EVENT DETAILS AND ABSTRACT

Title: Renormalization for the analysis of stability in Hamiltonian flows
Speaker: Professor C. Chandre
Speaker Info: CEA Saclay
Brief Description:
Special Note:
Abstract:

One of the main mechanism for the loss of stability in Hamiltonian flows is the break-up of invariant tori which plays an essential role in the large-scale dynamics. The aim of our approach is to determine the threshold of the break-up of invariant tori and its mechanism, by renormalization which is analogous to what has been done in the study of phase transition in statistical mechanics. The idea is to construct a renormalization transformation by a series of canonical changes of coordinates (performed on the Hamiltonian). This transformation combines an elimination of some non-resonant modes of the Hamiltonian (by perturbative techniques), with a rescaling of phase space that treats specifically the resonant modes of the Hamiltonian. This transformation acts as a microscope in phase space, i.e. it looks at the system at a smaller scale in phase space and at a longer time scale. From this renormalization transformation, one can deduce whether or not a torus exists, and the properties of critical tori (at the threshold of the break-up). For Hamiltonian systems with two degrees of freedom, the break-up of invariant tori is a universal mechanism and all the critical tori have statistically the same structure at all scales. The properties at criticality are deduced from a critical strange attractor of the renormalization transformation.