, May30}June 3. Computing normal forms for vibration analysis of nonlinear oscillators via multipletime scales.

13. P. YU Nonlinear Dynamics (in press). Analysis on double Hopf bifurcation using computer algebrawith the aid of multiple scales.

14. P. YU and Q. BI 1998 Journal of Sound and <ibration 217, 691}736. Analysis of non-lineardynamics and bifurcations of a double pendulum.

15. K. HUSEYIN 1986 Multiple-Parameter Stability ¹heory and its Applications. Oxford: OxfordUniversity Press.

16. P. YU and K. HUSEYIN 1993 Quarterly Applied Mathematics 51, 91}100. On phase-locked motionsassociated with strong resonance.

17. V. MANDADI and K. HUSEYIN 1980 International Journal of Non-linear Mechanics 15, 159}172.18. P. YU 2000 International Conference on Di+erential Equations (Berlin 1999), Vol. 2, 832}837.

Singapore: World Scienti"c. A method for computing center manifold and normal forms.

APPENDIX A: THE PROOF FOR EXAMPLE 2

To prove that normal form (50) obtained using the multiple scales is equivalent to normalform (49) obtained using PoincareH normal form theory, we "rst transform the two normalforms, which are given in polar co-ordinates, into Cartesian co-ordinates. To achieve this,applying the transformation

x1"r

1cos (h

1), x

2"r

1sin (h

1), x

3"r

2cos (h

2), x

4"r

2sin (h

2), (A.1)

to equation (50) yields

xR1"x

2#x

3!

21

2x31!9x3

2#

131

16x33!

22347

32x34!9x2

1x2#

83

4x21x3#

315

8x21x4

!

21

2x1x22#

129

4x22x3!

63

8x22x4!

63

4x1x23!43x

2x23!

22347

32x23x4!

147

4x1x24

#19x2x24#

131

16x3x24!

189

4x1x2x3!

23

2x1x2x4#62x

1x3x4#21x

2x3x4,


xR2"!x

1#x

4#9x3

1!

21

2x32#

22347

32x33#

131

16x34!

21

2x21x2#

63

8x21x3#

129

4x21x4

#9x1x22!

315

8x22x3#

83

4x22x4!19 x

1x23!

147

4x2x23#

131

16x23x4#43x

1x24

!

63

4x2x24#

22347

32x3x24!

23

2x1x2x3#

189

4x1x2x4#21x

1x3x4!62x

2x3x4,

xR3"x

4!5x3

1#

21

2x32!

1971

4x33!

63

4x34#

21

2x21x2#

63

2x21x3!14x2

1x4

!5x1x22#

105

2x22x3#6x2

2x4!

53

4x1x23#

735

8x2x23!

22347

32x23x4!

71

4x1x24

#

189

8x2x24!

1971

4x3x24#20x

1x2x3!21x

1x2x4!

273

4x1x3x4#

9

2x2x3x4,

xR4"!x

3!

21

2x31!5x3

2#

63

4x33!

1971

4x34!5x2

1x2!6x2

1x3#

105

2x21x4

!

21

2x1x22#14x2

2x3#

63

2x22x4!

189

8x1x23!

71

4x2x23!

1971

4x23x4!

735

8x1x24

!

53

4x4x24#

63

4x3x24!21x

1x2x3!20x

1x2x4#

9

2x1x3x4#

273

4x2x3x4. (A.2)

A similar transformation can be used for equation (49) to obtain

y51"y

2#y

3, yR

2"!y

1#y

4,

yR3"y

4!5y3

1#

21

2y32#

21

2y21y2!23y2

1y4!5y

1y22#42y2

2y3!21y2

2y4!58y

1y23

#63y2y23!58y

1y24#63y

2y24#2y

1y2y3!42y

1y2y4,

y54"!y

3!

21

2y31!5y3

2!5y2

1y2#21y2

1y3#42y2

1y4!

21

2y1y22#23y2

2y3

!63y1y23!58y

2y23!63y

1y24!58y

2y24!42y

1y2y3!2y

1y2y4. (A.3)

Then, with the aid of Maple, we can "nd the following near identity non-lineartransformation, given by

x1"y

1#

115

8y31#

147

16y32#

147

16y21y2!

1017

4y21y3!

483

8y21y4#

115

8y1y22#

1101

4y22y3!

13

8y22y4

#

131

16y1y23#

22347

32y2y23#

131

16y1y24#

22347

32y2y24#

235

4y1y2y3!

1059

2y1y2y4,

632 P. YU

x2"y

2!

147

16y31#

115

8y32#

115

8y21y2!

13

8y21y3!

1101

4y21y4!

147

16y1y22#

483

8y22y3!

1017

4y22y4

!

22347

32y1y23#

131

16y2y23!

22347

32y1y24#

131

16y2y24!

1059

2y1y2y3!

235

4y1y2y4,

x3"y

3#

21

2y31!9y3

2!9y2

1y2#

179

8y21y3#

777

16y21y4#

21

2y1y22!

143

8y22y3#

315

16y22y4

!

1971

4y1y23#

63

4y2y23!

1971

4y1y24#

63

4y2y24!

231

8y1y2y3#

161

4y1y2y4,

x4"y

4#9y3

1#

21

2y32#

21

2y21y2!

315

16y21y3!

143

8y21y4#9y

1y22!

777

16y22y3#

179

8y22y4

!

63

4y1y23!

1971

4y2y23!

63

4y1y24!

1971

4y2y24#

161

4y1y2y3#

231

8y1y2y4, (A.4)

to reduce equations (A.2) to equations (A.3). This proves that equations (A.2) and equations(A.3) (i.e., equations (49) and equations (50)) are indeed equivalent.

symbolic computation of normal forms for resonant double hopf bifurcations using a perturbation...

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