Extended systems: the impact of nonlinear science

Abstract

The transition to turbulence has been one of the most successful fields of application of nonlinear science in the past decade. Indeed, the identification of scenatios of transition to turbulence both in theory and experiment is an outstanding achievement. However this is believed to be relevant for “small geometries” or “small boxes” because of the restriction of the theory to dynamical systems with a few degrees of freedom only. It is perhaps not yet so widely known that remarkable progress has also occured in our understanding of the transition in extended systems. Two scenarios of transitions have been predicted theoretically and observed too: the transition by spatio-temporal intermittency and the transition by defect nucleation. Moreover we are beginning to understand key features of extended dynamical systems without gradient flow structures, that are relevant for nonequilibrium physics. I shall review the basic concept allowing to understand those nongradient flow systems, with a particular emphasis on the situation of subcritical bifurcation, and then show briefly how this does apply to a curious phenomenon where the direction of the motion of a front depends on its inner structure, contrary to one of the most fundamental properties of gradient flows systems. This is an example borrowed from solid state physics.