A Boundary Value Problem with a Nonlocal Condition for a by Sapagovas M.P.

By Sapagovas M.P.

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A. S. Salomao, Homoclinic orbits near saddle-center fixed points of Hamiltonian systems with two degrees of freedom, preprint, 2002. [4] M. Berti, L. Biasco, and P. Bolle, Drift in phase space: a new variational mechanism with optimal diffusion time, J. Math. Pures Appl. (9) 82 (2003), no. 6, 613–664. MR 1 996 776 [5] M. Berti and P. Bolle, A functional analysis approach to Arnold diffusion, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire 19 (2002), no. 4, 395–450. MR 2003g:37105 [6] U. Bessi, An approach to Arnold’s diffusion through the calculus of variations, 26 (1996), 1115–1135.

We also have that, since q˜N jumps from η0 to 2kπ − η0 in a subinterval of the compact set [0, 2k π/ω], q(t) ≡ 0 and satisfies point (b)–(d). 2 that y˜N converges weakly in H 1 (a, b) and uniformly on compact sets to y. It is then easy to prove that (y, q) is a solution of q¨(t) = (1 + δ(R cos ωt + y))V (q(t)) y¨ + ω 2 y = δ (R cos ωt + y)V (q(t)) Using also the fact that qN (t) is exponentially decreasing to 0 (increasing to 2kπ) outside the set [0, 2k π/ω], one then deduce that q(t) → 0 as t → −∞, q(t) → 2kπ as t → +∞, and that y(t − n) tends, as n → ±∞, to a solution of y¨ + ω 2 y = 0, say ρ± cos(ωt + ϕ± ) as n → ±∞.

H. Rabinowitz, A variational construction of chaotic trajectories for a reversible Hamiltonian system, J. Differential Equations 148 (1998), no. 2, 364–387. MR 99m:58143 [10] , Minimal heteroclinic geodesics for the n-torus, Calc. Var. Partial Differential Equations 9 (1999), no. 2, 125–139. MR 2000g:37100 [11] E. Bosetto and E. Serra, A variational approach to chaotic dynamics in periodically forced nonlinear oscillators, Ann. Inst. H. Poincar´ e Anal. Non Lin´ eaire 17 (2000), no. 6, 673–709.

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