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

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].


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