Rational solutions and limit cycles of polynomial and trigonometric Abel equations

TL;DR

This paper investigates the maximum number of rational solutions and limit cycles in polynomial and trigonometric Abel differential equations, providing bounds based on the coefficients' properties.

Contribution

It establishes bounds on the number of rational solutions and limit cycles for Abel equations with polynomial and trigonometric coefficients, advancing understanding of their solution structures.

Findings

Bounds on the number of rational solutions for polynomial Abel equations.

Bounds on the number of rational limit cycles for trigonometric Abel equations.

Enhanced understanding of solution constraints in Abel differential equations.

Abstract

We study the Abel differential equation x0 = A(t)x3 + B(t)x2 +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.