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Existence of homoclinic orbits for asymptotically periodic systems involving Duffing-like equation. (English) Zbl 1041.37032
Summary: We are concerned with perturbations of the Hamiltonian system \[ \ddot q-L(t)q+W_q(t,q)=0,\quad t\in \mathbb R, \tag{HS} \] where \(q = (q_1,\dots, q_N) \in \mathbb R^N\), \(W \in C^1 (\mathbb R \times \mathbb R^N, \mathbb R)\), and \(L(t) \in C (\mathbb R,\mathbb R^{N^2})\) is a positive definite symmetric matrix. Variational arguments are used to prove the existence of homoclinic solutions for system (HS).

MSC:
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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