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# Chaotic behavior of rapidly oscillating Lagrangian systems

• In the paper we prove that the Lagrangian system

$\ddot{q} = \alpha(\omega t) V'(q), \quad t \in \mathbb R, q \in \mathbb R^N,$ $\qquad\qquad (L_\omega)$

has, for some classes of functions $\alpha$, a chaotic behavior---more precisely the system has multi-bump solutions---for all $\omega$ large. These classes of functions include some quasi-periodic and some limit-periodic ones, but not any periodic function.
We prove the result using global variational methods.

Mathematics Subject Classification: 37J45, 37C29, 34C37, 70K44.

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