Existence of anticipatory, complete and lag synchronizations in time-delay systems

Existence of different kinds of synchronizations, namely anticipatory, complete and lag type synchronizations (both exact and approximate), are shown to be possible in time-delay coupled piecewise linear systems. We deduce stability condition for synchronization of such unidirectionally coupled systems following Krasovskii-Lyapunov theory. Transition from anticipatory to lag synchronization via complete synchronization as a function of coupling delay is discussed. The existence of exact synchronization is preceded by a region of approximate synchronization from desynchronized state as a function of a system parameter, whose value determines the stability condition for synchronization. The results are corroborated by the nature of similarity functions. A new type of oscillating synchronization that oscillates between anticipatory, complete and lag synchronization, is identified as a consequence of delay time modulation with suitable stability condition.Comment: 5 Figures 9 page

Delay time modulation induced oscillating synchronization and intermittent anticipatory/lag and complete synchronizations in time-delay nonlinear dynamical systems

Existence of a new type of oscillating synchronization that oscillates between three different types of synchronizations (anticipatory, complete and lag synchronizations) is identified in unidirectionally coupled nonlinear time-delay systems having two different time-delays, that is feedback delay with a periodic delay time modulation and a constant coupling delay. Intermittent anticipatory, intermittent lag and complete synchronizations are shown to exist in the same system with identical delay time modulations in both the delays. The transition from anticipatory to complete synchronization and from complete to lag synchronization as a function of coupling delay with suitable stability condition is discussed. The intermittent anticipatory and lag synchronizations are characterized by the minimum of similarity functions and the intermittent behavior is characterized by a universal asymptotic $-{3/2}$ power law distribution. It is also shown that the delay time carved out of the trajectories of the time-delay system with periodic delay time modulation cannot be estimated using conventional methods, thereby reducing the possibility of decoding the message by phase space reconstruction.Comment: accepted for publication in CHAOS, revised in response to referees comment

Phase Synchronization in Unidirectionally Coupled Ikeda Time-delay Systems

Phase synchronization in unidirectionally coupled Ikeda time-delay systems exhibiting non-phase-coherent hyperchaotic attractors of complex topology with highly interwoven trajectories is studied. It is shown that in this set of coupled systems phase synchronization (PS) does exist in a range of the coupling strength which is preceded by a transition regime (approximate PS) and a nonsynchronous regime. However, exact generalized synchronization does not seem to occur in the coupled Ikeda systems (for the range of parameters we have studied) even for large coupling strength, in contrast to our earlier studies in coupled piecewise-linear and Mackey-Glass systems \cite{dvskml2006,dvskml2008}. The above transitions are characterized in terms of recurrence based indices, namely generalized autocorrelation function $P(t)$, correlation of probability of recurrence (CPR), joint probability of recurrence (JPR) and similarity of probability of recurrence (SPR). The existence of phase synchronization is also further confirmed by typical transitions in the Lyapunov exponents of the coupled Ikeda time-delay systems and also using the concept of localized sets.Comment: 10 pages, 7 figure

Generalizing the transition from amplitude to oscillation death in coupled oscillators

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Transition from anticipatory to lag synchronization via complete synchronization in time-delay systems

The existence of anticipatory, complete and lag synchronization in a single system having two different time-delays, that is feedback delay $\tau_1$ and coupling delay $\tau_2$, is identified. The transition from anticipatory to complete synchronization and from complete to lag synchronization as a function of coupling delay $\tau_2$ with suitable stability condition is discussed. The existence of anticipatory and lag synchronization is characterized both by the minimum of similarity function and the transition from on-off intermittency to periodic structure in laminar phase distribution.Comment: 14 Pages and 12 Figure