Rational solutions and limit cycles of polynomial and trigonometric Abel equations

Abstract

e study the Abel differential equation $x'=A(t)x^3+B(t)x^2+C(t)x$. Specifically, we find bounds on the number of its rational solutions when $A(t), B(t)$ and $C(t)$ are polynomials with real or complex coefficients; and on the number of rational limit cycles when $A(t), B(t)$ and $C(t)$ are trigonometric polynomials with real coefficients.

Keywords

• limit cycle
• abel equation
• invariant curve