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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.

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