## Infinitely many homoclinic orbits of a Hamiltonian system with symmetry.(English)Zbl 0938.37034

The authors are interested in the multiplicity of homoclinic orbits of the Hamiltonian system $$\dot z=JH_z(t,z)$$, where $$z= (p,q) \in \mathbb{R}^N\times \mathbb{R}^N=\mathbb{R}^{2N}$$, $$J=\left( \begin{smallmatrix} 0 & -I\\ I & 0\end{smallmatrix} \right)$$ is the standard symplectic structure on $$\mathbb{R}^{2N}$$, $$H\in C(\mathbb{R}\times \mathbb{R}^{2N},\mathbb{R})$$ is 1-periodic in $$t\in \mathbb{R}$$ and $$H(t,z)={1\over 2} zL(t)z+W(t,z)$$, with $$L\in C(\mathbb{R},\mathbb{R}^{4N^2})$$ being $$2N\times 2N$$ symmetric valued function, and $$W\in C^1(R\times \mathbb{R}^{2N},\mathbb{R})$$ being superquadratic in $$z\in \mathbb{R}^{2N}$$. Under some natural (additional) conditions on the Hamiltonian, the authors prove the existence of infinitely many homoclinic orbits.

### MSC:

 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) 37N05 Dynamical systems in classical and celestial mechanics
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