Global isochronous potentials

TL;DR

This paper characterizes smooth nonlinear potentials that produce a globally isochronous center in a scalar differential equation, revisiting known potentials and introducing new explicit families with simple constructions.

Contribution

It provides a geometric characterization of such potentials and introduces new explicit examples, expanding the class of known isochronous systems.

Findings

Characterization of global isochronous potentials

Introduction of a new explicit family of potentials

Easy construction of implicit examples

Abstract

We present a geometric characterization of the nonlinear smooth functions $V: R\to R$ for which the origin is a global isochronous center for the scalar equation $\ddot x=-V'(x)$. We revisit Stillinger and Dorignac isochronous potentials $V$ and show a new simple explicit family. Implicit examples are easily produced.