Multibump solutions to possibly degenerate equilibria for almost periodic Lagrangian systems (Q1969069)

The authors consider Lagrangian systems described by a Lagrangian function \(L\) of the form \(L(q,v,t)={1\over 2}\langle A(q,t)v,v \rangle+ \langle b(q,t), v\rangle+ c(q,t)\), and study the existence of heteroclinic solutions, joining different equilibria which satisfy the asymptotic conditions \(q(t)\to \xi_\pm\) and \(\dot q(t)\to 0\) as \(t\to\pm \infty\) for any given \(\xi_-, \xi_+\in Z^N\). By using variational methods it is shown that the equations of motion corresponding to the above Lagrangian always admit infinitely many heteroclinic solutions, and that, given any two equilibria, there exist infinitely many chains of heteroclinics between them. Then the authors prove that, if a slowly oscillating term is added to the Lagrangian, the new system actually exhibits a multibump (chaotic) dynamics.