Homoclinic orbits for a class of Hamiltonian systems. (English) Zbl 0705.34054

Rabinowitz, Paul H.

Proc. R. Soc. Edinb., Sect. A 114, No. 1-2, 33-38 (1990).

The author proves, under certain conditions, the existence of homoclinic orbits emanating from 0 for the second order Hamiltonian systems \((*)\quad \ddot q+V_ q(t,q)=0,\) where \(q\in R^ n\) and \(V\in C^ 1(R\times R^ n,R)\) is T-periodic in t. The homoclinic solution q of (*) has been found as the limit, as \(k\to \infty\), of 2kT periodic solutions \(q_ k\). The approximating solutions \(q_ k\) are, in turn, obtained via the Mountain Pass Theorem.

Reviewer: N.Parhi

MSC:

34C37	Homoclinic and heteroclinic solutions to ordinary differential equations
34C05	Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C25	Periodic solutions to ordinary differential equations
70H05	Hamilton’s equations

Keywords:

homoclinic orbits; second order Hamiltonian systems; Mountain Pass Theorem

References:

[1]	DOI: 10.1016/0022-1236(73)90051-7 · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7
[2]	Rabinowitz, Anal. Nonlineaire 6 pp 331– (1989)
[3]	Lions, Rev. Mat. Iberoamericana 1 pp 145– (1985) · Zbl 0704.49005 · doi:10.4171/RMI/6
[4]	DOI: 10.1002/cpa.3160310203 · doi:10.1002/cpa.3160310203