A potential system with infinitely many critical periods

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)].