Secondary bifurcations in systems with all–to–all coupling

Abstract

We study the existence, branching geometry and stability of secondary branches of equilibria in all–to–all coupled systems of differential equations, that is, equations that are equivariant under the permutation action of the symmetric group S N . Specifically, we consider the most general cubic–order system of this type, which arises in models of polymorphism in evolutionary biology. Primary branches in such systems correspond to partitions of N into two parts, and secondary branches correspond to partitions of N into three parts of sizes a, b, c , respectively. If a = b = c , then the cubic–order system is too degenerate to provide secondary branches. The cases when one of a, b, c is equal to N /3 are special, and are not treated here. In all other cases, secondary branches exist, and the secondary branch corresponding to ( a,b,c ) intersects the primary branches corresponding to ( a + b,c ), ( a,b + c ) and ( a + c,b ). All such secondary branches are globally unstable in the cubic–order system. Abstract considerations suggest that such secondary branches are locally unstable, which would explain the common occurrence of jump bifurcations between primary branches in numerical simulations of the cubic–order system. However such considerations do not prove instability due to the possible existence of hidden symmetries. In this paper, we carry out the calculations required to verify that the secondary branches are unstable, and we show, moreover, that these branches are globally unstable.