On the rational limit cycles of Abel equations

Abstract

In this paper, we deal with Abel equations: dxdy=A(x)y2+B(x)y3, where A(x) and B(x) are real polynomials. If a solution y=φ(x) of the above equations satisfies that φ(0)=φ(1), then we say that it is a periodic solution. If a periodic solution is isolated, then we call it a limit cycle. If a limit cycle y=φ(x) is a rational function but not a polynomial, then we call it a nontrivial rational limit cycle.Firstly, we study the existence of nontrivial rational limit cycles. We prove that there exist Abel equations, which have at least two nontrivial rational limit cycles, and there also exists other Abel equations, which have at least one nontrivial rational limit cycle and one non-rational limit cycle. Secondly, we discuss the relation between the existence of nontrivial rational limit cycle and the degrees of A(x) and B(x). Finally we show that the multiplicity of a nontrivial rational limit cycle can be unbounded.