Entropy and stability of phase synchronisation of oscillators on networks

I examine the role of entropy in the transition from incoherence to phase synchronisation in the Kuramoto model of NN coupled phase oscillators on a general undirected network. In a Hamiltonian ‘action-angle’ formulation, auxiliary variables JiJi combine with the phases θiθi to determine a conserved system with a 2N2N dimensional phase space. In the vicinity of the fixed point for phase synchronisation, θi≈θjθi≈θj, which is known to be stable, the auxiliary variables JiJi exhibit instability  . This manifests Liouville’s Theorem in the phase synchronised regime in that contraction in the θiθi parts of phase space are compensated for by expansion in the auxiliary dimensions. I formulate an entropy rate based on the projection of the JiJi 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.