Henri Poincaré's neglected ideas

Abstract

The purpose of this article is to discuss two basic ideas of Henri Poincare in the theory of dynamical systems. The first one, the recurrence theorem, got at first a lot of attention but most scientists lost interest when finding out that long timescales were involved. We will show that recurrence can be a tool to find complex dynamics in resonance zones of Hamiltonian systems; this is related to the phenomenon of quasi-trapping. To demonstrate the use of recurrence phenomena we will explore the \begin{document}$ 2:2:3 $\end{document} Hamiltonian resonance near stable equilibrium. This will involve interaction of low and higher order resonance. A second useful idea is concerned with the characteristic exponents of periodic solutions of dynamical systems. If a periodic solution of a Hamiltonian system has more than two zero characteristic exponents, this points at the existence of an integral of motion besides the energy. We will apply this idea to examples of two and three degrees-of-freedom (dof), the Henon-Heiles (or Braun's) family and the \begin{document}$ 1:2:2 $\end{document} resonance.