A Novel hyperchaotic System and its Complex Dynamics

This paper is concerned with a novel four-dimensional continuous autonomous hyperchaotic system, which is obtained by adding a simple dynamical state-feedback controller to a Lorenz-like three-dimensional autonomous chaotic system. This new system contains three parameters and each equation of the system has one quadratic cross-product term. Some basic properties of the system are studied first. Its complex dynamic behaviors are then analyzed by means of Lyapunov exponent (LE) spectrum, bifurcation diagrams, phase portraits and Poincare sections. It is shown that the system has several large hyperchaotic regions. When the system is evolving in a hyperchaotic region, the two positive LEs are both large, which can be larger than 1 if the system parameters are taken appropriately. The pitchfork bifurcation of the system is finally analyzed by using the center manifold theorem.

[1]  Bing-Hong Wang,et al.  HYPERCHAOTIC SYNCHRONIZATION IN DETERMINISTIC SMALL-WORLD DYNAMICAL NETWORKS , 2004 .

[2]  Guanrong Chen,et al.  On a Generalized Lorenz Canonical Form of Chaotic Systems , 2002, Int. J. Bifurc. Chaos.

[3]  S. Mascolo,et al.  A system theory approach for designing cryptosystems based on hyperchaos , 1999 .

[4]  Xiao-Song Yang,et al.  Hyperchaos and bifurcation in a new class of four-dimensional Hopfield neural networks , 2006, Neurocomputing.

[5]  Svetoslav Nikolov,et al.  Hyperchaos–chaos–hyperchaos transition in modified Rössler systems , 2006 .

[6]  Stephen Wiggins.  Methods for Simplifying Dynamical Systems , 1990 .

[7]  Zhenya Yan,et al.  Controlling hyperchaos in the new hyperchaotic Chen system , 2005, Appl. Math. Comput..

[8]  O. Rössler.  An equation for hyperchaos , 1979 .

[9]  Guanrong Chen,et al.  Hyperchaos evolved from the generalized Lorenz equation , 2005, Int. J. Circuit Theory Appl..

[10]  L. Chua,et al.  Hyper chaos: Laboratory experiment and numerical confirmation , 1986 .

[11]  Guanrong Chen,et al.  A HYPERCHAOS GENERATED FROM CHEN'S SYSTEM , 2006 .

[12]  An-Pei Wang,et al.  Controlling hyperchaos of the Rossler system , 1999 .

[13]  L. Chua,et al.  Canonical realization of Chua's circuit family , 1990 .

[14]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[15]  Ming-Jyi Jang,et al.  Sliding mode control of hyperchaos in Rössler systems , 2002 .

[16]  Guangyi Wang,et al.  A new modified hyperchaotic Lü system , 2006 .

[17]  Guanrong Chen,et al.  Analysis of a new chaotic system , 2005 .

[18]  Simin Yu,et al.  Generating hyperchaotic Lü attractor via state feedback control , 2006 .

[19]  Guanrong Chen,et al.  YET ANOTHER CHAOTIC ATTRACTOR , 1999 .

[20]  R. Toral,et al.  Analysis and characterization of the hyperchaos generated by a semiconductor laser subject to a delayed feedback loop , 2005, IEEE Journal of Quantum Electronics.

[21]  Jinhu Lu,et al.  A New Chaotic Attractor Coined , 2002, Int. J. Bifurc. Chaos.

[22]  W. T. Rhodes,et al.  Communicating with hyperchaos: The dynamics of a DNLF emitter and recovery of transmitted information , 2003 .

[23]  Daizhan Cheng,et al.  Bridge the Gap between the Lorenz System and the Chen System , 2002, Int. J. Bifurc. Chaos.

[24]  Svetoslav Nikolov,et al.  Occurrence of regular, chaotic and hyperchaotic behavior in a family of modified Rossler hyperchaotic systems , 2004 .

[25]  K. Thamilmaran,et al.  Hyperchaos in a Modified Canonical Chua's Circuit , 2004, Int. J. Bifurc. Chaos.