transients that by diffusion smooth internal structures, the emergent behavior is one of relatively slow decay of the amplitude () at a rate controlled by the type of boundary condition (constant ). Notice that this slow manifold model is global in as each equilibria is necessarily in the slow subspace of each other equilibria, but is only local in parameter . We cannot yet be sure how large may be taken, but the theory assures us the results do hold for some finite parameter . Perhaps the simplest nontrivial stochastic slow manifold Stochastic modeling is much more complicated—this example illustrates just one such complication. Consider for small parameter the two variable dynamics of this linear system forced with noise from the random walk : One could simply notice that the Ornstein–Uhlenbeck process is formally the history integral and then assert that is simply the integral of this history integral. However, this solution then inappropriately contains fast time integrals, due to the in the integrand, in a supposedly long time model. Alternatively, a stochastic coordinate transform extracts a sound model for the long term dynamics. Change variables to where then the new variables evolve according to the simple In these new coordinates we readily deduce exponentially quickly, leaving undergoing a random walk to be the long term model of the stochastic dynamics on the stochastic slow manifold obtained by setting . A web service constructs such slow manifolds in finite dimensions, both deterministic and stochastic. See also Quasi-geostrophic equations References Dynamical systems