Dynamical analysis of the generalized Sprott C system with only two stable equilibria

A generalized Sprott C system with only two stable equilibria is investigated by detailed theoretical analysis as well as dynamic simulation, including some basic dynamical properties, Lyapunov exponent spectra, fractal dimension, bifurcations, and routes to chaos. In the parameter space where the equilibria of the system are both asymptotically stable, chaotic attractors coexist with period attractors and stable equilibria. Moreover, the existence of singularly degenerate heteroclinic cycles for a suitable choice of the parameters is investigated. Periodic solutions and chaotic attractors can be found when these cycles disappear.

[1]  Michael Peter Kennedy,et al.  Three steps to chaos. II. A Chua's circuit primer , 1993 .

[2]  Hiroshi Kokubu,et al.  Existence of a Singularly Degenerate Heteroclinic Cycle in the Lorenz System and Its Dynamical Consequences: Part I , 2004 .

[3]  Qigui Yang,et al.  Complex dynamics in the stretch-twist-fold flow , 2010 .

[4]  W. Steeb,et al.  The Rikitake Two-Disk Dynamo System and Domains with Periodic Orbits , 1999 .

[5]  Luis Fernando Mello,et al.  Degenerate Hopf bifurcations in the Lü system , 2009 .

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

[7]  Qigui Yang,et al.  Controlling the diffusionless Lorenz equations with periodic parametric perturbation , 2009, Comput. Math. Appl..

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

[9]  L. P. Šil'nikov,et al.  A CONTRIBUTION TO THE PROBLEM OF THE STRUCTURE OF AN EXTENDED NEIGHBORHOOD OF A ROUGH EQUILIBRIUM STATE OF SADDLE-FOCUS TYPE , 1970 .

[10]  Diyi Chen,et al.  Synchronization and circuit simulation of a new double-wing chaos , 2012 .

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

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

[13]  Albert C. J. Luo,et al.  The dynamics of a bouncing ball with a sinusoidally vibrating table revisited , 1996 .

[14]  Qigui Yang,et al.  Chaotic attractors of the Conjugate Lorenz-Type System , 2007, Int. J. Bifurc. Chaos.

[15]  Jianming Zhang,et al.  New Treatment on Bifurcations of Periodic Solutions and Homoclinic Orbits at High r in the Lorenz Equations , 1993, SIAM J. Appl. Math..

[16]  Jaume Llibre,et al.  Global dynamics of the Rikitake system , 2009 .

[17]  Guanrong Chen,et al.  Complex Dynamical Behaviors of the Chaotic Chen's System , 2003, Int. J. Bifurc. Chaos.

[18]  Julien Clinton Sprott,et al.  A new class of chaotic circuit , 2000 .

[19]  Zhen Wang,et al.  Existence of attractor and control of a 3D differential system , 2010 .

[20]  O. Rössler.  An equation for continuous chaos , 1976 .

[21]  Julien Clinton Sprott,et al.  Simplest dissipative chaotic flow , 1997 .

[22]  Luis Fernando Mello,et al.  Nonlinear analysis in a Lorenz-like system , 2010 .

[23]  Jaume Llibre,et al.  Invariant algebraic surfaces of the Rikitake system , 2000 .

[24]  Yongjian Liu,et al.  A new hyperchaotic system from the Lü system and its control , 2011, J. Comput. Appl. Math..

[25]  J. Sprott,et al.  Some simple chaotic flows. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[26]  Marcelo Messias,et al.  Dynamics at infinity and the existence of singularly degenerate heteroclinic cycles in the Lorenz system , 2009 .

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

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

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

[30]  Marcelo Messias,et al.  Bifurcation analysis of a new Lorenz-like chaotic system , 2008 .