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Homoclinic orbits for second order Hamiltonian systems with potential changing sign. (English) Zbl 0867.70012

We study the second order Hamiltonian system q ¨=-U ' (t,q), where q: N and U ' (t,q) denotes the gradient with respect to q of a smooth potential U:× N , T-periodic in time, having an unstable equilibrium point x ¯ for all t. Without loss of generality we can take T=1 and x ¯=0. Thus, q(t)0 is a trivial solution. We look for homoclinic orbits to 0, namely non-zero solutions of the problem

q ¨=-U ' (t,q),q(t)0ast±,q ˙(t)0ast±·(P)

The potential U has the form U(t,x)=-1 2x·L(t)x+V(t,x), where L and V satisfy some technical assumptions. We prove that the problem (P) admits infinitely many solutions.

70H05Hamilton’s equations
34C37Homoclinic and heteroclinic solutions of ODE
37J99Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems