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

The Hamiltonian system of ordinary differential equations

x ˙=J x H(t,x),(1)

where J denotes the 2n×2n matrix J=0I-I0 with J * =J -1 =-J, x H(t,x) denotes the vector H(t,x)/x, 1i2n, and the map H:× 2n belongs to the class αC 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)|0 when t±.

The author simplifies the problem by applying Floquet’s theory to the linearized system around xy ˙=JH '' (t,x(t))y using a variational approach for the solution.


MSC:
34C37Homoclinic and heteroclinic solutions of ODE
37J45Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
58E30Variational principles on infinite-dimensional spaces
34C25Periodic solutions of ODE
70H05Hamilton’s equations
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