Explicit limit cycles of a family of polynomial differential systems

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.

Keywords

• Planar polynomial differential system
• first integral
• Algebraic limit cycle
• non-algebraic limit cycle