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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.

##### MSC:
 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
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##### References:
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