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An estimate of the number of zeros of an Abelian integral depending on a parameter and limiting cycles. (Russian) Zbl 0545.58038
The author studies the zeros of the integral of n-1 differential forms with polynomial coefficients along the family of closed surfaces associated with some polynomial. The main theorem gives an estimate of the number of zeros of this integral. The estimate given is dependent only on the degree of polynomials. The main theorem was used for the investigation of limiting cycles of polynomial perturbation of polynomial Hamiltonian systems on the plane. The problem of the estimation of the number of zeros of an Abelian integral and the number of limiting cycles of Hamiltonian systems was set by V. I. Arnol’d in 1976.
Reviewer: G.Osipenko

##### MSC:
 37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations 49Q15 Geometric measure and integration theory, integral and normal currents in optimization 37C75 Stability theory for smooth dynamical systems 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems