Elsevier

Physics Letters A

Volume 366, Issue 3, 25 June 2007, Pages 217-222
Physics Letters A

Hyperchaos generated from the Lorenz chaotic system and its control

https://doi.org/10.1016/j.physleta.2007.02.024Get rights and content

Abstract

In this Letter, a hyperchaotic Lorenz system is constructed via state feedback control. Abundant dynamics of the hyperchaotic system is studied using the Lyapunov exponents, Poincaré section and bifurcation diagram. Furthermore, effective linear feedback controllers are designed for stabilizing hyperchaos to unstable equilibrium, periodic orbits and quasi-periodic orbit. Numerical simulations are given to illustrate and verify the results.

Introduction

In recent years, problems of chaos and chaos control, including chaotification of dynamical system, have attracted many physicists and mathematicians. Different methods and techniques have been proposed for chaotification and suppressing chaos [1], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], in which the linear feedback control method is simple but effective to chaotify or stabilize the chaotic system.

Possessing more than one positive Lyapunov exponent, the hyperchaotic systems, including hyperchaotic Rössler system [2] et al., have been studied for many years. In 2005, Y. Li, and G. Chen introduced a hyperchaotic Chen system [3] via state feedback control, in the same year, they drive a unified chaotic system to hyperchaotic using a sinusoidal parameter perturbation control input [4]. Very recently, A. Chen et al. constructed a new hyperchaotic system based on Lü system by using a state feedback [5]. More recent studies by F. Wang and C. Liu [6] realized hyperchaos based on the chaotic dynamical system they introduced in [7], using an additional state input.

A new hyperchaotic Lorenz system is constructed in this Letter, and stabilization of the hyperchaotic Lorenz system is achieved.

This Letter is presented as follows: in the next section, the controlled Lorenz system showing hyperchaotic behavior is constructed via introducing a state feedback. In the third section, properties and dynamics of the controlled system are investigated numerically via bifurcation diagram, Lyapunov exponents and Poincaré section. And in the last section, simple but effective controllers are designed for stabilizing the hyperchaotic system to unstable equilibrium, periodic-orbit and quasi-periodic orbit.

Section snippets

Construction of the hyperchaotic Lorenz system

The celebrated Lorenz system is described by{x˙=a(xy),y˙=xz+rxy,z˙=xybz, where a, r, b are system parameters. When a=10, r=28, b=8/3, it shows chaotic behavior. The strange attractor of the system is illustrated in Fig. 1.

With these parameters, the Lorenz attractor has the following Lyapunov exponents 0.906, 0, −14.572, the Kaplan–Yorke dimension of the attractor is2.062.

In light of the thought of G. Chen [3], we construct a hyperchaotic Lorenz system by introducing a state feedback

Dynamics analysis for the hyperchaotic Lorenz system

For arbitrary parameters a, b, d and r, E0=(0,0,0,0) is an equilibrium of the new system (2). When the parameters a, b, d, r satisfy abd(adr)(r1)>0 and ad1, the system (2) has two other nontrivial equilibriums:E1=(abd(adr)(r1)adr,abd(adr)(r1)ad1,ad(r1)ad1,a(r1)abd(adr)(r1)(adr)(ad1)),E2=(abd(adr)(r1)adr,abd(adr)(r1)ad1,ad(r1)ad1,a(r1)abd(adr)(r1)(adr)(ad1)).

For parameters a=10, r=28, b=83, when d(,0)(2.8,+), the three equilibriums exist, otherwise only E0

Hyperchaos control for the hyperchaotic Lorenz system

Consider a class of continuous time nonlinear chaotic (or hyperchaotic) systems in the form of{X˙=f(X,t),X(0)=X0Rn.

According to the thought of feedback control, our aim is to find a feedback controller v=v(X,k) such that the controlled system X˙=f(X˙,t)+v can be stabilized to unstable equilibriums and period orbits. If the controller satisfies v=0, the controlled system becomes the original chaotic (or hyperchaotic) system.

For simplicity, we always suppose the parameters a=10, r=28, b=8/3 and d

Conclusion

In this Letter, hyperchaos generated from the Lorenz system is studied numerically and analytically. Bifurcation diagram, Lyapunov exponents and Poincaré section are employed in the investigation. Effective linear feedback controllers are designed for stabilizing hyperchaos to unstable equilibrium, periodic and quasi-periodic orbits. Numerical simulations are proposed to verify and illustrate the effectiveness of these controllers.

Acknowledgements

The author would thank Dr. Mei Sun for her valuable comments and helps.

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