Conservative dynamics in phase oscillator networks

TL;DR

This paper investigates conservative interactions in phase oscillator networks, deriving conditions for pairwise couplings to be conservative and analyzing the spectral properties of such systems.

Contribution

It introduces a general condition based on a pair-Hamiltonian for conservative couplings and explores their properties in various oscillator network models.

Findings

Conservative networks exhibit nearly symmetric Lyapunov spectra.

Conditions for pairwise couplings to be conservative are derived.

Generalization to triplet and quadruplet couplings is demonstrated.

Abstract

The interaction between phase oscillators is conservative if the phase volume is conserved throughout the dynamics. We derive a general condition, based on the notion of a pair-Hamiltonian, for the pairwise couplings to be conservative. The conservative networks with Winfree-type and Kuramoto-Daido-type couplings are also discussed. It is demonstrated that although, in contradistinction to genuine Hamiltonian dynamics, there is no exact pairwise symmetry of the Lyapunov exponents, the Lyapunov spectrum for a large network is nearly symmetric. The concept is also generalized to triplet and quadruplet couplings.