Transition from phase to generalized synchronization in time-delay systems

TL;DR

This paper demonstrates the transition from non-synchronized to phase and generalized synchronization in coupled time-delay systems with complex attractors, introducing a new phase extraction method and analyzing the transitions with recurrence-based indices and Lyapunov exponents.

Contribution

It introduces a novel transformation to extract phase in non-phase-coherent time-delay systems and characterizes synchronization transitions using recurrence indices and Lyapunov exponents.

Findings

Identification of phase synchronization in non-phase-coherent time-delay systems.

Transition from non-synchronized to phase and generalized synchronization with increasing coupling.

Validation of the phase extraction method across different time-delay systems.

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 piece-wise linear and in coupled Mackey-Glass time-delay systems with highly non-phase-coherent regimes. We show that there is a transition from non-synchronized 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 (CPR), joint probability of recurrence (JPR) and similarity of probability of recurrence (SPR). 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.