Bifurcations from nondegenerate families of periodic solutions in Lipschitz systems

Adriana Buicǎ, Jaume Llibre, Oleg Makarenkov

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9 Citations (Scopus)

Abstract

The paper addresses the problem of bifurcation of periodic solutions from a normally nondegenerate family of periodic solutions of ordinary differential equations under perturbations. The approach to solve this problem can be described as transforming (by a Lyapunov-Schmidt reduction) the initial system into one which is in the standard form of averaging, and subsequently applying the averaging principle. This approach encounters a fundamental problem when the perturbation is only Lipschitz (nonsmooth) as we do not longer have smooth Lyapunov-Schmidt projectors. The situation of Lipschitz perturbations has been addressed in the literature lately and the results obtained conclude the existence of the bifurcated branch of periodic solutions. Motivated by recent challenges in control theory, we are interested in the uniqueness problem. We achieve this in the case when the Lipschitz constant of the perturbation obeys a suitable estimate. © 2011 Elsevier Inc.
Original languageEnglish
Pages (from-to)3899-3919
JournalJournal of Differential Equations
Volume252
Issue number6
DOIs
Publication statusPublished - 15 Mar 2012

Keywords

  • Lipschitz differential equations
  • Lyapunov-Schmidt reduction
  • Malkin bifurcation function
  • Periodic solution
  • Perturbation theory

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