Extension of Poincare's program for integrability, chaos and bifurcations

Abstract

We will review the achievements of Henri Poincar e in the theory of dy- namical systems and will add a number of extensions and generalizations of his results. It is pointed out that the attention given to two degrees-of-freedom Hamiltonian sys- tems is rather deceptive as near stable equilibrium such systems play a special part. We illustrate Poincar e's theory of critical exponents for the Hamiltonian (1 : 2 : 2)- resonance. To assess the measures of chaos, asymptotic estimates in terms of magni- tude and timescales can be given. Another of Poincar e's topic, bifurcations, is brie y reviewed.