Simultaneous Bifurcation of Limit Cycles and Critical Periods

• Oliveira, Regilene D. S. ^[1] ; Sánchez-Sánchez , Iván ^[2] ; Torregrosa, Joan ^[2]
  1. [1] Universidade de São Paulo

     Universidade de São Paulo

     Brasil

     
  2. [2] Universitat Autònoma de Barcelona

     Universitat Autònoma de Barcelona

     Barcelona, España

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 21, Nº 1, 2022
• Idioma: inglés
• Enlaces
  • Texto completo
• Resumen

  • 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.

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