Patterns of patterns of synchronization: Noise induced attractor switching in rings of coupled nonlinear oscillators

TL;DR

This paper studies how noise causes transitions between different synchronized states in a ring of coupled nonlinear oscillators, revealing phase crossing events and exponential waiting times, and constructs a switching network to describe the system's attractor architecture.

Contribution

It introduces a detailed analysis of noise-induced attractor switching in ring networks of oscillators, connecting phase crossing events to state transitions and modeling the switching dynamics.

Findings

Attractor switching always involves crossing of two adjacent oscillators' phases.

The distribution of escape times from a state is exponential.

A coarse-grained switching network effectively describes the attractor-basin structure.

Abstract

Following the long-lived qualitative-dynamics tradition of explaining behavior in complex systems via the architecture of their attractors and basins, we investigate the patterns of switching between qualitatively distinct trajectories in a network of synchronized oscillators. Our system, consisting of nonlinear amplitude-phase oscillators arranged in a ring topology with reactive nearest neighbor coupling, is simple and connects directly to experimental realizations. We seek to understand how the multiple stable synchronized states connect to each other in state space by applying Gaussian white noise to each of the oscillators' phases. To do this, we first identify a set of locally stable limit cycles at any given coupling strength. For each of these attracting states, we analyze the effect of weak noise via the covariance matrix of deviations around those attractors. We then explore the noise-induced attractor switching behavior via numerical investigations. For a ring of three oscillators we find that an attractor-switching event is always accompanied by the crossing of two adjacent oscillators' phases. For larger numbers of oscillators we find that the distribution of times required to stochastically leave a given state falls off exponentially, and we build an attractor switching network out of the destination states as a coarse-grained description of the high-dimensional attractor-basin architecture.