Electron. J. Differential Equations, Vol. 2017 (2017), No. 217, pp. 1-7.

Explicit limit cycles of a family of polynomial differential systems

Rachid Boukoucha

Abstract:
We consider the family of polynomial differential systems
$$\displaylines{ x' = x+( \alpha y-\beta x) (ax^2-bxy+ay^2) ^{n}, \cr y' = y-( \beta y+\alpha x) (ax^2-bxy+ay^2) ^{n}, }$$
where a, b, $\alpha $, $\beta $ are real constants and n is positive integer. We prove that these systems are Liouville integrable. Moreover, we determine sufficient conditions for the existence of an explicit algebraic or non-algebraic limit cycle. Examples exhibiting the applicability of our result are introduced.

Submitted October 4, 2016. Published September 13, 2017.
Math Subject Classifications: 34A05, 34C05, 34CO7, 34C25.
Key Words: Planar polynomial differential system; first integral; Algebraic limit cycle; non-algebraic limit cycle.

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Rachid Boukoucha
Department of Technology
Faculty of Technology
University of Bejaia
06000 Bejaia, Algeria
email: rachid_boukecha@yahoo.fr

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