Integrability and attractors

Abstract

We discuss global and local integrability and also attractors from a physicist's standpoint. A local Lie method is used to form an analytic and geometric picture of integrability via conservation laws. The starting point is Lie's proof that two-dimensional flows, even dissipative ones, have a conservation law and so are integrable. Through conservation laws, integrable higher-dimensional flows reduce to the two-dimensional case. We consider a class of models defined by the Euler-Lagrange equations for a set of nonintegrable velocities (nonholonomic coordinates). The destruction of the global conservation laws of this conservative system by the inclusion of linear damping and constant external driving leads to self-confinement and attractors in phase space. In particular, the Lorenz model belongs to this class and can be seen as a damped, driven symmetric top.