Influence of a small perturbation on Poincare-Andronov operators with not well defined topological degree

TL;DR

This paper investigates how small perturbations affect the topological degree of Poincare-Andronov operators near a limit cycle, proving Mawhin's conjecture that the degree can be any integer despite a measure-zero fixed point set.

Contribution

It provides an explicit formula linking the degree to bifurcation function zeros and proves Mawhin's conjecture about the degree's possible values.

Findings

Derived an explicit formula for the topological degree.

Proved Mawhin's conjecture on the degree's range.

Demonstrated the degree can be any integer despite measure-zero fixed points.

Abstract

Let $P_e\in C^0(R^n,R^n)$ be the Poincare-Andronov operator over period $T>0$ of the $T$-periodically perturbed autonomous system $x'=f(x)+e g(t,x,e),$ where $e>0$ is small. Assuming that for $e=0$ this system has a $T$-periodic limit cycle $x_0$ we evaluate the topological degree $d(I-P_e,U)$ of $I-P_e$ on an open bounded set $U$ whose boundary contains $x_0([0,T])$ and does not contain other fixed points of $P_0.$ We give an explicit formula connecting $d(I-P_e,U)$ with topological indexes of zeros of the associated Malkin's bifurcation function. The goal of the paper is to prove Mawhin's conjecture which claims that $d(I-P_e,U)$ can be any integer in spite of the fact that the measure of the set of fixed points of $P_0$ on $\partial U$ is zero.