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Multiple homoclinics for a class of singular Hamiltonian systems. (English) Zbl 0887.58017
The authors investigate a second order Hamiltonian system of the form $\ddot u + \nabla V(u) = 0$ in $\bbfR^n$, where the potential has a unique strict global maximum at the origin $p$ and a singular set $S \not\ni p$ such that $\bbfR^n \backslash S$ is open, path-connected and has non-trivial fundamental group $\pi_1 = G$. Some additional conditions on behaviour of $V$ near $S$ and $p$ are also assumed so that the class of systems under consideration includes, for instance, the classical $n$-body problem. Under some suitable conditions on $V$ and $G$ the authors prove the existence of multiple homoclinics, each one belonging to a different class of $G$.

MSC:
37J99Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
34C37Homoclinic and heteroclinic solutions of ODE
70F10$n$-body problems
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References:
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