hyperchaos in a new four-dimensional autonomous system

January 13, 2011 9:24 WSPC/S0218-1274 S0218127410027684

International Journal of Bifurcation and Chaos, Vol. 20, No. 10 (2010) 3295–3301c© World Scientific Publishing CompanyDOI: 10.1142/S0218127410027684

HYPERCHAOS IN A NEW FOUR-DIMENSIONALAUTONOMOUS SYSTEM

MARCOS J. CORREIA∗ and PAULO C. RECH†Departamento de Fısica,

Universidade do Estado de Santa Catarina,89223-100 Joinville, Brazil∗[email protected]†[email protected]

Received October 21, 2009; Revised February 22, 2010

In this letter we report a new four-dimensional autonomous system, constructed from a chaoticLorenz system by introducing an adequate feedback controller to the third equation. We showthat when parameters are conveniently chosen, the control method can drive the chaotic Lorenzsystem to hyperchaotic regions. Analytical and numerical procedures are conducted to studythe dynamical behaviors of the proposed new system.

Keywords : Phase-space; Lyapunov exponents; hyperchaos; Lorenz system.

1. Introduction

The first hyperchaotic system was reported byRossler in 1979 [Rossler, 1979], and it is a chaoticsystem that has more than one positive Lyapunovexponent. It means that the dynamics of a hyper-chaotic system is expanded in two or more direc-tions simultaneously as the time increases, resultingin a more complex chaotic attractor when the com-parison is made with a chaotic system with only onepositive Lyapunov exponent. This property makeshyperchaotic systems have better performance thanchaotic systems in many chaos-based fields, includ-ing several technological applications. For instance,hyperchaotic systems, due to its higher unpre-dictability, can be used to improve the security inchaotic communication systems, where a chaoticsignal is used to mask the message to be transmit-ted, once messages masked by chaotic systems arenot always secure [Perez & Cerdeira, 1995].

As we know, these does not exist a standardprocedure to theoretically construct a hyperchaoticsystem. An usual and efficient method considersat least a fourth dimension in a three-dimensionalchaotic system, by the introduction of feedback

controllers. It was recently created from a Lorenzsystem [Barboza, 2007; Cai et al., 2008], from Chensystem [Gao et al., 2006; Gao et al., 2009; Li et al.,2005], from Lu system [Bao & Liu, 2008; Chen etal., 2006; Wang et al., 2006], and from a Chua sys-tem with a cubic nonlinearity [Rech & Albuquerque,2009]. In this letter we report a new hyperchaoticsystem, constructed from a Lorenz system by intro-ducing an adequate feedback controller to the thirdequation, as explained in Sec. 2.

The letter is organized as follows. In Sec. 2, wepresent the new four-dimensional system, obtainedfrom a three-dimensional Lorenz system by theinclusion of a fourth variable. In Sec. 3, we dis-cuss some basic properties of this new system, andshow that the trivial fixed point can be unstable. InSec. 4, numerical results involving Lyapunov expo-nents spectrum and phase-space portraits are pre-sented. Finally, the paper is summarized in Sec. 5.

2. The New System

The paradigmatic Lorenz system [Lorenz, 1963],which is a model for atmospheric convection, is

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January 13, 2011 9:24 WSPC/S0218-1274 S0218127410027684

3296 M. J. Correia & P. C. Rech

described by

x = −σ(x − y),

y = −xz + rx − y,

z = xy − bz,

(1)

where x, y, z represent dynamical variables, andσ > 0, r > 0, b > 0 are parameters. By introduc-ing an adequate feedback controller w to the thirdequation in Lorenz system (1), we obtain a four-dimensional system described by

x = −σ(x − y),

y = −xz + rx − y,

z = xy − bz + w,

w = −a(x + y),

(2)

where a is another parameter. As we will see inSec. 4, when the parameters are properly chosen,system (2) can exhibit hyperchaos.

3. Some Analytical Results

The divergence [Ott, 2000] of the new system (2) isgiven by

∂x

∂x+

∂y

∂y+

∂z

∂z+

∂w

∂w= −(1 + b + σ),

and from this negative result we conclude thatsystem (2) is always dissipative signifying, there-fore, that the phase-space contracts volumes atan exponential rate equal to e−(1+b+σ)t, as timet increases. In consequence, all the system trajec-tories finally converge to an attractor in a four-dimensional phase-space.

