Hopf bifurcation

Abstract

If the center subspace is finite dimensional, the center manifold theorem enables us to reduce an infinite dimensional dynamical system to a finite dimensional one. As a consequence, results on bifurcation from equilibria (e.g., saddle node, Hopf, Takens-Bogdanov bifurcation [159])) can be “lifted” from the theory of ODE to the theory of RFDE. The alternative is to study such bifurcations directly in the infinite dimensional setting and to “imitate” the ODE proof. However, this usually entails a number of technical complications (which are, in one way or another, connected with lack of smoothness) and, therefore, the alternative is, in our opinion, less attractive.