Attractivity AMP Hopf Bifurcation

Abstract

**Please note that the full text is embargoed** ABSTRACT: Consider the one-parameter family of differential equations [see pdf for notation] where [see pdf for notation] and [see pdf for notation]. Here [see pdf for notation] and [see pdf for notation]. Denoting by [see pdf for notation] the eigenvalues of [see pdf for notation] we shall suppose throughout the paper that [see pdf for notation] and [see pdf for notation]. We are concerned with the general problem of asymptotic stability of the periodic orbits arising in the Hopf bifurcation for (1.1). Such property is related to the asymptotic behaviour of the flow relative to 0 (the critical value of the parameter) near the origin [see pdf for notation] of [see pdf for notation]. Actually the bifurcating periodic orbits are found to be attracting under the general assumption that [see pdf for notation] is asymptotically stable for [see pdf for notation], and there exists an odd integer [see pdf for notation] such that the above character of [see pdf for notation] is recognizable in a suitable sense by the terms of [see pdf for notation] of degree [see pdf for notation] (h-asymptotic stability). Denoting this property by [see pdf for notation], we point out some relevant aspect of our analysis: