Bifurcations of limit cycles from cubic Hamiltonian systems with a center and a homoclinic saddle-loop

Zhao, Yulin; Zhang, Zhifen

doi:10.5565/publmat_44100_08

Cited by 6 publications

(2 citation statements)

References 17 publications

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“…Obviously, system ð1:1Þ E is the perturbations of Hamiltonian vector field, whose Hamiltonian has the form H nþ1 ¼ y 2 2 þ P nþ1 ðxÞ; where P nþ1 ðxÞ is a polynomial in x of degree n þ 1: Many results can be found for n ¼ 1; 2; 3; see [12][13][14][15][16] for example. For nX5; there are few results.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Bifurcations of limit cycles from Quintic Hamiltonian systems with a double figure eight loop

Hong

Zhang

Chen

2004

Journal of Complexity

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This paper deals with Lie´nard equations of the form 'x ¼ y; ' y ¼ PðxÞ þ yQðx; yÞ; with P and Q polynomial of degree 5 and 4, respectively. Attention goes to perturbations of the Hamiltonian vector fields with an elliptic Hamiltonian of degree six, exhibiting a double figure eight-loop. It is proved that the Hopf cyclicity is two, and it is also given by the new configurations of the limit cycles bifurcated from the homoclinic loop or heteroclinic loop for quintic system with quintic perturbations by using the methods of bifurcation theory and qualitative analysis. r

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Bifurcations of limit cycles from Quintic Hamiltonian systems with a double figure eight loop

Hong

Zhang

Chen

2004

Journal of Complexity

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“…Clearly, a homoclinic loop can be either stable or unstable. A nonisolated homoclinic loop may appear as the boundary curve of period annuli [4,6,22]. In many cases a nonisolated homoclinic loop can generate an isolated loop under perturbations on a codimension one surfaces in the parameter space.…”

Section: Introductionmentioning

confidence: 99%

The loop quantities and bifurcations of homoclinic loops

Han

Zhu

2007

Journal of Differential Equations

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The stability and bifurcations of a homoclinic loop for planar vector fields are closely related to the limit cycles. For a homoclinic loop of a given planar vector field, a sequence of quantities, the homoclinic loop quantities were defined to study the stability and bifurcations of the loop. Among the sequence of the loop quantities, the first nonzero one determines the stability of the homoclinic loop. There are formulas for the first three and the fifth loop quantities. In this paper we will establish the formula for the fourth loop quantity for both the single and double homoclinic loops. As applications, we present examples of planar polynomial vector fields which can have five or twelve limit cycles respectively in the case of a single or double homoclinic loop by using the method of stability-switching.

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Linear estimate of the number of zeros of Abelian integrals for quadratic centers having almost all their orbits formed by cubics

Zhao

et al. 2002

Sci. China Ser. A-Math.

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Bifurcations of limit cycles from cubic Hamiltonian systems with a center and a homoclinic saddle-loop

Abstract: It is proved in this paper that the maximum number of limit cycles of systemis equal to two in the finite plane, where k >, 0 < | | 1, |α| + |β| + |γ| = 0. This is partial answer to the seventh question in [2], posed by Arnold.

Cited by 6 publications

References 17 publications

Bifurcations of limit cycles from Quintic Hamiltonian systems with a double figure eight loop

Bifurcations of limit cycles from Quintic Hamiltonian systems with a double figure eight loop

The loop quantities and bifurcations of homoclinic loops

Linear estimate of the number of zeros of Abelian integrals for quadratic centers having almost all their orbits formed by cubics

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