The Hopf Bifurcation Theorem for Diffeomorphisms

Let X be a vector field and let γ be a closed orbit of the flow ⌽t of X. Let P be a Poincare map associated with γ. (See §2B). Suppose there is a circle σ that is invariant under P. Then it is clear that Ut⌽t(σ) is an invariant torus for the flow of X (see Figure 6.1).If we have a one parameter family of vector fields and closed orbits Xµ and γµ , it is quite conceivable that for small µ, γµ might be stable, but for large µ it might become unstable and a stable invariant torus take its place. Recall that γµ is stable (unstable) if the eigenvalues of the derivative of the Poincare map Pµ have absolute value 1). (See §2B). The Hopf Bifurcation Theorem for diffeomorphisms gives conditions under which we may expect bifurcation to stable invariant tori after loss of stability of γµ. The theorem we present is due to Ruelle-Takens [1]; we follow the exposition of Lanford [1] for the proof.