## 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 $$\text{sp}(JL)\cap i\mathbb{R}= \emptyset$$.

### MSC:

 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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