Parametric Resonance in Networked Oscillators

TL;DR

This paper explores how periodic changes in connection strengths can induce resonance phenomena in networked oscillators, revealing complex interactions between network structure and dynamic responses.

Contribution

It introduces a novel multiple-scale perturbation analysis for high-dimensional oscillator networks under parametric forcing, extending resonance theory to complex networks.

Findings

Identification of resonance modes linked to graph Laplacian eigenvalues

Analysis of resonance behavior for different forcing scenarios

Development of a new perturbation method for high-dimensional systems

Abstract

We investigate parametric resonance in oscillator networks subjected to periodically time-varying oscillations in the edge strengths. Such models are inspired by the well-known parametric resonance phenomena for single oscillators, as well as the potential rich phenomenology when such parametric excitations are present in a variety of applications like deep brain stimulation, AC power transmission networks, as well as vehicular flocking formations. We consider cases where a single edge, a subgraph, or the entire network is subjected to forcing, and in each case, we characterize an interesting interplay between the parametric resonance modes and the eigenvalues/vectors of the graph Laplacian. Our analysis is based on a novel treatment of multiple-scale perturbation analysis that we develop for the underlying high-dimensional dynamic equations.