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Existence of infinitely many homoclinic orbits in Hamiltonian systems. (English) Zbl 0725.58017
We consider a Hamiltonian system in ${\bbfR}\sp{2N}$, $z'=J\nabla\sb zH(t,z)$, H being 1-periodic in time, and 0 being a hyperbolic rest point. Under global assumptions on H, we prove that there are always infinitely many orbits homoclinic to 0, i.e. such that $z(\pm \infty)=0$. Those orbits are geometrically distinct, in the following sense: $$ (x,y\text{ are geometrically distinct}),\quad \Leftrightarrow \quad (\forall n\in {\bbfZ}:\ x(.)\ne y(.-n)). $$ The approach we use here is variational, and no transversality hypothesis is needed.
Reviewer: E.Séré (Paris)

MSC:
37J99Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems
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