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 wide 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 - we characterise the more generic multi-way interactions between oscillators that will typically be present. We examine a generic system of weakly coupled oscillators that are close to a supercritical Hopf bifurcation by considering two parameters, $\epsilon$ to express the strength of coupling and $\lambda$ to unfold the Hopf bifurcation with a small amplitude stable limit cycle for $\lambda>0$. Using the equivariant normal form we derive a generic normal form for a system of coupled phase oscillators with $S_N$ symmetry and show that for fixed $N$, in the limit $\lambda \rightarrow 0$ with $\lambda>0$ and $\epsilon=o(\lambda^{3/2})$, that the attracting dynamics of the system can be expressed just in terms of phases. Assuming the lowest order nontrivial terms, we show that these coupled phase equations generically include terms involving (for $N\geq 4$) up to four interacting phases, regardless of $N$. For a normalization that maintains nontrivial interactions in the limit $N\rightarrow \infty$ we show that the additional terms can lead to new phenomena in existence of more than two-cluster states with the same phase difference but different size of clusters.