Hopf bifurcation and heteroclinic orbit in a 3D autonomous chaotic system

In this paper we revisit a 3D autonomous chaotic system, which contains both the modified Lorenz system and the conjugate Chen system, presented in [Huang and Yang, Chaos Solitons Fractals 39:567–578, 2009]. First by citing two examples to show the errors and limitations for the local stability of the equilibrium point S+ obtained in this literature, we formulate a complete determining criterion for the local stability of S+ of this system. Although the local bifurcation problem of this system, mainly for Hopf bifurcation, etc., has been studied, the invoking of incorrect proposition leads to an incorrect result for Hopf bifurcation. We then renew the study of the Hopf bifurcation of this system by utilizing the Project Method. The global bifurcation problem, relatively speaking, should be more difficult than the local bifurcation problem for a given system. However, the global bifurcation problem of this system, to the best of our knowledge, has not been investigated yet in the literatures. So next we consider the global bifurcation problem for this system, mainly for the existence of homoclinic and heteroclinic orbits. Our results, one of which shows the existence of two heteroclinic orbits, not only correct and further supplement the ones obtained in the literature, but also give something new to theoretically help fully understand the occurrence of chaos.

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

[2]  C. P. Silva,et al.  Shil'nikov's theorem-a tutorial , 1993 .

[3]  Haijun Wang,et al.  Homoclinic and heteroclinic orbits and bifurcations of a New Lorenz-Type System , 2011, Int. J. Bifurc. Chaos.

[4]  Leo R. M. Maas,et al.  The diffusionless Lorenz equations; Shil'nikov bifurcations and reduction to an explicit map , 2000 .

[5]  Xianyi Li,et al.  Dynamical properties and simulation of a new Lorenz-like chaotic system , 2011 .

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

[7]  A. Mees,et al.  Homoclinic and heteroclinic orbits in the double scroll attractor , 1987 .

[8]  Y. Kuznetsov.  Elements of applied bifurcation theory (2nd ed.) , 1998 .

[9]  Qigui Yang,et al.  A Chaotic System with One saddle and Two Stable Node-Foci , 2008, Int. J. Bifurc. Chaos.

[10]  Guanrong Chen,et al.  On the generalized Lorenz canonical form , 2005 .

[11]  Qigui Yang,et al.  Dynamics of a new Lorenz-like chaotic system , 2010 .

[12]  Kuifei Huang,et al.  Stability and Hopf bifurcation analysis of a new system , 2009 .

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

[14]  Leon O. Chua,et al.  Chaos Synchronization in Chua's Circuit , 1993, J. Circuits Syst. Comput..

[15]  Gheorghe Tigan,et al.  Heteroclinic orbits in the T and the Lü systems , 2009 .

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

[17]  Y. Kuznetsov.  Elements of Applied Bifurcation Theory , 2023, Applied Mathematical Sciences.

[18]  Guanrong Chen,et al.  On homoclinic and heteroclinic orbits of Chen's System , 2006, Int. J. Bifurc. Chaos.

[19]  Guanrong Chen,et al.  A Unified Lorenz-Type System and its Canonical Form , 2006, Int. J. Bifurc. Chaos.