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Arnold diffusion: a variational construction. (English) Zbl 0910.58015
The author proposes a new method for studying the phenomenon of Arnold diffusion for perturbed integrable Hamiltonian systems. Usually, one tries to construct (lower dimensional) transition tori to create order one changes in the action variable for arbitrary small perturbations, but nearby resonances may cause breaking of the transition chain. The author’s variational approach is based on a generalization of the Aubry-Mather theory for twist maps and Mather’s theory on connecting orbits of action-minimizing sets. In such an approach, also cantori are used to describe transition and diffusion. To construct connecting orbits between action-minimizing orbits, a generalization is introduced of Peierl’s barrier functions, measuring the splitting of the stable and unstable manifold of normally hyperbolic invariant tori and cantori, and whose gradient (approximately) is the PoincarĂ©-Melnikov function.
The present paper does not contain the proofs of the theorems; they are announced for publication elsewhere.
Reviewer: W.Sarlet (Gent)

MSC:
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
70H05 Hamilton’s equations
70H30 Other variational principles in mechanics
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