## 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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