Bifurcation of limit cycles in small perturbations of a hyper-elliptic Hamiltonian system with two nilpotent saddles. (English) Zbl 1319.34067
Summary: We study the first-order Melnikov function for a planar near-Hamiltonian system near a heteroclinic loop connecting two nilpotent saddles. The asymptotic expansion of this Melnikov function and formulas for the first seven coefficients are given. Next, we consider the bifurcation of limit cycles in a class of hyper-elliptic Hamiltonian systems which has a heteroclinic loop connecting two nilpotent saddles. It is shown that this system can undergo a degenerate Hopf bifurcation and Poincaré bifurcation, which emerges at most four limit cycles in the plane for sufficiently small positive \(\varepsilon\).
The number of limit cycles which appear near the heteroclinic loop is discussed by using the asymptotic expansion of the first-order Melnikov function. Furthermore we give all possible distribution of limit cycles bifurcated from the period annulus.
The number of limit cycles which appear near the heteroclinic loop is discussed by using the asymptotic expansion of the first-order Melnikov function. Furthermore we give all possible distribution of limit cycles bifurcated from the period annulus.
MSC:
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34C07 | Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations |
| 34C08 | Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.) |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34C37 | Homoclinic and heteroclinic solutions to ordinary differential equations |
| 37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |
