A global isochronous center is linear

TL;DR

This paper proves that a polynomial vector field with a punctured neighborhood of infinity foliated by closed orbits and a bounded period function must be linear, confirming that a global isochronous center is necessarily linear.

Contribution

It establishes that under certain conditions, a polynomial vector field with a global isochronous center must be conjugate to a linear center, answering a previously posed question.

Findings

A polynomial vector field with a bounded period function near infinity is linear.

A global isochronous center must be linear.

The result confirms that non-linear polynomial centers cannot have a bounded period function.

Abstract

Let $X$ be a polynomial vector field in $\mathbb{R}^2$ which, after one-point compactification of the plane, has a punctured neighbourhood $\dot U$ of the point at infinity which is foliated by closed orbits of $X$. If the period function of $X$ in $\dot U$ is bounded from below by a positive constant, $X$ is necessarily linear, hence conjugated, up to a nonzero constant factor, to $-y \frac{\partial}{\partial x} + x \frac{\partial}{\partial y}$. This result answers to a question posed by J. Llibre , proving e.g. that a global isochronous center is linear.