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Zbl 1117.37032
Ding, Yanheng; Jeanjean, Louis
Homoclinic orbits for a nonperiodic Hamiltonian system.
(English)
[J] J. Differ. Equations 237, No. 2, 473-490 (2007). ISSN 0022-0396

Consider the first order Hamiltonian system $\overset{.}\to{z}={\cal J}H_{z}(t,z)$ where $z=(p,q)\in {\mathbb R}^{2N},{\cal J} =\left( \matrix 0 & -I \\ I & 0 \endmatrix \right)$ 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, $R_{z}\left( t,z\right) =o\left(\vert z\vert \right)$ as $z\rightarrow 0$ and asymptotically linear as $\vert z\vert \rightarrow \infty$. A solution $z$ of this system is a homoclinic orbit if $z\neq 0$ and $z\left( t\right) \rightarrow 0$ as $\vert t\vert \rightarrow \infty$. The existence and multiplicity of homoclinic orbits is studied without assuming periodicity conditions.
[Mircea Crâşmăreanu (Iaşi)]
MSC 2000:
*37J45 Periodic, homoclinic and heteroclinic orbits, etc.

Keywords: Hamiltonian system; asymptotically linear

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