Existence of infinitely many homoclinic orbits in Hamiltonian systems. (English) Zbl 0725.58017

We consider a Hamiltonian system in \({\mathbb{R}}^{2N}\), \(z'=J\nabla_ zH(t,z)\), 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: \[ (x,y\text{ are geometrically distinct}),\quad \Leftrightarrow \quad (\forall n\in {\mathbb{Z}}:\;x(.)\neq y(.-n)). \] The approach we use here is variational, and no transversality hypothesis is needed.
Reviewer: E.Séré (Paris)


37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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