Global heteroclinic rebel dynamics among large 2-clusters in permutation equivariant systems

Abstract

We explore equivariant dynamics under the symmetric group $S_N$ of all permutations of $N$ elements. Specifically we study one-parameter vector fields, up to cubic order, which commute with the standard real $(N-1)$-dimensional irreducible representation of $S_N$. The parameter is the linearization at the trivial 1-cluster equilibrium of total synchrony. All equilibria are cluster solutions involving up to three clusters. The resulting global dynamics is of gradient type: all bounded solutions are cluster equilibria and heteroclinic orbits between them. In the limit of large $N$, we present a detailed analysis of the web of heteroclinic orbits among the plethora of 2-cluster equilibria. Our focus is on the global dynamics of 3-cluster solutions with one rebel cluster of small size. These solutions describe slow relative growth and decay of 2-cluster states. For $N\rightarrow\infty$, the limiting heteroclinic web defines an integrable \emph{rebel flow} in the space of 2-cluster equilibrium configurations. We identify and study the seven qualitatively distinct global rebel flows which arise in this setting. Applications include oscillators with all-to-all coupling, and electrochemistry. For illustration we consider synchronization clusters among $N$ complex Stuart-Landau oscillators with complex linear global coupling.