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No Access Submitted: 24 January 2008 Accepted: 27 March 2008 Published Online: 20 May 2008

Transition from phase to generalized synchronization in time-delay systems

Chaos 18, 023118 (2008); https://doi.org/10.1063/1.2911541
D. V. Senthilkumar1, a), M. Lakshmanan1, b), and J. Kurths2, c)
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ABSTRACT
The notion of phase synchronization in time-delay systems, exhibiting highly non-phase-coherent attractors, has not been realized yet even though it has been well studied in chaotic dynamical systems without delay. We report the identification of phase synchronization in coupled nonidentical piecewise linear and in coupled Mackey–Glass time-delay systems with highly non-phase-coherent regimes. We show that there is a transition from nonsynchronized behavior to phase and then to generalized synchronization as a function of coupling strength. We have introduced a transformation to capture the phase of the non-phase-coherent attractors, which works equally well for both the time-delay systems. The instantaneous phases of the above coupled systems calculated from the transformed attractors satisfy both the phase and mean frequency locking conditions. These transitions are also characterized in terms of recurrence-based indices, namely generalized autocorrelation function P(t), correlation of probability of recurrence, joint probability of recurrence, and similarity of probability of recurrence. We have quantified the different synchronization regimes in terms of these indices. The existence of phase synchronization is also characterized by typical transitions in the Lyapunov exponents of the coupled time-delay systems.

ACKNOWLEDGMENTS

The work of D. V. S. and M. L. has been supported by a Department of Science and Technology, Government of India sponsored research project. The work of M. L. is supported by a DST Ramanna Fellowship. J. K. has been supported by his Humboldt-CSIR research award and NoE BIOSIM (EU) Contract No. LSHB-CT-2004-005137.

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