Hopf Bifurcation with Symmetry

Abstract

In the previous chapter we studied the symmetry properties of time-periodic states of equivariant dynamical systems. We did not enquire how such states might arise. In this chapter we develop the theory of one of the most widespread routes to time-periodicity: Hopf bifurcation. The physical characteristics of Hopf bifurcation are the loss of stability of a steady state, as a parameter is varied, leading to a bifurcation to small-amplitude periodic states with `finite period’. That is, the limiting period as the amplitude tends to zero (immediately after bifurcation) is finite and nonzero