Conditions to the existence of center in planar systems and center for Abel equations

TL;DR

This paper investigates conditions under which planar systems and Abel equations have a center at the origin, providing new criteria and subclasses of systems with centers, especially when certain symmetry conditions are met.

Contribution

It establishes that Abel equations with odd functions have a center at the origin and introduces a new subclass of polynomial systems with centers based on these results.

Findings

Abel equations with odd functions have a center at the origin.

New subclass of polynomial systems with centers identified.

Conditions linking symmetry in functions to the existence of centers.

Abstract

Abel equations of the form $x'(t)=f(t)x^3(t)+g(t)x^2(t)$, $t \in [-a,a]$, where $a>0$ is a constant, $f$ and $g$ are continuous functions, are of interest because of their close relation to planar vector fields. If $f$ and $g$ are odd functions, we prove, in this paper, that the Abel equation has a center at the origin. We also consider a class of polynomial differential equations $\dot{x} = -y+P_n(x,y)$ and $\dot{y} = x+Q_n(x,y)$, where $P_n$ and $Q_n$ are homogeneous polynomials of degree $n$. Using the results obtained for Abel's equation, we obtain a new subclass of systems having a center at the origin.