A potential system with infinitely many critical periods

TL;DR

This paper introduces a non-polynomial potential system with infinitely many critical periodic orbits, demonstrating the existence of infinitely many periodic solutions using variational methods and Bessel functions, thus supporting Dumortier's conjecture.

Contribution

It presents a novel analytical potential system with infinitely many critical orbits, providing a concrete example related to a longstanding conjecture.

Findings

Existence of infinitely many 2π-periodic solutions

Use of variational methods and Bessel functions in proof

Supports Dumortier's conjecture

Abstract

In this paper, we propose an analytical non-polynomial potential system which has infinitely many critical periodic orbits in phase plane. By showing the existence of infinitely many $2\pi-$ periodic solutions, the proof bases on variational methods and the properties of Bessel function. The result provides an affirmative example to Dumortier's conjecture [Nonlinear Anal. 20(1993)].