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Bifurcation of homoclinic orbits for Hamiltonian systems. (Bifurcation d’orbites homoclines pour les systèmes hamiltoniens.) (French) Zbl 0780.58034

The author considers a Hamiltonian ordinary differential equation imbedded in a dissipative family of nonautonomous differential equations depending periodically on the time. Otherwise a nonautonomous non- Hamiltonian perturbation of an autonomous Hamiltonian system is studied.
It is supposed that the nonperturbed system admits a hyperbolic periodic solution and homoclinic solutions to such a periodic orbit. A necessary and sufficient condition of bifurcation in the perturbated system, is given (existence of doubly-asymptotic solutions). An asymptotic development of the bifurcated solution and two examples are presented.
Since a long time this subject has been very popular in the research groups on “Nonlinear Dynamics” of the former Soviet Union. The author’s references do not know this relatively ancient contribution.
Reviewer: C.Mira (Toulouse)

MSC:

37G99 Local and nonlocal bifurcation theory for dynamical systems
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
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References:

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