Rational limit cycles of Abel equations (Q2032077)

The study deals with Abel equations \[\frac{dy}{dx}=A(x)y^2+B(x)y^3,\] where \(x \in [0, 1]\) and \(y\) are real variables and \(A(x)\) and \(B(x)\) are polynomials. The authors prove that these Abel equations can have at most two rational limit cycles, that is, cycles representable in the form \(y(x)=q(x)/p(x)\), where \(p\) and \(q\) are coprime polynomials. This bound can be reached with nonhyperbolic limit cycles. Moreover, examples of these Abel equations with two nontrivial rational limit cycles are derived.