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The cyclicity of period annulus of degenerate quadratic Hamiltonian systems with polycycles \(S^{(2)}\) or \(S^{(3)}\) under perturbations of piecewise smooth polynomials. (English) Zbl 1462.34068

Summary: In this paper, by using Picard-Fuchs equations and Chebyshev criterion, we study the bifurcation of limit cycles for degenerate quadratic Hamilton systems with polycycles \(S^{(2)}\) or \(S^{(3)}\) under the perturbations of piecewise smooth polynomials with degree \(n\). Roughly speaking, for \(n\in\mathbb{N}\), a polycycle \(S^{(n)}\) is cyclically ordered collection of \(n\) saddles together with orbits connecting them in specified order. The discontinuity is on the line \(y=0\). If the first order Melnikov function is not equal to zero identically, it is proved that the upper bounds of the number of limit cycles bifurcating from each of the period annuli with the boundary \(S^{(2)}\) and \(S^{(3)}\) are respectively \(25n+149\) and \(25n+115\) (taking into account the multiplicity).

MSC:

34C23 Bifurcation theory for ordinary differential equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
34E10 Perturbations, asymptotics of solutions to ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
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