Linear and nonlinear diffusion approximation of the slow motion in systems with two time scales

摘要

For a dynamical system with slow and fast variables the popular method of averaging enables one to derive an equation for the slow variables alone whose solution approximates the original slow motion on a finite time interval. If the fast variables are sufficiently "random" the error term in the averaging procedure is described by a central limit theorem, i.e., the scaled error is Gaussian in the weak limit and satisfies a linear SDE (linear diffusion, or Gaussian approximation). We will present an approximation of the slow motion by the solution of a nonlinear SDE (in meteorology known as Hasselmann's equation) which was recently proved by Yuri Kifer. Although this nonlinear diffusion approximation holds in general, as the previous one's, only on a finite time interval, it is at least in principle capable of correctly describing important long-term qualitative features of the slow motion. We present several examples which support the usefulness of the nonlinear diffusion approximation, including the Lorenz-Maas model from climatology.