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Bifurcations of limit cycles from a quintic Hamiltonian system with a figure double-fish. (English) Zbl 1275.34049

Summary: In this paper, we consider Liénard systems of the form \[ \frac{dx}{dt}=y, \quad \frac{dy}{dt}=-\bigg(-x+\frac{21}{10}x^3\bigg)+\epsilon(\alpha+\beta x^2+\gamma x^4)y, \] where \(0 < |\epsilon| \ll 1\) and \((\alpha, \beta, \gamma) \in \mathbb R^3\). We prove that the least upper bound of the number of isolated zeros of the related abelian integrals \[ I(h)=\oint_{\Gamma_{h}}(\alpha+\beta x^2+\gamma x^4)ydx \] is 4 (counting the multiplicity) and this upper bound is a sharp one.

MSC:

34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.)
34E10 Perturbations, asymptotics of solutions to ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
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References:

[1] DOI: 10.1016/j.na.2007.02.039 · Zbl 1158.34024
[2] DOI: 10.1016/j.na.2007.10.054 · Zbl 1161.34329
[3] DOI: 10.1016/j.camwa.2009.12.024 · Zbl 1189.34072
[4] DOI: 10.1016/0022-0396(87)90122-7 · Zbl 0622.34033
[5] DOI: 10.1006/jdeq.2000.3977 · Zbl 1004.34018
[6] DOI: 10.1006/jdeq.2000.3978 · Zbl 1034.34036
[7] DOI: 10.1016/S0022-0396(02)00110-9 · Zbl 1056.34044
[8] DOI: 10.1016/S0022-0396(02)00111-0 · Zbl 1057.34015
[9] DOI: 10.1007/s10884-008-9108-3 · Zbl 1165.34016
[10] DOI: 10.1112/plms/s3-69.1.198 · Zbl 0802.58046
[11] DOI: 10.1006/jdeq.1996.0017 · Zbl 0849.34022
[12] DOI: 10.1088/0951-7715/15/6/310 · Zbl 1219.34042
[13] Pontryagin L., Zh. Exp. Th. Phys. 4 pp 234– (1934)
[14] DOI: 10.1016/j.jde.2010.11.004 · Zbl 1217.34051
[15] DOI: 10.1016/j.physleta.2006.05.031 · Zbl 1142.34327
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