Homoclinics for a singular Hamiltonian system. (English) Zbl 0936.37035

Rabinowitz, Paul H.

Jost, Jürgen (ed.), Geometric analysis and the calculus of variations. Dedicated to Stefan Hildebrandt on the occasion of his 60th birthday. Cambridge, MA: International Press. 267-296 (1996).

The paper studies the existence of homoclinics for a family of Hamiltonian systems described by the differential equation \(\ddot q+V_{q}(t,q)=0\), where \(q\in \mathbb{R}^{2}\) and the potential \(V\) satisfies some conditions. In particular \(V(t,x)\) has a singularity as \(x\to \xi\) uniformly in \(t\). Using elementary minimization arguments the author shows that the system has a pair of solutions that are homoclinic to \(0\) and wind about \(\xi\) in a positive and a negative sense. Some criteria for the existence of further homoclinic solutions are given. When \(V\) is autonomous these solutions can be represented by a simple curve. Besides, the more general situation when the potential possesses multiple singularities is treated. It is shown that if \(V\) has \(k\) strong singularities at \(\xi_{1},\dots,\xi_{k}\) there exist at least \(k\) geometrically distinct solutions of the system homoclinic to \(0\).
For the entire collection see [Zbl 0914.00109].

Reviewer: Alexei Ivanov (St.Peterburg)

MSC:

37J45	Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
34C37	Homoclinic and heteroclinic solutions to ordinary differential equations

Keywords:

homoclinics; singular Hamiltonian system; geometrically distinct solutions