an:04196981
Zbl 0725.58017
S??r??, Eric
Existence of infinitely many homoclinic orbits in Hamiltonian systems
EN
Math. Z. 209, No. 1, 27-42 (1992).
00007656
1992
j
37J99
Hamiltonian systems; homoclinic orbits; chaos; variational problems; Palais-Smale condition; concentration-compactness
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.
E.S??r?? (Paris)