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Homoclinic orbits of a Hamiltonian system. (English) Zbl 0997.37041
The authors are interested in the existence of homoclinic orbits of the Hamiltonian system \(\dot x= JH_z(t,z)\) where \(z=(p,q)\in \mathbb{R}^N\times \mathbb{R}^N\), \(J\) is the standard symplectic matrix in \(\mathbb{R}^{2N}\), \(J= \left( \begin{smallmatrix} 0 &-\text{Id}\\ \text{Id} &0 \end{smallmatrix} \right)\), and \(H\in C(\mathbb{R}\times \mathbb{R}^{2N},\mathbb{R})\) is 1-periodic in \(t\in \mathbb{R}\) and is of the form \(H(t,z)= \frac{1}{2} z\cdot L(t)z+ W(t,z)\) with \(L\in C(\mathbb{R}, \mathbb{R}^{4N^2})\) being \(2N\times 2N\) symmetric matrix valued function and \(W(t,z)\) satisfying a certain global superquadratic condition. The authors partly relax the assumption often used before: \(L\) is independent of \(t\) and \(\text{sp}(JL)\cap i\mathbb{R}= \emptyset\).

37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
70H05 Hamilton’s equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
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