SN-equivalent symmetry-breaking bifurcations

Abstract

I analyze generic symmetry-breaking bifurcations of SN-equivariant vector fields (where genericity is in the restricted space of SN-equivariant systems). The normal form for the Liapunov–Schmidt reduced bifurcation problem is derived, and its codimension 1 symmetry-breaking bifurcations are investigated. Branch equations are given along with the conditions for stability, and the parameter space is partitioned accordingly. Applications in complex systems theory are suggested in which the dynamics of SN-equivariant systems are used to model the emergent behavior of systems of multiple interacting agents.