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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 $q\in \bbfR^{2}$ 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].

MSC:
37J45Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
34C37Homoclinic and heteroclinic solutions of ODE