Linear estimate for the number of zeros of Abelian integrals for quadratic isochronous centres. (English) Zbl 0999.34032
The authors provide an explicit and fairly accurate upper bound on the number of zeros of Abelian integrals defined by quadratic isochronous centres when they are perturbed within the class of all polynomial systems of degree \(n.\) The technique that the authors use is a classical one. It consists in writing the non-Hamiltonian quadratic isochronous centre in a Hamiltonian form, multiplying the non-Hamiltonian system by an integrating factor, and then to use the method based on computing zeros of an Abelian integral to determine the limit cycles bifurcating from periodic orbits of the centre. The key point in this approach is that, by using Green’s theorem, the authors compute the Abelian integral as a double integral.
Reviewer: V.A.Gaiko (Minsk)
MSC:
| 34C08 | Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.) |
| 37G15 | Bifurcations of limit cycles and periodic orbits in dynamical systems |
| 37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
