A new method to determine isochronous center conditions for polynomial differential systems

Liu, Yirong; Huang, Wentao

doi:10.1016/s0007-4497(02)00006-4

Cited by 63 publications

(41 citation statements)

References 13 publications

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“…were defined in [9,12,13,15]. The following theorems could be used to compute focal values and periodic constants.…”

Section: Some Preliminary Resultsmentioning

confidence: 99%

Center and pseudo-isochronous conditions in a quasi analytic system

Zheng¹,

Li²

2016

J. Nonlinear Sci. Appl.

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The center conditions and pseudo-isochronous center conditions at origin or infinity in a class of non-analytic polynomial differential system are classified in this paper. By proper transforms, the quasi analytic system can be changed into an analytic system, and then the first 77 singular values and periodic constants are computed by Mathematics. Finally, we investigate the center conditions and pseudo-isochronous center conditions at infinity for the system. Especially, this system was investigated when λ

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“…were defined in [9,12,13,15]. The following theorems could be used to compute focal values and periodic constants.…”

Section: Some Preliminary Resultsmentioning

confidence: 99%

Center and pseudo-isochronous conditions in a quasi analytic system

Zheng¹,

Li²

2016

J. Nonlinear Sci. Appl.

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“…In [25,27], the authors defined complex center and complex isochronous center for system (2.8) and gave recursive algorithms to determine necessary conditions for an isochronous center. We now restate the definitions and algorithms.…”

Section: )mentioning

confidence: 99%

Bifurcation of Limit Cycles and Pseudo-Isochronicity at Degenerate Singular Point in a Septic System

Zhang

2010

Results. Math.

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This paper deals with the problems of bifurcation of limit cycles and pseudo-isochronous center conditions at degenerate singular point in a class of septic polynomial differential system. We solve the problems by an indirect method, i.e., we transform the degenerate singular point into an elementary singular point. Then we construct a septic system which allows the appearance of eight limit cycles in the neighborhood of degenerate singular point. Finally, we investigate the pseudo-isochronous center conditions at degenerate singular point for the system. As far as we know, this is the first time that an example of septic system with eight limit cycles bifurcating from degenerate singular point is given, and it is also the first time the pseudo-isochronous center conditions at degenerate singular point in a septic system are discussed. Mathematics Subject Classification (2000). Primary 34C05; Secondary 34C07.

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“…In [13], μ k is called the k-th singular point quantity of the origin of system (69). In [12], τ k is called the k-th period constant of the origin of system (69).…”

Section: Isochronicity Of Infinitymentioning

confidence: 99%

Bifurcation and Isochronicity at Infinity in a Class of Cubic Polynomial Vector Fields

Wang

Liu

2007

Acta Mathematicae Applicatae Sinica, English Series

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In this paper, we study the appearance of limit cycles from the equator and isochronicity of infinity in polynomial vector fields with no singular points at infinity. We give a recursive formula to compute the singular point quantities of a class of cubic polynomial systems, which is used to calculate the first seven singular point quantities. Further, we prove that such a cubic vector field can have maximal seven limit cycles in the neighborhood of infinity. We actually and construct a system that has seven limit cycles. The positions of these limit cycles can be given exactly without constructing the Poincaré cycle fields. The technique employed in this work is essentially different from the previously widely used ones. Finally, the isochronous center conditions at infinity are given.

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A new method to determine isochronous center conditions for polynomial differential systems

Cited by 63 publications

References 13 publications

Center and pseudo-isochronous conditions in a quasi analytic system

Center and pseudo-isochronous conditions in a quasi analytic system

Bifurcation of Limit Cycles and Pseudo-Isochronicity at Degenerate Singular Point in a Septic System

Bifurcation and Isochronicity at Infinity in a Class of Cubic Polynomial Vector Fields

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