Methods of topological degree theory in Malkin I. G. - Melnikov V. K.'s problems for periodically perturbed systems

TL;DR

This paper discusses a topological degree approach to analyzing bifurcations of periodic solutions in non-linear systems under periodic perturbations, extending classical results without assuming system differentiability.

Contribution

It introduces a topological degree-based averaging method for bifurcation analysis in non-differentiable systems, generalizing classical results by Malkin and Melnikov.

Findings

Topological degree provides conditions for bifurcation of periodic solutions.

The index of bifurcated solutions is expressed via topological degree.

Convergence rates of solutions as perturbation vanishes are analyzed.

Abstract

A topological degree based averaging principle has been proposed by J. Mawhin in his PhD thesis [J. Mawhin, Le Probleme des Solutions Periodiques en Mecanique non Lineaire, These de doctorat en sciences, Universite de Liege, 1969]. In his thesis the author gives analogous topological degree versions of classical bifurcation results due to I.G. Malkin and V.K. Melnikov, namely the conditions for bifurcation of periodic solutions from families are expressed in term of the topological degree of the bifurcation function. Moreover, the topological index of bifurcated periodic solutions is evaluated over that degree. A third part of the thesis is devoted to the rate the bifurcated periodic solutions converge when the perturbation vanishes. The differentiability of perturbed systems is not assumed.