Inverse synchronizations in coupled time-delay systems with inhibitory coupling

TL;DR

This paper investigates various inverse synchronization phenomena in coupled time-delay systems with inhibitory coupling, establishing stability conditions and confirming different synchronization types through multiple analytical and numerical methods.

Contribution

It introduces a unified stability condition applicable to diverse inverse synchronization types in time-delay systems with inhibitory coupling.

Findings

Transitions between inverse synchronization types depend on coupling delay.

The same stability condition applies under different coefficient time-dependencies.

Different inverse synchronizations are confirmed via similarity functions and Lyapunov exponents.

Abstract

Transitions between inverse anticipatory, inverse complete and inverse lag synchronizations are shown to occur as a function of the coupling delay in unidirectionally coupled time-delay systems with inhibitory coupling. We have also shown that the same general asymptotic stability condition obtained using the Krasovskii-Lyapunov functional theory can be valid for the cases where (i) both the coefficients of the $\Delta$ and $\Delta_\tau$ terms in the error equation corresponding to the synchronization manifold are time independent and (ii) the coefficient of the $\Delta$ term is time independent while that of the $\Delta_\tau$ term is time dependent. The existence of different kinds of synchronization are corroborated using similarity function, probability of synchronization and also from changes in the spectrum of Lyapunov exponents of the coupled time-delay systems.