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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.

Some multiplicity results of homoclinic solutions for second order Hamiltonian systems