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On the number of zeros of abelian integral for a class of cubic Hamiltonian systems. (English) Zbl 1440.34035

The author considers a class of cubic Hamiltonian systems and, using perturbation methods, determines the maximal number of zeros of the Abelian integral for such a system. The obtained results are applied for solving the limit cycle problem of polynomial dynamical systems.

MSC:

34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.)
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
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