Isochronicity and linearizability of a planar cubic system

Fernandes, Wilker; Romanovski, Valery G.; Sultanova, Marzhan; Tang, Yilei

doi:10.1016/j.jmaa.2017.01.058

Cited by 8 publications

(10 citation statements)

References 19 publications

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“…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. Consider the following Hamiltonian differential system on R 2…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the Topology of Isochronous Centers of Hamiltonian Differential Systems

Dong

Liu

Yang

2019

Int. J. Bifurcation Chaos

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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.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

On the Topology of Isochronous Centers of Hamiltonian Differential Systems

Dong

Liu

Yang

2019

Int. J. Bifurcation Chaos

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“…Isochronous center is one kind of the most interesting singularities of integrable differential systems and has been studied extensively for decades (see, e.g., [1,6,10] and references therein), especially for polynomial Hamiltonian differential systems as follows:…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On the transformations linearizing isochronous centers of Hamiltonian systems

Dong,

Zhang

2019

Preprint

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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.

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“…Differentiating with respect to t both sides of each equation of (2.2) we obtain Substituting in the above equations the expressions from (2.2) and (2.1), one computes the linearizability quantities i k , j k step-by-step (see e.g. [21] for more details). From (2.3) it is easy to see that the linearizability quantities i k , j k are polynomials in parameters a p,q , b p,q of system (2.1).…”

Section: Linearizability Of System (13)mentioning

confidence: 99%

Linearizability and critical period bifurcations of a generalized Riccati system

et al. 2017

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Abstract. In this paper we investigate the isochronicity and linearizability problem for a cubic polynomial differential system which can be considered as a generalization of the Riccati system. Conditions for isochronicity and linearizability are found. The global structure of systems of the family with an isochronous center is determined. Furthermore, we find the order of weak center and study the problem of local bifurcation of critical periods in a neighborhood of the center.

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Isochronicity and linearizability of a planar cubic system

Cited by 8 publications

References 19 publications

On the Topology of Isochronous Centers of Hamiltonian Differential Systems

On the Topology of Isochronous Centers of Hamiltonian Differential Systems

On the transformations linearizing isochronous centers of Hamiltonian systems

Linearizability and critical period bifurcations of a generalized Riccati system

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