Next, we discuss the stability of the fixed pointsof the nonlinear new system (2). The fixed pointsare obtained by taking x = 0, y = 0, z = 0, andw = 0 in Eqs. (2), that is, by solving the set ofalgebraic equations

σ(y − x) = 0,

rx − y − xz = 0,

xy − bz + w = 0,

a(x + y) = 0,

for x, y, z, and w. We can easily obtain that theorigin P0 ≡ (0, 0, 0, 0) is the only fixed point. TheJacobian matrix for the system (2) at P0, denoted

by J0, is given by

J0 =

−σ σ 0 0r −1 0 00 0 −b 1

−a −a 0 0

,

and the corresponding characteristic equation, cal-culated using det(J0 − mI) = 0, where I is the4 × 4 identity matrix and m represents the eigen-values, can be written as

m(m + b)(m2 + (1 + σ)m + (1 − r)σ) = 0. (3)

The roots of polynomial (3), that is, the eigenvaluesassociated with the origin P0 are

m1 = 0, m2 = −b,

m± =−σ − 1 ± √

(σ − 1)2 + 4σr

2.

As σ and r are always positive, m+ is a positivenumber for r > 1. Hence, the trivial fixed point P0

is unstable for r > 1, and we conclude that hyper-chaos, chaos, or limit cycles can be displayed bysystem (2), depending on parameters σ > 0, b > 0,r > 1, and a.

4. Some Numerical Results

In our numerical investigations, parameters σ, b,and r are fixed in σ = 10, b = 8/3, and r = 200, andthe dynamics of the new system (2) is examined asparameter a is varied. The spectrum of Lyapunovexponents λ1 > λ2 > λ3 > λ4, for system (2) alwaysintegrated with a fourth-order Runge–Kutta algo-rithm with a fixed step size equal to 10−3, and con-sidering 5×105 steps to compute each one exponent,is shown in Fig. 1, as a function of the parametera, from a = −38 to a = 62. It can be observed inFig. 1 that the parameter a covers two regions, froma ≈ −38 to a ≈ −28, and from a ≈ 28 to a ≈ 54,for which the system (2) has two positive Lyapunovexponents, λ1 and λ2, and, as a consequence, themotion in these intervals is hyperchaotic. For theinterval −28 � a � 26 there is only one positiveLyapunov exponent, λ1, and the system is chaotic.In the other two regions, for 26 � a � 28 and54 � a � 60, the greater Lyapunov exponent, λ1, isequal to zero, and the motion is periodic. Therefore,system (2) evolves on a hyperchaotic attractor inthe interval a ∈ (−38,−28) ∪ (28, 54), on a chaoticattractor in the interval a ∈ (−28, 26), and on a

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Hyperchaos in a New Four-Dimensional Autonomous System 3297

-20 0 20 40 60a

-3

-2

-1

0

1

2

λ

λ2

λ1

λ3

Fig. 1. The Lyapunov exponents spectrum of system (2), asa function of a with −38 < a < 62, σ = 10, b = 8/3, r = 200.The smaller exponent λ4 does not appear because it is alwaysless than −8.

periodic attractor in the interval a ∈ (26, 28) ∪(54, 60). Hereafter, we concentrate our attention ontwo regions as defined by the interval a ∈ (−38,−28)∪(28, 54), where the dynamics is hyperchaotic.

Typical three-dimensional projections of thefour-dimensional hyperchaotic attractor, for a =−33.1951952, therefore in the interval −38 �a � −28 are shown in Fig. 2. The correspond-ing Lyapunov exponents are λ1 = 0.5230, λ2 =0.1583, λ3 = −0.0150, λ4 = −20.3831, and theLyapunov dimension is DL = 3.0326. This fractalnature of the attractor means that the system (2)has a nonperiodic orbit. In Fig. 2(a) we can seethat the projection of this hyperchaotic trajectoryin the three-dimensional xyz phase-space resemblesa chaotic Lorenz attractor, which is typical of theLorenz system (1). Note that this Lorenz-like struc-ture does not appear in other projections, as shownin Figs. 2(b)–2(d).

In Fig. 3, one sees all the two-dimensional pro-jections of the same hyperchaotic attractor shownin Fig. 2. Our motivation to include these por-traits here, is that they give us another opportunityto observe the structure of the orbit more clearly.For instance, note in Fig. 3(a) that the trajectoryreally is not periodic, the assertion based on the factof the trajectory is not completed. The Lorenz-likestructure of the plane projections is clearly visible

-80 -40 0 40 80x -300

-100

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300

y

0100200300400

z

-80 -40 0 40 80x -300

-100

100

300

y

-2000

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-800

w

(a) (b)

-80 -40 0 40 80x 0100

200300

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-1200

-800

w

-300-100

100300y 0

100200

300400

z

-2000

-1600

-1200

-800

w

(c) (d)

Fig. 2. Three-dimensional projections of the hyperchaotic attractor generated by system (2), for a = −33.1951952 and fromthe initial condition (x, y, z, w) = (10, 10, 10, 10). All diagrams were constructed with 2 × 104 points. (a) xyz phase-space. (b)xyw phase-space. (c) xzw phase-space. (d) yzw phase-space.

