General study on limit cycle bifurcation near a double homoclinic loop. (English) Zbl 1516.34068
In this paper, the authors study the maximal number of limit cycles bifurcating from a general double homoclinic loop. They assume that the unperturbed system is a Hamiltonian system containing a double homoclinic loop and there exist three families of periodic orbits located inside and outside the double loop. Upon perturbation, the number of zeros of three Melnikov functions and the coefficients of some analytic functions are computed in order to find a lower bound of the maximal number of limit cycles.
An example on finding the maximal number of limit cycles of a particular polynomial differential system is provided to illustrate the application of the theory. The results obtained in this paper are useful in the investigation of the second part of Hilbert’s 16th problem.
An example on finding the maximal number of limit cycles of a particular polynomial differential system is provided to illustrate the application of the theory. The results obtained in this paper are useful in the investigation of the second part of Hilbert’s 16th problem.
Reviewer: Kwok-wai Chung (Hong Kong)
MSC:
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34C37 | Homoclinic and heteroclinic solutions to 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 |
| 37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |
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