SN-equivariant symmetry-breaking bifurcations

I analyze generic symmetry-breaking bifurcations of S-N-equivariant vector fields (where genericity is in the restricted space of S-N-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 S-N-equivariant systems are used to model the emergent behavior of systems of multiple interacting agents.