Adaptive coupling induced multi-stable states in complex networks

TL;DR

This paper investigates how adaptive coupling in complex networks of identical oscillators leads to multi-stable states, including synchronized clusters and desynchronization, influenced by coupling dynamics, asymmetries, and time scales.

Contribution

It demonstrates the emergence of multi-stable states in adaptively coupled oscillators and analyzes the effects of asymmetries and time scales both numerically and analytically.

Findings

Multi-stable states include two-cluster synchronization and desynchronization.

Phase relationships are asymptotically stable regardless of synchronization.

Coupling and plasticity asymmetries influence state transitions.

Abstract

Adaptive coupling, where the coupling is dynamical and depends on the behaviour of the oscillators in a complex system, is one of the most crucial factors to control the dynamics and streamline various processes in complex networks. In this paper, we have demonstrated the occurrence of multi-stable states in a system of identical phase oscillators that are dynamically coupled. We find that the multi-stable state is comprised of a two cluster synchronization state where the clusters are in anti-phase relationship with each other and a desynchronization state. We also find that the phase relationship between the oscillators is asymptotically stable irrespective of whether there is synchronization or desynchronization in the system. The time scale of the coupling affects the size of the clusters in the two cluster state. We also investigate the effect of both the coupling asymmetry and plasticity asymmetry on the multi-stable states. In the absence of coupling asymmetry, increasing the plasticity asymmetry causes the system to go from a two clustered state to a desynchronization state and then to a two clustered state. Further, the coupling asymmetry, if present, also affects this transition. We also analytically find the occurrence of the above mentioned multi-stable-desynchronization-multi-stable state transition. A brief discussion on the phase evolution of nonidentical oscillators is also provided. Our analytical results are in good agreement with our numerical observations.