Rational Limit Cycles of Abel Differential Equations

TL;DR

This paper investigates the maximum number of rational limit cycles in Abel differential equations with trigonometric polynomial coefficients, establishing an upper bound based on the degree of the polynomial A(t).

Contribution

It provides a new upper bound on the number of rational limit cycles for Abel equations with trigonometric polynomial coefficients.

Findings

Maximum number of rational limit cycles is at most degree of A(t) plus one.

The result applies to Abel equations with real trigonometric polynomial coefficients.

Abstract

We study the number of rational limit cycles of the Abel equation $x'=A(t)x^3+B(t)x^2$, where $A(t)$ and $B(t)$ are real trigonometric polynomials. We show that this number is at most the degree of $A(t)$ plus one.