Hopf normal form with $S_N$ symmetry and reduction to systems of nonlinearly coupled phase oscillators

Abstract

Coupled oscillator models where $N$ oscillators are identical and symmetrically coupled to all others with full permutation symmetry $S_N$ are found in a variety of applications. Much, but not all, work on phase descriptions of such systems consider the special case of pairwise coupling between oscillators. In this paper, we show this is restrictive - and we characterise generic multi-way interactions between oscillators that are typically present, except at the very lowest order near a Hopf bifurcation where the oscillations emerge. We examine a network of identical weakly coupled dynamical systems that are close to a supercritical Hopf bifurcation by considering two parameters, $\epsilon$ (the strength of coupling) and $\lambda$ (an unfolding parameter for the Hopf bifurcation). For small enough $\lambda>0$ there is an attractor that is the product of $N$ stable limit cycles; this persists as a normally hyperbolic invariant torus for sufficiently small $\epsilon>0$. Using equivariant normal form theory, we derive a generic normal form for a system of coupled phase oscillators with $S_N$ symmetry. For fixed $N$ and taking the limit $0<\epsilon\ll\lambda\ll 1$, we show that the attracting dynamics of the system on the torus can be well approximated by a coupled phase oscillator system that, to lowest order, is the well-known Kuramoto-Sakaguchi system of coupled oscillators. The next order of approximation genericlly includes terms with up to four interacting phases, regardless of $N$. Using a normalization that maintains nontrivial interactions in the limit $N\rightarrow \infty$, we show that the additional terms can lead to new phenomena in terms of coexistence of two-cluster states with the same phase difference but different cluster size.