On the cubic Kukles systems with an algebraic limit cycle of degree two

Abstract

This paper is concerned with cubic Kukles system of the form x˙=−y, y˙=x+λy+a1x2+a2xy+a3y2+a4x3+a5x2y+a6xy2+a7y3, where λ,ai∈R, i=1,2,…,7, under the assumption that the above system has an algebraic periodic orbit of degree 2. It is shown that such an algebraic periodic orbit surrounds a center if and only if λ=0. If λ≠0, then it is the unique limit cycle. Moreover, we provide all the possibilities for the global phase portrait in the Poincaré disk of cubic Kukles system with an algebraic periodic orbit of degree 2.