99-365 Patrick BERNARD

Homoclinic orbits in families of hypersurfaces with hyperbolic periodic orbits. (57K, Latex 2e with PS figure) Sep 30, 99

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Abstract. We consider a Hamiltonian system in C^n having an invariant plane with harmonic oscillations on it. We assume that this plane is hyperbolic, which means that the oscillations are hyperbolic with respect to their energy shell. We use variational theory to prove that, under global hypothesis, many of these oscillations have an homoclinic orbit, more precisely we prove that the periodic orbits having an homoclinic are dense in the invariant plane outside of a compact set. The major difficulty here is that the periodic orbits are not hyperbolic with repect to the full phase space.

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