On Degenerate Planar Hopf Bifurcations

TL;DR

This paper investigates degenerate Hopf bifurcations in planar dynamical systems, classifying degeneracy types, analyzing their properties, and providing conditions for second-kind degeneracies with complex bifurcation behaviors.

Contribution

It introduces a two-type classification of degeneracy in Hopf bifurcations and offers conditions and typical forms for second-kind degeneracies in planar systems.

Findings

Degeneracy of first kind prevents local limit cycle emergence.

Degeneracy of second kind involves multiple or semistable cycles.

The radius of emerging cycles can tend to zero with an anomalous order.

Abstract

Our concern is the study of degenerate Hopf bifurcation of smooth planar dynamical systems near isolated singular points. To do so, we propose to split up the definition of degeneracy into two types. Degeneracy of first kind shall means that no limit cycle surrounding the steady state can emerge after or before the critical point, with the possible emergence of limit cycles surrounding the point at infinity. Degeneracy of second kind shall means that either several limit cycles or semistable cycles as a limiting case, emerge surrounding the steady state super or subcritically. In degenerate bifurcation of second kind we also show that the radius of the emerging cycle tends to zero with an "anomalous" order as the bifurcation parameter tends to the critical value. Finally, we give a sufficient condition for degenerate bifurcations of second kind up to 6-jet-equivalence, and show some "typical" forms for degenerate bifurcations.