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A variational approach to homoclinic orbits in Hamiltonian systems. (English) Zbl 0731.34050

The Hamiltonian system of ordinary differential equations

$\stackrel{˙}{x}=J{\nabla }_{x}H\left(t,x\right),\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $J$ denotes the $2n×2n$ matrix $J=\left(\begin{array}{cc}0& I\\ -I& 0\end{array}\right)$ with ${J}^{*}={J}^{-1}=-J$, ${\nabla }_{x}H\left(t,x\right)$ denotes the vector $\partial H\left(t,x\right)/\partial x$, $1\le i\le 2n$, and the map $H:ℝ×{ℝ}^{2n}\to ℝ$ belongs to the class $\alpha {C}^{2}$, are considered in this paper.

The main purpose is to study the so-called doubly asymptotic solutions. We recall that if $x$ is a periodic solution of the system (1), another solution $z$ will be called doubly asymptotic to $x$ if $|z\left(t\right)-x\left(t\right)|\to 0$ when $t\to ±\infty$.

The author simplifies the problem by applying Floquet’s theory to the linearized system around $x\stackrel{˙}{y}=J{H}^{\text{'}\text{'}}\left(t,x\left(t\right)\right)y$ using a variational approach for the solution.

##### MSC:
 34C37 Homoclinic and heteroclinic solutions of ODE 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods 58E30 Variational principles on infinite-dimensional spaces 34C25 Periodic solutions of ODE 70H05 Hamilton’s equations
##### References:
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