Nonlinear normal mode-based study of synchronization in delay coupled limit cycle oscillators

Abstract

Nonlinear normal modes (NNMs), a powerful analytical tool for studying nonlinear systems is used here to study synchronization dynamics in delay coupled oscillators, which exhibit limit cycle oscillations (LCO) and hysteresis when uncoupled. A method described by Shaw and Pierre [1] is used to arrive at the NNMs which are found to be two dimensional invariant manifolds containing closed curves corresponding to either the in-phase or out-of-phase synchronized oscillations for both delayed and non-delayed, reactively or dissipatively coupled systems. Motion initiated on these NNMs, is found to evolve strictly on them, settling onto a stable orbit if existent or on to a quasi-periodic orbit lying between the two manifolds in the absence of one. The NNMs are used to decouple the governing equations of the system, giving the in-phase and out-of-phase synchronization frequencies directly. Parametric study in terms of strength of coupling, frequency detuning and delay time is performed using the averaged equations of the system, aided by continuation software AUTO, which also reveals the bifurcations in the system. These are verified using NNM, direct numerical integration and shooting method based Floquent analysis of the original system and are found to match well. The results obtained here agree with prior work in literature, at the same time extending them to hitherto unexplored parameter range and provide a new perspective on the same from the point of view of NNMs. Hence this study demonstrates the utility of NNMs for the study of synchronization dynamics in both delayed and non delay mutually coupled LCOs and can be useful for studying larger arrays of the same.