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

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.

$T$-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

#### Full Text:

FULL TEXT### Refbacks

- There are currently no refbacks.