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On the Topology of Isochronous Centers of Hamiltonian Differential Systems

In this paper, we study the topology of isochronous centers of Hamiltonian differential systems with polynomial Hamiltonian functions [Formula: see text] such that the isochronous center lies on the level curve [Formula: see text]. We prove that, in the one-dimensional homology group of the Riemann surface (removing the points at infinity) of level curve [Formula: see text], the vanishing cycle of an isochronous center cannot belong to a subgroup generated by those small loops such that each of them is centered at a removed point at infinity of having one of the two special types described in the paper, where [Formula: see text] is sufficiently close to [Formula: see text]. Besides, we present some topological properties of isochronous centers for a large class of Hamiltonian systems of degree [Formula: see text], whose homogeneous parts of degree [Formula: see text] contain factors with multiplicity of no more than [Formula: see text]. As applications, we study the nonisochronicity for some Hamiltonian systems with quite complicated forms which are usually very hard to handle by the classical tools.