Hopf's Bifurcation Theorem and the Center Theorem of Liapunov with resonance cases

Abstract

In recent years numerous papers have dealt with the bifurcation of periodic orbits from an equilibrium point. The starting point for most investigations is the Liapunov Center Theorem [8] or the Hopf Bifurcation Theorem [6]. Local results concerning these theorems were published among others by Chafee [2] who investigated in detail the structure of the periodic orbits in the vicinity of an equilibrium point and by Henrard [4], Schmidt and Sweet [lo] who looked at resonance cases in the Liapunov Center Theorem. Global results on the bifurcation of periodic orbits were obtained by Alexander and Yorke [I]. They showed in their paper that Liapunov’s Center Theorem can be derived as a consequence of Hopf’s Bifurcation Theorem. Based on a suggestion by J. A. Yorke we show on a local level that Liapunov’s theorem can be derived from the one of Hopf. For this we provide an analytic proof of Hopf’s theorem based on the method used in [9], which is a special case of the alternative method, which is described in [3]. Our proof is general enough to include the Center Theorem as a corollary, and in addition it is simple enough to allow us to discuss some exceptional cases of the Hopf Bifurcation Theorem.