Melnikov functions for period annulus, nondegenerate centers, heteroclinic and homoclinic cycles

Weigu Li, Jaume Llibre, Xiang Zhang

Research output: Contribution to journalArticleResearchpeer-review

11 Citations (Scopus)

Abstract

We give sufficient conditions in terms of the Melnikov functions in order that an analytic or a polynomial differential system in the real plane has a period annulus. We study the first nonzero Melnikov function of the analytic differential systems in the real plane obtained by perturbing a Hamiltonian system having either a nondegenerate center, a heteroclinic cycle, a homoclinic cycle, or three cycles obtained connecting the four separatrices of a saddle. All the singular points of these cycles are hyperbolic saddles. Finally, using the first nonzero Melnikov function we give a new proof of a result of Roussarie on the finite cyclicity of the homoclinic orbit of the integrable system when we perturb it inside the class of analytic differential systems.
Original languageEnglish
Pages (from-to)49-77
JournalPacific Journal of Mathematics
Volume213
Issue number1
DOIs
Publication statusPublished - 1 Jan 2004

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