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Limit cycles bifurcating from a degenerate center

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  • Llibre, Jaume
  • Pantazi, Chara

Abstract

We study the maximum number of limit cycles that can bifurcate from a degenerate center of a cubic homogeneous polynomial differential system. Using the averaging method of second order and perturbing inside the class of all cubic polynomial differential systems we prove that at most three limit cycles can bifurcate from the degenerate center. As far as we know this is the first time that a complete study up to second order in the small parameter of the perturbation is done for studying the limit cycles which bifurcate from the periodic orbits surrounding a degenerate center (a center whose linear part is identically zero) having neither a Hamiltonian first integral nor a rational one. This study needs many computations, which have been verified with the help of the algebraic manipulator Maple.

Suggested Citation

  • Llibre, Jaume & Pantazi, Chara, 2016. "Limit cycles bifurcating from a degenerate center," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 120(C), pages 1-11.
    • Handle: RePEc:eee:matcom:v:120:y:2016:i:c:p:1-11
    DOI: 10.1016/j.matcom.2015.05.005
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    Cited by:

    1. Giné, Jaume, 2016. "Center conditions for nilpotent cubic systems using the Cherkas method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 129(C), pages 1-9.
    2. Dias, Fabio Scalco & Llibre, Jaume & Valls, Claudia, 2018. "Polynomial Hamiltonian systems of degree 3 with symmetric nilpotent centers," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 144(C), pages 60-77.
    3. Lavinia Bîrdac & Eva Kaslik & Raluca Mureşan, 2022. "Dynamics of a Reduced System Connected to the Investigation of an Infinite Network of Identical Theta Neurons," Mathematics, MDPI, vol. 10(18), pages 1-17, September.

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