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

Oleg Makarenkov

DOI: http://dx.doi.org/10.12775/TMNA.2008.043

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.

Keywords


Topological degree; perturbed Poincaré-Andronov map; zero measure singularities

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