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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References:

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