Some multiplicity results of homoclinic solutions for second order Hamiltonian systems

where q : R → R , W ∈ C1(R × R ,R) and ∇W (t, q) denotes the gradient of W with respect to q ∈ R for every t ∈ R. We look for homoclinic solutions of (1.1), i.e., solutions q to (1.1) such that q(t)→ 0 as |t| → +∞. The existence of homoclinic solutions for Hamiltonian systems and their importance in the study of the behavior of dynamical systems have been recognized from Poincaré [12]. From their existence, one may, under certain conditions, infer the existence of chaos nearby on the bifurcation behavior of periodic orbits.