On a family of maps with multiple chaotic attractors

Abstract Multistability is characterized by the occurrence of multiple coexisting attractors. We introduce a family of maps that possess this property and in particular exhibits coexisting chaotic attractors. In this family not only the maps’ parameters can be varied but also their dimension. So, four types of multistable attractors, equilibria, periodic orbits, quasi-periodic orbits and chaotic attractors can be found for a given dimension.

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

[2]  Katsuhiko Ogata,et al.  Discrete-time control systems , 1987 .

[3]  Gaston H. Gonnet,et al.  On the LambertW function , 1996, Adv. Comput. Math..

[4]  Robert M Corless,et al.  Some applications of the Lambert W  function to physics , 2000, Canadian Journal of Physics.

[5]  S. Rajasekar,et al.  Coexisting chaotic attractors, their basin of attractions and synchronization of chaos in two coupled Duffing oscillators , 1999 .

[6]  Celso Grebogi,et al.  Why are chaotic attractors rare in multistable systems? , 2003, Physical review letters.

[7]  Tomasz Kapitaniak,et al.  Noise-induced basin hopping in a vibro-impact system , 2007 .

[8]  Dmitry E. Postnov,et al.  CHAOTIC HIERARCHY IN HIGH DIMENSIONS , 2000 .

[9]  J. Sprott.  Chaos and time-series analysis , 2001 .

[10]  Julien Clinton Sprott,et al.  A comparison of correlation and Lyapunov dimensions , 2005 .

[11]  A. d’Onofrio.  Fractal growth of tumors and other cellular populations: Linking the mechanistic to the phenomenological modeling and vice versa , 2009, 1309.3329.

[12]  H. Kantz,et al.  Nonlinear time series analysis , 1997 .

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

[14]  Zhang Yi,et al.  Dynamical properties of background neural networks with uniform firing rate and background input , 2007 .

[15]  Th. Meyer,et al.  HYPERCHAOS IN THE GENERALIZED ROSSLER SYSTEM , 1997, chao-dyn/9906028.

[16]  Hendrik Richter,et al.  The Generalized HÉnon Maps: Examples for Higher-Dimensional Chaos , 2002, Int. J. Bifurc. Chaos.

[17]  Guanrong Chen,et al.  Coexisting chaotic attractors in a single neuron model with adapting feedback synapse , 2005 .

[18]  Hendrik Richter,et al.  Control of the triple chaotic attractor in a Cournot triopoly model , 2004 .

[19]  Hendrik Richter.  Controlling chaotic systems with multiple strange attractors , 2002 .

[20]  Observation of intermingled basins in coupled oscillators exhibiting synchronized chaos. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[21]  Ulrike Feudel,et al.  Multistability, noise, and attractor hopping: the crucial role of chaotic saddles. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  G. Baier,et al.  Maximum hyperchaos in generalized Hénon maps , 1990 .

[23]  A N Pisarchik,et al.  Controlling the multistability of nonlinear systems with coexisting attractors. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  K. Aihara,et al.  Crisis-induced intermittency in two coupled chaotic maps: towards understanding chaotic itinerancy. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  Hendrik Richter,et al.  Controlling Chaos in Maps with Multiple Strange attractors , 2003, Int. J. Bifurc. Chaos.

[26]  Abraham C.-L. Chian,et al.  Complex economic dynamics: Chaotic saddle, crisis and intermittency , 2006 .

[27]  O. Rössler.  The Chaotic Hierarchy , 1983 .