Simultaneous Bifurcation of Limit Cycles and Critical Periods

Regilene D. S. Oliveira, Iván Sánchez-Sánchez, Joan Torregrosa

Research output: Contribution to journalArticleResearchpeer-review

Abstract

The present work introduces the problem of simultaneous bifurcation of limit cycles and critical periods for a system of polynomial differential equations in the plane. The simultaneity concept is defined, as well as the idea of bi-weakness in the return map and the period function. Together with the classical methods, we present an approach which uses the Lie bracket to address the simultaneity in some cases. This approach is used to find the bi-weakness of cubic and quartic Liénard systems, the general quadratic family, and the linear plus cubic homogeneous family. We finish with an illustrative example by solving the problem of simultaneous bifurcation of limit cycles and critical periods for the cubic Liénard family.

Original languageEnglish
Article number20
Number of pages35
JournalQualitative Theory of Dynamical Systems
Volume21
Issue number1
DOIs
Publication statusPublished - 1 Mar 2022

Keywords

  • Critical periods
  • Limit cycles
  • Simultaneous bifurcation

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