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Global Phase Portraits of Kukles Differential Systems with Homogeneous Polynomial Nonlinearities of Degree 6 Having a Center and Their Small Limit Cycles

Jaume Llibre

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193, Bellaterra, Barcelona, Catalonia, Spain

E-mail Address: jllibre@mat.uab.cat

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and

Maurício Fronza da Silva

Departamento de Matemática, Universidade Federal de Santa Maria, 97110-820, Santa Maria, RS, Brazil

E-mail Address: mauriciofronzadasilva@hotmail.com

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https://doi.org/10.1142/S0218127416500449Cited by:6 (Source: Crossref)

Abstract

We provide the nine topological global phase portraits in the Poincaré disk of the family of the centers of Kukles polynomial differential systems of the form $ẋ = - y,$ $ẏ = x + a x^{5} y + b x^{3} y^{3} + c x y^{5},$ where $x, y \in ℝ$ and $a, b, c$ are real parameters satisfying $a^{2} + b^{2} + c^{2} \neq 0 .$ Using averaging theory up to sixth order we determine the number of limit cycles which bifurcate from the origin when we perturb this system first inside the class of all homogeneous polynomial differential systems of degree $6,$ and second inside the class of all polynomial differential systems of degree $6 .$

Keywords:

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