## Generalized Hopf Bifurcation and h-Asymptotic stability

##### Abstract

The prevalent approach to the Hopf bifurcation problem is to prove directly the existence of the bifurcating periodic orbits by using such standard
procedures as the implicit function theorem, the LiapunovSchmidt method and its known variants, and topological degree arguments (see [7]).
The phenomenon of Hopf bifurcation often occurs because of exchange of stability properties of the equilibrium under perturbations (see for instance,
Chafee in [7] p. 85-88,Andronov et. al. [1], Marchetti et. al. [6] and Negrini and Salvadori [8]). This connection between the exchange of stability of
the equilibrium and the appearance of bifurcating periodic orbits can be carefully investigated in order to develop a different approach for obtaining
existence results and qualitative properties of these orbits. Now we want to provide a systematic development of the procedure sketched in [6] and [8]
by considering the generalized Hopf bifurcation as was studied by Chafee [3] who used the alternative method as described by Hale [4]. In particular consider an n dimensional system of differential equations [see pdf for notation]. Assume the Jacobian matrix if [see pdf for notation]
has a complex conjugate pair of eigenvalues ±i and that all other eigenvalues [see pdf for notation].