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

The Hamiltonian system of ordinary differential equations \[ \dot x=J\nabla_ xH(t,x), \tag{1} \] where \(J\) denotes the \(2n\times 2n\) matrix \(J=\begin{pmatrix} 0 & I \\ -I & 0 \end{pmatrix}\) with \(J^*=J^{-1}=-J\), \(\nabla_ x H(t,x)\) denotes the vector \(\partial H(t,x)/\partial x\), \(1\leq i\leq 2n\), and the map \(H: \mathbb R\times \mathbb R^{2n}\to \mathbb R\) 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 \(| z(t)-x(t)| \to 0\) when \(t\to\pm\infty\).
The author simplifies the problem by applying Floquet’s theory to the linearized system around \(x\dot y=JH''(t,x(t))y\) using a variational approach for the solution.

MSC:

34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
58E30 Variational principles in infinite-dimensional spaces
34C25 Periodic solutions to ordinary differential equations
70H05 Hamilton’s equations
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References:

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