Splitting of Asymptotic Surfaces

Abstract

Non-degenerate hyperbolic invariant toriof Hamiltonian systems have asymptotic invariant manifolds filled by trajectories which tend to quasi-periodic orbits in the hyperbolic tori as t → ±∞. In integrable Hamiltonian systems such manifolds, as a rule, coincide. In the nonintegrable cases, the situation is different: asymptotic surfaces can have transverse intersections forming a complicated tangle. “One will be struck by the complexity of this figure which I do not even attempt to draw. Nothing more properly gives us an idea of the complication of the problem of three bodies and, in general, of all problems in Dynamics where there is no uniform integral…” (Poincaré [203]).