Polynomial Differential Equations with Small coefficients

TL;DR

This paper establishes bounds on polynomial differential equations' coefficients to determine the number of complex and real periodic solutions, proving a recent conjecture and applying results to limit cycle counts.

Contribution

It provides explicit bounds on coefficients ensuring a specific number of periodic solutions and proves a recent conjecture on their count.

Findings

Explicit bounds guarantee n complex periodic solutions.

Improved bounds for real periodic solutions in most classes.

Proof of a recent conjecture on the number of periodic solutions.

Abstract

Classes of polynomial differential equations of degree n are considered. An explicit upper bound on the size of the coefficients are given which implies that each equation in the class has exactly n complex periodic solutions. In most of the classes the upper bound can be improved when we consider real periodic solutions. We present a proof to a recent conjecture on the number of periodic solutions. The results are used to give upper bounds for the number of limit cycles of polynomial two-dimensional systems.