Higher Dimensional Limit Cycles and Coupling Induced Synchronization in Dynamical Systems

TL;DR

This paper explores higher dimensional limit cycles in coupled oscillators, demonstrating how coupling can induce synchronization and entrainment in systems like Van der Pol and Kuznetsov oscillators using a renormalization group approach.

Contribution

It introduces the concept of higher dimensional limit cycles and applies a renormalization group method to analyze quasi periodic orbits in coupled oscillators.

Findings

Coupling can induce entrainment in oscillators with different frequency behaviors.

Renormalization group approach effectively analyzes quasi periodic analogues of limit cycles.

Coupled oscillators can synchronize through linear coupling even if only one is initially entrained.

Abstract

Limit cycles (attractors for neighbouring periodic orbits in a dissipative dynamical system) have been widely studied but the corresponding generalization for quasi periodic orbits have rarely been discussed. Here we investigate "higher dimensional limit cycles" by analysing a pair of coupled non identical Van der Pol oscillators and also the Kuznetsov oscillator. We find that the renormalization group based approach introduced by Chen, Goldenfeld and Oono is ideally suited for analysing the quasi periodic analogues of limit cycles. We also address entrainment issues in a pair of forced and coupled Van der Pol oscillators. Our principle finding there is that if two such independent oscillators one with frequency entrainment and the other without are coupled linearly, then it is possible to produce an entrained state via the coupling.