Design of coupling for synchronization in time-delayed systems

TL;DR

This paper presents a method for designing delay coupling in dynamical systems to achieve various synchronization states, including mixed and generalized synchronization, with stability analysis and numerical validation.

Contribution

It introduces a novel coupling design enabling multiple synchronization regimes and stability conditions in mismatched delay systems.

Findings

Achieved synchronization, antisynchronization, and lag-synchronization in delay systems.

Demonstrated mixed synchronization with coexisting states.

Validated stability conditions through numerical examples.

Abstract

We report a design of delay coupling for targeting desired synchronization in delay dynamical systems. We target synchronization, antisynchronization, lag-, antilag- synchronization, amplitude death (or oscillation death) and generalized synchronization in mismatched oscillators. A scaling of the size of an attractor is made possible in different synchronization regimes. We realize a type of mixed synchronization where synchronization, antisynchronization coexist in different pairs of state variables of the coupled system. We establish the stability condition of synchronization using the Krasovskii-Lyapunov function theory and the Hurwitz matrix criterion. We present numerical examples using the Mackey-Glass system and a delay R\"{o}ssler system.