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

2007/09/29 by Oleg Makarenkov, Makarenkov, Oleg
Computer Science · Mathematics · #14E20 #54C40 #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.0710.0052

openalex publication_date 2007/09/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let P_e∈ C⁰(Rⁿ,Rⁿ) 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 x₀ we evaluate the topological degree d(I-P_e,U) of I-P_e on an open bounded set U whose boundary contains x₀([0,T]) and does not contain other fixed points of P₀. 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 P₀ on ∂ U is zero.

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