Electron. J. Diff. Equ., Vol. 2015 (2015), No. 246, pp. 1-11.

Limit cycles bifurcating from the period annulus of a uniform isochronous center in a quartic polynomial differential system

Jackson Itikawa, Jaume Llibre

Abstract:
We study the number of limit cycles that bifurcate from the periodic solutions surrounding a uniform isochronous center located at the origin of the quartic polynomial differential system
$$ \dot{x}=-y+xy(x^2+y^2),\quad \dot{y}=x+y^2(x^2+y^2), $$
when perturbed in the class of all quartic polynomial differential systems. Using the averaging theory of first order we show that at least 8 limit cycles bifurcate from the period annulus of the center. Recently this problem was studied by Peng and Feng [9], where the authors found 3 limit cycles.

Submitted October 15, 2014. Published September 22, 2015.
Math Subject Classifications: 34A36, 34C07, 34C25, 37G15.
Key Words: Polynomial vector field; limit cycle; averaging method; periodic orbit; uniform isochronous center.

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  Jackson Itikawa
Departament de Matemàtiques
Universitat Autònoma de Barcelona
08193 Bellaterra, Barcelona, Catalonia, Spain
Fax +34 935812790. Phone +34 93 5811303
email: itikawa@mat.uab.cat
Jaume Llibre
Departament de Matemàtiques
Universitat Autònoma de Barcelona
08193 Bellaterra, Barcelona, Catalonia, Spain
Fax +34 935812790. Phone +34 93 5811303
email: jllibre@mat.uab.cat

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