Homoclinic orbits to invariant tori of Hamiltonian systems. (English) Zbl 0847.58024

Kozlov, V. V. (ed.), Dynamical systems in classical mechanics. Transl., Ser. 2, Am. Math. Soc. 168(25), 21-90 (1995).
The author tries to extend a work by Poincaré on periodic orbits near an invariant circle to the multidimensional case. To prove the existence of a homoclinic orbit, instead of using the conservation of area as Poincaré did, the author reduces a smooth area-conserving integrable twist map to the Poincaré map of a time-periodic positive definite Lagrangian system by means of classical perturbation theory of Delaunay. The author also considers invariant tori of positive definite Lagrangian systems that are supports of action-minimizing invariant measures and proves the existence of semi-asymptotic and semi-homoclinic orbits to the invariant tori. Then the existence of homoclinics to perturbed invariant tori follows.
For the entire collection see [Zbl 0827.00018].
Reviewer: Y.Kozai (Tokyo)


37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion