Generalized Hopf Bifurcation for planar vector fields via the inverse integrating factor

TL;DR

This paper investigates the maximum number of limit cycles bifurcating from focus singular points in planar systems using inverse integrating factors, providing new bounds and existence results for various singular point types.

Contribution

It introduces the concept of vanishing multiplicity of inverse integrating factors to determine bifurcation limits and proves their existence near multiple singular point types.

Findings

Maximum number of bifurcating limit cycles determined by vanishing multiplicity.

Existence of inverse integrating factors near focus and certain singular points.

Provides bounds and conditions for cyclicity in planar differential systems.

Abstract

In this paper we study the maximum number of limit cycles that can bifurcate from a focus singular point $p_0$ of an analytic, autonomous differential system in the real plane under an analytic perturbation. We consider $p_0$ being a focus singular point of the following three types: non-degenerate, degenerate without characteristic directions and nilpotent. In a neighborhood of $p_0$ the differential system can always be brought, by means of a change to (generalized) polar coordinates $(r, \theta)$, to an equation over a cylinder in which the singular point $p_0$ corresponds to a limit cycle $\gamma_0$. This equation over the cylinder always has an inverse integrating factor which is smooth and non--flat in $r$ in a neighborhood of $\gamma_0$. We define the notion of vanishing multiplicity of the inverse integrating factor over $\gamma_0$. This vanishing multiplicity determines the maximum number of limit cycles that bifurcate from the singular point $p_0$ in the non-degenerate case and a lower bound for the cyclicity otherwise. Moreover, we prove the existence of an inverse integrating factor in a neighborhood of many types of singular points, namely for the three types of focus considered in the previous paragraph and for any isolated singular point with at least one non-zero eigenvalue.