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)


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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[1] Coti Zelati, V.), Ekeland, I.) et Séré, E.) . - A variational approach to homoclinic orbits in Hamiltonian systems, Math. Ann.288 (1990), pp. 133-160. · Zbl 0731.34050
[2] Séré, E.) . - Une approche variationnelle au problème des orbites homoclines de systèmes hamiltonien, Preprint.
[3] Coti Zelati, V.) et Rabinowitz, P.H.) .- Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, Preprint. · Zbl 0744.34045
[4] Rabinowitz, P.H.) . - Homoclinic orbits for a class of Hamiltonian systems, Proc. Roy. Soc. Edinburgh114A (1990), pp. 33-38. · Zbl 0705.34054
[5] Ekeland, I.) .- A perturbation theory near convex Hamiltonian systems, Jour. Diff. Equat.50 (1983), pp. 407-440. · Zbl 0476.34035
[6] ARNOL’D .- Méthodes mathématiques de la mécanique classique, Editions Mir, Moscou, 1976. · Zbl 0385.70001
[7] Blot, J.) . - Thèse de 3ème cycle, Université Paris IX-Dauphine (1981).
[8] Albizzatti, A.) .- Sélection de phase par un terme d’excitation pour les solutions périodiques de certaines équations différentielles, C.R.A.S.Paris, 296 (1983), pp. 259-262. · Zbl 0524.34048
[9] Bahri, A.) et Beresticky, H.) .- Existence offorced oscillations for some nonlinear differential systems, Comm. Pure and App. Math. (1983). · Zbl 0588.34028
[10] Gaussens, E.) . - Thèse d’Etat, Université Paris IX-Dauphine (1984).
[11] Ambrosetti, E.), Coti Zelati, V.) et Ekeland, I.) .- Symmetry breaking in critical point theory and applications, Jour. Diff. Equat.67 (1987), pp. 165-184. · Zbl 0606.58043
[12] Lassoued, L.) . - Homogénéisation pour des système hamiltoniens, Cahiers de Ceremade n° 8606.
[13] Poincaré, H.) . - Les méthodes nouvelles de la mécanique celeste, Gauthier-Villars, Paris (1899). · JFM 30.0834.08
[14] Greenspan, B.D.) et Holmes, P.J.) .- Homoclinic orbits, subharmonics and global bifurcations in forced oscillations, (G.I Barenblatt, G. Iooss and D.D. Joseph, eds.) Pitman, Boston-London-Melburne (1983). · Zbl 0532.58019
[15] Melikov, V.K.) . - On the stability of the center for periodic perturbations, Transactions of the Moscow Mathematical Society12 (1963), pp. 1-57. · Zbl 0135.31001
[16] Hénon, M.) et Heiles, C.) . - The applicability of the third integral of motion : some numerical experiments, Astron. J.69 (1973).
[17] Smale, S.) .- Diffeomorphisms with many periodic points, Differential and Combinatorial Topology, S.S. Cairns (ed.), pp. 63-80. Princeton University Press, Princeton. · Zbl 0142.41103
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