CHAOS

We study the stability of Hamiltonian systems in classical mechanics with two degrees of freedom by renormalization-group methods. One of the key mechanisms of the transition to chaos is the break-up of invariant tori, which plays an essential role in the large scale and long-term behavior. The aim is to determine the threshold of break-up of invariant tori and its mechanism. The idea is to construct a renormalization transformation as a canonical change of coordinates, which deals with the dominant resonances leading to qualitative changes in the dynamics. Numerical results show that this transformation is an efficient tool for the determination of the threshold of the break-up of invariant tori for Hamiltonian systems with two degrees of freedom. The analysis of this transformation indicates that the break-up of invariant tori is a universal mechanism. The properties of invariant tori are described by the renormalization flow. A trivial attractive set of the renormalization transformation characterizes the Hamiltonians that have a smooth invariant torus. The set of Hamiltonians that have a non-smooth invariant torus is a fractal surface. This critical surface is the stable manifold of a single strange set encompassing all irrational frequencies. This hyperbolic strange set characterizes the Hamiltonians that have an invariant torus at the threshold of the break-up. From the critical strange set, one can deduce the critical properties of the tori (self-similarity, universality classes).

The connection between the thermodynamic description of transport phenomena and a microscopic description of the underlying chaotic motion has recently received new attention due to the convergence of ongoing developments in the theory of deterministic chaotic systems, in the foundation of non-equilibrium statistical physics and of non-equilibrium molecular dynamics simulations. An overview of these developments is given with an emphasis on explicit calculations on exactly solvable models, that may serve as paradigms for this approach to model transport.