Multi-wing hyperchaotic attractors from coupled Lorenz systems

Abstract This paper illustrates an approach to generate multi-wing attractors in coupled Lorenz systems. In particular, novel four-wing (eight-wing) hyperchaotic attractors are generated by coupling two (three) identical Lorenz systems. The paper shows that the equilibria of the proposed systems have certain symmetries with respect to specific coordinate planes and the eigenvalues of the associated Jacobian matrices exhibit the property of similarity. In analogy with the original Lorenz system, where the two-wings of the butterfly attractor are located around the two equilibria with the unstable pair of complex-conjugate eigenvalues, this paper shows that the four-wings (eight-wings) of these attractors are located around the four (eight) equilibria with two (three) pairs of unstable complex-conjugate eigenvalues.

[1]  Guanrong Chen,et al.  Chaotic behaviors and toroidal/spherical attractors generated by discontinuous dynamics , 2006 .

[2]  Guanrong Chen,et al.  Generation and Control of Spherical and Circular attractors Using Switching Schemes , 2007, Int. J. Bifurc. Chaos.

[3]  Wajdi M. Ahmad.  A simple multi-scroll hyperchaotic system , 2006 .

[4]  Guanrong Chen,et al.  Generation of n-scroll attractors via sine function , 2001 .

[5]  Giuseppe Grassi,et al.  Hyperchaotic Coupled Chua Circuits: an Approach for Generating New n×m-scroll attractors , 2003, Int. J. Bifurc. Chaos.

[6]  M. Brucoli,et al.  a Method for the Synchronization of Hyperchaotic Circuits , 1996 .

[7]  J. Thompson,et al.  Nonlinear Dynamics and Chaos: Geometrical Methods for Engineers and Scientists , 1986 .

[8]  GUANRONG CHEN,et al.  Four-Wing attractors: from Pseudo to Real , 2006, Int. J. Bifurc. Chaos.

[9]  B. Mandelbrot.  Fractal Geometry of Nature , 1984 .

[10]  G. C. Joshi,et al.  Octonionic Julia sets , 1992 .

[11]  Guanrong Chen,et al.  Analysis and circuit implementation of a new 4D chaotic system , 2006 .

[12]  Daizhan Cheng,et al.  A New Chaotic System and Beyond: the Generalized Lorenz-like System , 2004, Int. J. Bifurc. Chaos.

[13]  Johan A. K. Suykens,et al.  Families of scroll Grid attractors , 2002, Int. J. Bifurc. Chaos.

[14]  Giuseppe Grassi,et al.  Experimental realization of observer-based hyperchaos synchronization , 2001 .

[15]  Guanrong Chen,et al.  Generating chaotic attractors with multiple merged basins of attraction: a switching piecewise-linear control approach , 2003 .

[16]  Mustak E. Yalcin,et al.  Multi-scroll and hypercube attractors from a general jerk circuit using Josephson junctions , 2007 .

[17]  M. Naschie.  Dimensional symmetry breaking and gravity in cantorian space , 1997 .

[18]  Giuseppe Grassi,et al.  New 3D-scroll attractors in hyperchaotic Chua's Circuits Forming a Ring , 2003, Int. J. Bifurc. Chaos.

[19]  Associators in generalized octonionic maps , 1993 .

[20]  Ahmed S. Elwakil,et al.  1-d digitally-Controlled multiscroll Chaos Generator , 2007, Int. J. Bifurc. Chaos.

[21]  Wajdi M. Ahmad.  Generation and control of multi-scroll chaotic attractors in fractional order systems , 2005 .

[22]  G. C. Joshi,et al.  Transition points in octonionic Julia sets , 1993 .

[23]  Ahmed S. Elwakil,et al.  Creation of a complex butterfly attractor using a novel Lorenz-Type system , 2002 .

[24]  Guanrong Chen,et al.  On a four-dimensional chaotic system , 2005 .

[25]  G. C. Joshi,et al.  Bifurcation phenomena of the non-associative octonionic quadratic , 1995 .

[26]  Xinghuo Yu,et al.  Design and analysis of multiscroll chaotic attractors from saturated function series , 2004, IEEE Transactions on Circuits and Systems I: Regular Papers.

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

[28]  M. Naschie.  Fractal gravity and symmetry breaking in a hierachical Cantorian space , 1997 .