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-50 0 50x

-300

-200

-100

0

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y

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0

100

200

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400

500

z

(a) (b)

-50 0 50x

-2000

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-1400

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-1000

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w

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0

100

200

300

400

500

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(c) (d)

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w

100 2000 300 400 500z

-2000

-1800

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-1400

-1200

-1000

-800

-600

w

(e) (f)

Fig. 3. Two-dimensional projections of the hyperchaotic attractor generated by system (2), for a = −33.1951952 and fromthe initial condition (x, y, z, w) = (10, 10, 10, 10). All diagrams were constructed with 2 × 104 points. (a) xy phase-plane. (b)xz phase-plane. (c) xw phase-plane. (d) yz phase-plane. (e) yw phase-plane. (f) zw phase-plane.

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-80 -40 0 40 80x -300-100

100 300

y

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z

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(a) (b)

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w

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100 300y -100

100

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500

z

-2200

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-1400

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w

(c) (d)

Fig. 4. Three-dimensional projections of the chaotic attractor generated by system (2), for a = 37.1751752 and from theinitial condition (x, y, z, w) = (10, 10, 10, 10). All diagrams were constructed with 2× 104 points. (a) xyz phase-space. (b) xywphase-space. (c) xzw phase-space. (d) yzw phase-space.

in Fig. 3(a) (xy plane), Fig. 3(b) (xz plane), andFig. 3(d) (yz plane).

Now we consider a = 37.1751752, therefore inthe interval 28 � a � 54 where the motion is again

hyperchaotic. For this case, the Lyapunov expo-nents are λ1 = 0.6312, λ2 = 0.1480, λ3 = 0.0001,λ4 = −20.4962, and the Lyapunov dimension isDL = 3.0380. Three-dimensional projections of

-50 0 50 100x

-300

-200

-100

0

100

200

300

y

-50 0 50 100x

0

100

200

300

400

500

z

(a) (b)

Fig. 5. Two-dimensional projections of the hyperchaotic attractor generated by system (2), for a = 37.1751752 and from theinitial condition (x, y, z, w) = (10, 10, 10, 10). All diagrams were constructed with 2 × 104 points. (a) xy phase-plane. (b) xzphase-plane. (c) xw phase-plane. (d) yz phase-plane. (e) yw phase-plane. (f) zw phase-plane.

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0

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(c) (d)

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w

0 100 200 300 400z

-2000

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w

(e) (f)

Fig. 5. (Continued)

the corresponding four-dimensional chaotic attrac-tor are shown in Fig. 4. Again we observe, nowlooking for Fig. 4(a), that there is only one three-dimensional projection of the hyperchaotic orbitthat resembles the Lorenz attractor, namely thatrepresented by the xyz phase-space.

Two-dimensional projections of the samehyperchaotic attractor whose three-dimensionalprojections are shown in Fig. 4, appear in Fig. 5.Our motivation to include these portraits here isthe same as before, that is, they give us the oppor-tunity to observe the structure of the hyperchaoticorbit more clearly. Again the Lorenz-like structureof the plane projections is clearly visible in Fig. 5(a)(xy plane), Fig. 5(b) (xz plane), and Fig. 5(d) (yzplane).

5. Summary

In summary, in this letter we have reported a newfour-dimensional autonomous system, constructedfrom a particular chaotic Lorenz system with σ =10, b = 8/3, and r = 200, by using a statefeedback controller. We have shown that whena fourth parameter a is changed in an adequaterange, the control method can drive this chaoticsystem to hyperchaotic regions. Analytical inves-tigations concerning dissipation, fixed points, andstability of the trivial fixed point have been realized.Besides, numerical procedures involving Lyapunovexponents spectrum and phase-space portraits havebeen utilized to characterize this hyperchaoticsystem.

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Acknowledgments

The authors thank Conselho Nacional de Desen-volvimento Cientıfico e Tecnologico-CNPq andCoordenacao de Aperfeicoamento de Pessoalde Nıvel Superior-CAPES, Brazil, for financialsupport.

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hyperchaos in a new four-dimensional autonomous system

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