Dynamical Systems Seminar

Consider a fast-slow system of ordinary differential equations of the form \[ \dot x=a(x,y)+\epsilon^{-1}b(x,y), \qquad \dot y=\epsilon^{-2}g(y), \] where it is assumed that \(b\) averages to zero under the fast flow generated by \(g\). Here \(x\in\mathbb{R}^d\) and \(y\) lies in a compact manifold. We give conditions under which solutions \(x\) to the slow equations converge to solutions \(X\) of a \(d\)-dimensional stochastic differential equation as \(\epsilon\) goes to \(0\). The limiting SDE is given explicitly.

Our theory applies when the fast flow is Anosov or Axiom A, as well as to a large class of nonuniformly hyperbolic fast flows (including the one defined by the well-known Lorenz equations), and our main results do not require any mixing assumptions on the fast flow.

This is joint work with David Kelly and combines methods from smooth ergodic theory with methods from rough path theory.