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Inverse synchronizations in coupled time-delay systems with inhibitory coupling

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 $Δ$ and $Δ_τ$ terms in the error equation corresponding to the synchronization manifold are time independent and (ii) the coefficient of the $Δ$ term is time independent while that of the $Δ_τ$ 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.