Global 2-cluster dynamics under large symmetric groups

We explore equivariant dynamics under the symmetric group $S_N$ of all permutations of $N$ elements. Specifically we study cubic vector fields which commute with the standard real $(N-1)$-dimensional irreducible representation of $S_N$. All stationary solutions are cluster solutions of 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. Applications include oscillators with all-to-all coupling and electrochemistry. For illustration we consider synchronization clusters among $N$ Stuart-Landau oscillators with complex linear global coupling.