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Homoclinic orbits for a nonperiodic Hamiltonian system. (English) Zbl 1117.37032
Consider the first order Hamiltonian system \(\overset{.}{z}={\mathcal J}H_{z}(t,z) \) where \(z=(p,q)\in {\mathbb R}^{2N},{\mathcal J} =\left( \begin{matrix} 0 & -I \\ I & 0 \end{matrix} \right) \) and \(H\in C^{1}({\mathbb R}\times {\mathbb R}^{2N},{\mathbb R}) \) has the form \( H\left( t,z\right)=\frac{1}{2}L\left( t\right) z\cdot z+R\left( t,z\right) \) with \(L\left( t\right) \) a continuous symmetric \(2N\times 2N\) matrix-valued function, \(R_{z}\left( t,z\right) =o\left(| z| \right) \) as \(z\rightarrow 0\) and asymptotically linear as \(| z| \rightarrow \infty \). A solution \(z\) of this system is a homoclinic orbit if \(z\neq 0\) and \(z\left( t\right) \rightarrow 0\) as \(| t| \rightarrow \infty \). The existence and multiplicity of homoclinic orbits is studied without assuming periodicity conditions.

MSC:
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
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