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

Abstract

Let ${\mathcal P}_\varepsilon\in C^0({\mathbb R}^n,{\mathbb R}^n)$ be the Poincaré-Andronov operator over period $T> 0$ of $T$-periodically perturbed autonomous system $\dot x=f(x)+\varepsilon g(t,x,\varepsilon)$, where $\varepsilon> 0$ is small. Assuming that for $\varepsilon=0$ this system has a $T$-periodic limit cycle $x_0$ we evaluate the topological degree $d(I-{\mathcal P}_\varepsilon,U)$ of $I-{\mathcal P}_\varepsilon$ on an open bounded set $U$ whose boundary $\partial U$ contains $x_0([0,T])$ and ${\mathcal P}_0(v)\not=v$ for any $v\in \partial U\setminus x_0([0,T])$. We give an explicit formula connecting $d(I-{\mathcal P}_\varepsilon,U)$ with the topological indices of zeros of the associated Malkin's bifurcation function. The goal of the paper is to prove the Mawhin's conjecture claiming that $d(I-{\mathcal P}_\varepsilon,U)$ can be any integer in spite of the fact that the measure of the set of fixed points of ${\mathcal P}_0$ on $\partial U$ is zero.