Multistability, Phase Diagrams and Statistical Properties of the Kicked Rotor: a Map with Many Coexisting attractors

We investigate the prevalence of multistability in the parameter space of the kicked rotor map. We report high-resolution phase diagrams showing how the density of attractors and the density of periods vary as a function of both model parameters. Our diagrams illustrate density variations that exist when moving between the familiar conservative and strongly dissipative limits of the map. We find the kicked rotor to contain multistability regions with more than 400 coexisting attractors. This fact makes the rotor a promising high-complexity local unit to investigate synchronization in networks of chaotic maps, in both regular and complex topologies.

[1]  J. Gallas,et al.  Conjugacy classes and chiral doublets in the Hénon Hamiltonian repeller , 2006 .

[2]  M. Newman,et al.  Scaling and percolation in the small-world network model. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[3]  R. Toral,et al.  Dynamical mechanism of anticipating synchronization in excitable systems. , 2004, Physical review letters.

[4]  C Masoller,et al.  Random delays and the synchronization of chaotic maps. , 2005, Physical review letters.

[5]  J. Gallas,et al.  Structure of the parameter space of the Hénon map. , 1993, Physical review letters.

[6]  L S Tsimring,et al.  Dynamics of an ensemble of noisy bistable elements with global time delayed coupling. , 2003, Physical review letters.

[7]  Cristina Masoller,et al.  Chaotic maps coupled with random delays: Connectivity, topology, and network propensity for synchronization , 2005, nlin/0512075.

[8]  Mingzhou Ding,et al.  Will a large complex system with time delays be stable? , 2004, Physical review letters.

[9]  A. Lichtenberg,et al.  Regular and Chaotic Dynamics , 1992 .

[10]  Jason A. C. Gallas,et al.  Dissecting shrimps: results for some one-dimensional physical models , 1994 .

[11]  Grebogi,et al.  Controlling complexity. , 1995, Physical review letters.

[12]  J. Gallas,et al.  Reductions and simplifications of orbital sums in a Hamiltonian repeller , 2006 .

[13]  J. G. Freire,et al.  Synchronization and predictability under rule 52, a cellular automaton reputedly of class 4 , 2007 .

[14]  V. Astakhov,et al.  Multistability formation and synchronization loss in coupled Hénon maps: two sides of the single bifurcational mechanism. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[15]  Imre M. Jánosi,et al.  Globally coupled multiattractor maps: Mean field dynamics controlled by the number of elements , 1999 .

[16]  Jason A. C. Gallas,et al.  Mandelbrot-like sets in dynamical systems with no critical points , 2006 .

[17]  Mark E. J. Newman,et al.  Structure and Dynamics of Networks , 2009 .

[18]  J. Gallas,et al.  Exploring collective behaviors with a multi-attractor quartic map , 1998 .

[19]  Wenzel,et al.  Periodic orbits in the dissipative standard map. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[20]  V. I. Arnolʹd.  Bifurcation theory and catastrophe theory , 1994 .

[21]  Y. ZOU,et al.  Shrimp Structure and Associated Dynamics in Parametrically excited oscillators , 2006, Int. J. Bifurc. Chaos.

[22]  C. DaCamara,et al.  Multistability, phase diagrams, and intransitivity in the Lorenz-84 low-order atmospheric circulation model. , 2008, Chaos.

[23]  Gesine Reinert,et al.  Small worlds , 2001, Random Struct. Algorithms.

[24]  Hans J Herrmann,et al.  Coherence in scale-free networks of chaotic maps. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[25]  J. Gallas,et al.  Accumulation horizons and period adding in optically injected semiconductor lasers. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[26]  T. Morrison,et al.  Dynamical Systems , 2021, Nature.

[27]  Schmidt,et al.  Dissipative standard map. , 1985, Physical review. A, General physics.

[28]  J. Gallas,et al.  Accumulation boundaries: codimension-two accumulation of accumulations in phase diagrams of semiconductor lasers, electric circuits, atmospheric and chemical oscillators , 2008, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[29]  Jürgen Jost,et al.  Delays, connection topology, and synchronization of coupled chaotic maps. , 2004, Physical review letters.

[30]  Pedro G. Lind,et al.  Impact of bistability in the synchronization of chaotic maps with delayed coupling and complex topologies , 2006 .

[31]  J. Gallas,et al.  Basin size evolution between dissipative and conservative limits. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[32]  B. Chirikov.  A universal instability of many-dimensional oscillator systems , 1979 .

[33]  Celso Grebogi,et al.  Bifurcation rigidity , 1999 .

[34]  Grebogi,et al.  Map with more than 100 coexisting low-period periodic attractors. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[35]  Jürgen Kurths,et al.  Coupled Bistable Maps: a Tool to Study convection Parameterization in Ocean Models , 2004, Int. J. Bifurc. Chaos.

[36]  R. Chacón,et al.  Well-behaved dynamics in a dissipative nonideal periodically kicked rotator. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[37]  Celso Grebogi,et al.  Dynamical properties of a simple mechanical system with a large number of coexisting periodic attractors , 1998 .

[38]  S. Boccaletti,et al.  Synchronization of chaotic systems , 2001 .

[39]  Cristian Bonatto,et al.  Self-similarities in the frequency-amplitude space of a loss-modulated CO2 laser. , 2005, Physical review letters.

[40]  J S Andrade,et al.  Periodic neural activity induced by network complexity. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[41]  Owen J. Brison,et al.  Spatial updating, spatial transients, and regularities of a complex automaton with nonperiodic architecture. , 2007, Chaos.

[42]  G. Zaslavsky.  The simplest case of a strange attractor , 1978 .