## 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 $$\mathbb{R}^n$$, where the potential has a unique strict global maximum at the origin $$p$$ and a singular set $$S \not\ni p$$ such that $$\mathbb{R}^n \backslash S$$ 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:

 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations 70F10 $$n$$-body problems

### Keywords:

Hamiltonian system; homoclinic solution
Full Text:

### References:

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