The Boundaries of Synchronization in Oscillators Networks

TL;DR

This paper investigates how small initial perturbations in oscillator networks can lead to either synchronization or desynchronization, revealing fractal basin boundaries and chaotic dynamics that determine long-term network states.

Contribution

It introduces a detailed analysis of the boundary between synchronized and desynchronized states, highlighting the role of fractal basin boundaries and chaotic sets in network dynamics.

Findings

Existence of fractal basin boundaries separating states

Perturbations can cause transitions between synchronization and desynchronization

Chaotic sets influence long-term network behavior

Abstract

We analyze the final state sensitivity of nonlocal networks with respect to initial conditions of their units. By changing the initial conditions of a single network unit, we perturb an initially synchronized state. Depending on the perturbation strength, we observe the existence of two possible network long-term states: (i) The network neutralizes the perturbation effects and returns to its synchronized configuration. (ii) The perturbation leads the network to an alternative desynchronized state. By computing uncertainty exponents of a two-dimensional cross section of the state space, we find the existence of fractal basin boundaries separating synchronized solutions from desynchronized ones. We attribute these features to an unstable chaotic set in which trajectories persist for times indefinitely long in the network.