Order and chaos, invariant tori, KAM theory, resonances, Arnold web, diffusion …, these are “keywords” in the theory of dynamical systems. But for mathematicians who are not directly involved in this area they may sound a bit vague. To grasp what they are about, consider the following question: Given a conservative mechanical system, thus without dissipative forces, what is its ultimate fate? Without further information the answer probably cannot be given in general. However, if we restrict ourselves to the special, but very important, case of a
slightly perturbed integrable system
, we can claim that crucial progresses have been achieved. An example is our solar system, in which the perturbations are the interactions between planets. In this book we will suggest how to guess an answer, and on the way all these keywords will come into play.