noisy dynamical systems (stochastic differential equation), as do also stochastic center, stable and unstable manifolds. Such stochastic slow manifolds are similarly useful in modeling emergent stochastic dynamics, but there are many fascinating issues to resolve such as history and future dependent integrals of the noise. Examples Simple case with two variables The coupled system in two variables and has the exact slow manifold on which the evolution is . Apart from exponentially decaying transients, this slow manifold and its evolution captures all solutions that are in the neighborhood of the origin. The neighborhood of attraction is, roughly, at least the half-space . Slow dynamics among fast waves Edward Norton Lorenz introduced the following dynamical system of five equations in five variables to explore the notion of a slow manifold of quasi-geostrophic flow Linearized about the origin the eigenvalue zero has multiplicity three, and there is a complex conjugate pair of eigenvalues, . Hence there exists a three-dimensional slow manifold (surrounded by 'fast' waves in the and variables). Lorenz later argued a slow manifold did not exist! But normal form arguments suggest that there is a dynamical system that is exponentially close to the Lorenz system for which there is a good slow manifold. Eliminate an infinity of variables In modeling we aim to simplify enormously. This example uses a slow manifold to simplify the 'infinite dimensional' dynamics of a partial differential equation to a model of one ordinary differential equation. Consider a field undergoing the nonlinear diffusion with Robin boundary conditions Parametrising the boundary conditions by empowers us to cover the insulating Neumann boundary condition case , the Dirichlet boundary condition case , and all cases between. Now for a marvelous trick, much used in exploring dynamics with bifurcation theory. Since parameter is constant, adjoin the trivially true differential equation Then in the extended state space of the evolving field and parameter, , there exists an infinity of equilibria, not just one equilibrium, with (insulating) and constant, say . Without going into details, about each and every equilibria the linearized diffusion has two zero eigenvalues and for all the rest are negative (less than ). Thus the two-dimensional dynamics on the slow manifolds emerge (see emergence) from the nonlinear diffusion no matter how complicated the initial conditions. Here one can straightforwardly verify the slow manifold to be precisely the field where amplitude evolves according to That is, after the initial