Symbolic dynamics for Henon--Heiles Hamiltonian on the critical level. A computer assisted proof.

Piotr Zgliczynski, CDSNS, School of Mathematics, Georgia Institute of Technology

The Henon-Heiles hamiltonian is given by

\begin{equation} H=\frac{p_{x}^{2}+p_{y}^{2}}{2}+\frac{x^{2}+y^{2}}{2}+x^{2}y-\frac{y^{3}}{3} \label{eq:ham}% \end{equation}

We present a computer assisted proof of the existence of a rich symbolic dynamic structure for the H\'{e}non--Heiles Hamiltonian system at the critical energy $E=1/6$.

We construct a symbolic dynamics on 20+ symbols. We give an explicit positive bound for topological entropy and we show that the suitable Poincare map has periodic points of all periods but 3. We use the topological method which we call {\em topological hyperbolicity}. A basic feature of our procedure is that we do not have any assumption involving derivatives. In fact this method has a lot in common with the classical approach used for dealing with Smale's horseshoes. From the assumptions used there, we drop all those involving derivatives (cone conditions). We keep the assumptions concerning how the image of some sets is located with respect to other sets. By this method, given an apparent transversal intersection of the approximate stable and unstable manifolds, we are able to explicitly build the sets on which the symbolic dynamics is obtained, but we do not need to verify what is the exact relations between our approximate invariant manifolds and the true invariant manifolds.