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Homoclinic and heteroclinic orbits for a class of Hamiltonian systems. (English) Zbl 0791.34042
The goal of this paper is to establish the existence of a rich structure of homoclinic and heteroclinic solutions for the Hamiltonian system \(\ddot q+V_ q(t,q)=0\), where the potential \(V(t,q)\) satisfies the conditions \((\text{V}_ 1)\;V \in C^ 2(\mathbb{R}^ 1 \times \mathbb{R}^ n, \mathbb{R}^ 1)\) and is \(T\)-periodic in \(t\); \((\text{V}_ 2)\) There is a point \(\xi \in \mathbb{R}^ n \backslash \{0\}\) such that \(\forall t \in \mathbb{R}^ 1\), (i) \(V(t,0)=0=V(t,\xi)\), (ii) \(V_ q (t,0)= 0= V_ q(t,\xi)\), (iii) \(V_{qq} (t,0)\) and \(V_{qq} (t,\xi)\) are negatively definite; \((\text{V}_ 3)\;V(t,q)<0\) for \(q \neq 0\), \(\xi\); \((\text{V}_ 4)\) There is a constant \(V_ 0<0\) such that \(\varlimsup_{| q | \to \infty} V(t,q) \leq V_ 0\). The technique of the proof is based on the calculus of variations and is useful to finding actual solutions of an equation near an approximate solution.

34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
58E99 Variational problems in infinite-dimensional spaces
Full Text: DOI
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