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.

Number of Critical Periods for Perturbed Rigidly Isochronous Centers

This paper deals with the bifurcation of critical periods from a rigidly quartic isochronous center. It shows that under any small homogeneous perturbation of degree four, up to any order in [Formula: see text], there are at most two critical periods bifurcating from the periodic orbits of the unperturbed system, and the upper bound is sharp. In addition, we further prove that under any small polynomial perturbation of degree [Formula: see text], up to the first order in [Formula: see text], there are at most [Formula: see text] critical periods bifurcating from the periodic orbits of the unperturbed quartic system.

Limit Cycles for a Class of Piecewise Smooth Quadratic Differential Systems with Multiple Parameters

This paper considers a class of quadratic differential systems with an isochronous center under small piecewise smooth perturbations. Two perturbation parameters at different scales are included in the system. By using the first order Melnikov function, we obtain some new results on the number of small-amplitude limit cycles bifurcating around an isochronous center.

On the Limit Cycles for Continuous and Discontinuous Cubic Differential Systems

We study the number of limit cycles for the quadratic polynomial differential systemsx˙=-y+x2,y˙=x+xyhaving an isochronous center with continuous and discontinuous cubic polynomial perturbations. Using the averaging theory of first order, we obtain that 3 limit cycles bifurcate from the periodic orbits of the isochronous center with continuous perturbations and at least 7 limit cycles bifurcate from the periodic orbits of the isochronous center with discontinuous perturbations. Moreover, this work shows that the discontinuous systems have at least 4 more limit cycles surrounding the origin than the continuous ones.

Critical Period Bifurcation by Perturbing a Reversible Rigidly Isochronous Center with Multiple Parameters

This paper focuses on bifurcation of critical periods by perturbing a rigidly isochronous center with multiple parameters. First, we give expressions of period bifurcation functions (PBF for short) in the form of integrals, and then study the first PBF T1(ρ, λ) with a new method. Compared with the result in [Liu & Han, 2014], more critical periods can be found by our method.