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Bifurcations of limit cycles from a quintic Hamiltonian system with a heteroclinic cycle. (English) Zbl 1301.34057

The authors consider Liénard systems of the form \[ \frac{dx}{dt}=y, \frac{dy}{dt}=-(x+bx^3-x^5)+\varepsilon(\alpha+\beta x^2+\gamma x^4)y, \] where \(b\in \mathbb{R}\), \(0<|\varepsilon|\ll 1\) and triple \((\alpha,\beta,\gamma)\) belongs to bounded region \(D\subset \mathbb{R}^3\). Applying the investigation of isolated zeros for the related abelian integrals it is proved that for \(|b|\gg 1(b<0)\) the least upper bound for the number of limit cycles bifurcating from period annulus equals \(2\) (counting the multiplicity) and this upper bound is a sharp one.

MSC:

34C23 Bifurcation theory for ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
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