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 N$N$ coupled phase oscillators on a general undirected network. In a Hamiltonian ‘action-angle’ formulation, auxiliary variables J_i$J_i$ combine with the phases θ_i${\theta }_i$ to determine a conserved system with a 2N$2N$ dimensional phase space. In the vicinity of the fixed point for phase synchronisation, θ_i≈θ_j${\theta }_i\approx {\theta }_j$, which is known to be stable, the auxiliary variables J_i$J_i$ exhibit instability  . This manifests Liouville’s Theorem in the phase synchronised regime in that contraction in the θ_i${\theta }_i$ parts of phase space are compensated for by expansion in the auxiliary dimensions. I formulate an entropy rate based on the projection of the J_i$J_i$ 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.