Entropy and stability of phase synchronisation of oscillators on networks

• Published in: Annals of physics 2014-09, Vol.348, p.127-143
• Main Author: Kalloniatis, Alexander C.
• Format: Article
• Language: English
• Subjects: CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; EIGENVECTORS; ENTROPY; HAMILTONIANS; Kuramoto; Laplace transforms; LAPLACIAN; Neural networks; Oscillator; OSCILLATORS; Pesin theorem; PHASE OSCILLATIONS; PHASE SPACE; Phase transitions; STABILITY; Synchronisation; SYNCHRONIZATION
• ISSN: 0003-4916; 1096-035X
• DOI: 10.1016/j.aop.2014.05.012

I examine the role of entropy in the transition from incoherence to phase synchronisation in the Kuramoto model of N coupled phase oscillators on a general undirected network. In a Hamiltonian ‘action-angle’ formulation, auxiliary variables Ji combine with the phases θi to determine a conserved system with a 2N dimensional phase space. In the vicinity of the fixed point for phase synchronisation, θi≈θj, which is known to be stable, the auxiliary variables Ji exhibit instability. This manifests Liouville’s Theorem in the phase synchronised regime in that contraction in the θi parts of phase space are compensated for by expansion in the auxiliary dimensions. I formulate an entropy rate based on the projection of the Ji onto eigenvectors of the graph Laplacian that satisfies Pesin’s Theorem. This leads to the insight that the evolution to phase synchronisation of the Kuramoto model is equivalent to the approach to a state of monotonically increasing entropy. Indeed, for unequal intrinsic frequencies on the nodes, the networks that achieve the closest to exact phase synchronisation are those which enjoy the highest entropy production. I compare numerical results for a range of networks.