A Simple Proof of Stability on the Center Manifold for Hopf Bifurcation.

Abstract

A simple proof is presented for a well known fact about Hopf bifurcation: if the loss of an equilibrium point results in periodic solutions via Hopf bifurcation, then the stability ot these periodic solutions is determined by their stability on an associated center manifold. More precisely, it is shown that the characteristic exponent determining the stability of the periodic solutions is the same whether computed for the original systems or the system restricted to the center manifold. Attention is focused on the finite dimensional case of a one parameter family of ordinary differential equations. The proof consists of exhibiting a similarity transformation which uncovers the relationship between the linearized flow of the original system and that of its restriction to the center manifold. (This paper combines and revises papers TR-85-5 and TR-85-6; to appear in IEEE Trans. Automat. Cont.).