An estimate of the number of zeros of an Abelian integral depending on a parameter and limiting cycles. (Russian) Zbl 0545.58038

Varchenko, A. N.

Funkts. Anal. Prilozh. 18, No. 2, 14-25 (1984).

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