Desynchronization of chaos in coupled logistic maps

Yu. L. Maistrenko, V. L. Maistrenko, O. Popovych, and E. Mosekilde
Phys. Rev. E 60, 2817 – Published 1 September 1999
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Abstract

When identical chaotic oscillators interact, a state of complete or partial synchronization may be attained in which the motion is restricted to an invariant manifold of lower dimension than the full phase space. Riddling of the basin of attraction arises when particular orbits embedded in the synchronized chaotic state become transversely unstable while the state remains attracting on the average. Considering a system of two coupled logistic maps, we show that the transition to riddling will be soft or hard, depending on whether the first orbit to lose its transverse stability undergoes a supercritical or subcritical bifurcation. A subcritical bifurcation can lead directly to global riddling of the basin of attraction for the synchronized chaotic state. A supercritical bifurcation, on the other hand, is associated with the formation of a so-called mixed absorbing area that stretches along the synchronized chaotic state, and from which trajectories cannot escape. This gives rise to locally riddled basins of attraction. We present three different scenarios for the onset of riddling and for the subsequent transformations of the basins of attraction. Each scenario is described by following the type and location of the relevant asynchronous cycles, and determining their stable and unstable invariant manifolds. One scenario involves a contact bifurcation between the boundary of the basin of attraction and the absorbing area. Another scenario involves a long and interesting series of bifurcations starting with the stabilization of the asynchronous cycle produced in the riddling bifurcation and ending in a boundary crisis where the stability of an asynchronous chaotic state is destroyed. Finally, a phase diagram is presented to illustrate the parameter values at which the various transitions occur.

  • Received 22 January 1999

DOI:https://doi.org/10.1103/PhysRevE.60.2817

©1999 American Physical Society

Authors & Affiliations

Yu. L. Maistrenko1, V. L. Maistrenko1, O. Popovych1, and E. Mosekilde2

  • 1Institute of Mathematics, National Academy of Sciences, Kiev, 252601, Ukraine
  • 2Department of Physics, Technical University of Denmark, 2800 Lyngby, Denmark

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Vol. 60, Iss. 3 — September 1999

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