On the transformations linearizing isochronous centers of Hamiltonian systems

Abstract

In this paper we study the transformations linearizing isochronous centers of planar Hamiltonian differential systems with polynomial Hamiltonian functions H(x, y) having only isolated singularities. Assuming the origin is an isochronous center lying on the level curve L 0 defined by H(x, y) = 0, we prove that, there exists a canonical linearizing transformation analytic on a simply-connected open set Ω with closure Ω = R 2 , if and only if, L 0 consists of only isolated points; furthermore, if the origin is the unique center, then the condition that L 0 consists of only isolated points implies that the corresponding canonical linearizing transformation can be analytically defined on the whole plane.