Riddling bifurcations in coupled piecewise linear maps

Abstract A mechanism for riddling bifurcations in the system of coupled piecewise linear maps is described. We give sufficient conditions for the occurrence of locally and globally riddled basins based on the properties of absorbing areas of the chaotic attractors on the invariant manifold. It is also shown that riddled basins are preserved upon bifurcation of the chaotic attractors as long as the attractor after bifurcation is located in the absorbing area of the attractor before bifurcation.

[1]  Alan V. Oppenheim,et al.  Circuit implementation of synchronized chaos with applications to communications. , 1993, Physical review letters.

[2]  W. D. Melo,et al.  ONE-DIMENSIONAL DYNAMICS , 2013 .

[3]  Carroll,et al.  Synchronization in chaotic systems. , 1990, Physical review letters.

[4]  Ott,et al.  Transitions to Bubbling of Chaotic Systems. , 1996, Physical review letters.

[5]  Carroll,et al.  Experimental and Numerical Evidence for Riddled Basins in Coupled Chaotic Systems. , 1994, Physical review letters.

[6]  E. Ott,et al.  Blowout bifurcations: the occurrence of riddled basins and on-off intermittency , 1994 .

[7]  James A. Yorke,et al.  Dynamics: Numerical Explorations , 1994 .

[8]  Kapitaniak,et al.  Different types of chaos synchronization in two coupled piecewise linear maps. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[9]  Leon O. Chua,et al.  SYNCHRONIZING CHAOS FROM ELECTRONIC PHASE-LOCKED LOOPS , 1991 .

[10]  Matthew Nicol,et al.  On the unfolding of a blowout bifurcation , 1998 .

[11]  Leon O. Chua,et al.  Cycles of Chaotic Intervals in a Time-delayed Chua's Circuit , 1993, Chua's Circuit.

[12]  Leon O. Chua,et al.  EXPERIMENTAL SYNCHRONIZATION OF CHAOS USING CONTINUOUS CONTROL , 1994 .

[13]  Christian Mira,et al.  Some Properties of a Two-Dimensional Piecewise-Linear Noninvertible Map , 1996 .

[14]  Tomasz Kapitaniak,et al.  LOCALLY AND GLOBALLY RIDDLED BASINS IN TWO COUPLED PIECEWISE-LINEAR MAPS , 1997 .

[15]  T. Kapitaniak,et al.  Synchronization of chaos using continuous control. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[16]  Tomasz Kapitaniak,et al.  Loss of Chaos Synchronization through the Sequence of Bifurcations of Saddle Periodic Orbits , 1997 .

[17]  M. Rabinovich,et al.  Stochastic synchronization of oscillation in dissipative systems , 1986 .

[18]  Erik Mosekilde,et al.  Role of the Absorbing Area in Chaotic Synchronization , 1998 .

[19]  E. Mosekilde,et al.  TRANSVERSE INSTABILITY AND RIDDLED BASINS IN A SYSTEM OF TWO COUPLED LOGISTIC MAPS , 1998 .

[20]  J. Milnor.  On the concept of attractor , 1985 .

[21]  I. Stewart,et al.  Bubbling of attractors and synchronisation of chaotic oscillators , 1994 .

[22]  H. Fujisaka,et al.  Stability Theory of Synchronized Motion in Coupled-Oscillator Systems. II: The Mapping Approach , 1983 .

[23]  Grebogi,et al.  Synchronization of chaotic trajectories using control. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[24]  Louis M. Pecora,et al.  Synchronizing chaotic circuits , 1991 .

[25]  J. Yorke,et al.  The transition to chaotic attractors with riddled basins , 1994 .

[26]  T. Kapitaniak,et al.  MONOTONE SYNCHRONIZATION OF CHAOS , 1996 .

[27]  Christian Mira,et al.  Chaotic Dynamics in Two-Dimensional Noninvertible Maps , 1996 .

[28]  James A. Yorke,et al.  BORDER-COLLISION BIFURCATIONS FOR PIECEWISE SMOOTH ONE-DIMENSIONAL MAPS , 1995 .

[29]  Leon O. Chua,et al.  Transmission of Digital signals by Chaotic Synchronization , 1992, Chua's Circuit.

[30]  Grebogi,et al.  Riddling Bifurcation in Chaotic Dynamical Systems. , 1996, Physical review letters.