On certain generalizations of the Hopf bifurcation

Abstract

New sufficient conditions for the existence of generalized Hopf bifurcations are given in the context of asymptotically compact dynamical or semidynamical systems on a metric space. These conditions weaken the hypotheses of previous contributions to the subject.

The Hopf bifurcation in the Shimizu–Morioka system

“…Note that for α < 0 the solutions in the invariant z-axis go away from the origin. If α = 0 the invariant z-axis is filled by singular points of system (1). Then the origin is a non-isolated degenerate singular point.…”

“…In this note we perform an analytic bifurcation analysis of dynamical aspects of the solutions of system (1), when the parameters vary, aiming to give a contribution to the understanding of its complex behavior. Our approach permits a geometric synthesis of the bifurcation analysis, based on the algebraic expression and geometric location of the codimension 2 Hopf point leading to the bifurcation of periodic orbits.…”

“…These methods are used in Section 5 to prove statements (a) and (b) of Theorem 1. For some extensions of the Hopf bifurcation see [1].…”