Abstract
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.
Highlights
Introduction and Main ResultsIsochronous center is one of the most interesting singularities of planar integrable differential systems and has been studied extensively for decades, especially for Hamiltonian differential systems
We study the topology of isochronous centers of Hamiltonian differential systems with polynomial Hamiltonian functions H(x, y) such that the isochronous center lies on the level curve H(x, y) = 0
In the one-dimensional homology group of the Riemann surface of level curve H(x, y) = h, 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 h is sufficiently close to 0
Summary
Isochronous center is one of the most interesting singularities of planar integrable differential systems and has been studied extensively for decades (see, e.g. [Chavarriga & Sabatini, 1999; Fernandes et al, 2017; Han & Romanovski, 2012; Llibre & Valls, 2011] and references therein), especially for Hamiltonian differential systems. A basic problem is to describe which element the vanishing cycle of an isochronous center represents in the homology group of Ch. It is closely related to the properties of points at infinity on Ch. In reference [Gavrilov, 1997], the author proved that the vanishing cycle of an Morse isochronous center is homologous to a zero cycle on the Riemann surface of Ch, under certain assumptions on the Hamiltonian function H(x, y). In reference [Gavrilov, 1997], the author proved that if H(x, y) is a good polynomial having only simple singularities (classified according to the Coxeter groups Ak, Dk, E6, E7 and E8 (see [Arnold et al, 1985])), and the origin is a single critical point of Morse type which is isochronous on C0, the corresponding vanishing cycle represents a zero homology cycle on the Riemann surface of Ch. The precise definition of a good polynomial is given shortly below. Let us first provide some preliminaries such as definitions, notation and important lemmas, give the detailed proof of the main results
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