Existence of symmetric homoclinic orbits for systems of Euler-Lagrange equations. (English) Zbl 0589.34029

Toland, J. F.

Nonlinear functional analysis and its applications, Proc. Summer Res. Inst., Berkeley/Calif. 1983, Proc. Symp. Pure Math. 45/2, 447-459 (1986).

[For the entire collection see Zbl 0583.00018.]
This paper gives the original version of some theorems in \({\mathbb{R}}^ 2\) which were subsequently generalised. The basic observation is that certain dynamical systems with indefinite Hamiltonian structure have orbits which are monotone in position space (not configuration or phase space), and that as a consequence the existence of homoclinic orbits can be inferred for the general theory [see H. Hofer and the author, Math. Ann. 268, 387-403 (1984; Zbl 0569.70017)].

MSC:

34C15	Nonlinear oscillations and coupled oscillators for ordinary differential equations
37-XX	Dynamical systems and ergodic theory
34A34	Nonlinear ordinary differential equations and systems, general theory
34C28	Complex behavior and chaotic systems of ordinary differential equations

Keywords:

monotone trajectory; Hamiltonian structure; homoclinic orbits