Constraint-defined Manifolds: a Legacy Code Approach to Low-dimensional Computation

Abstract

If the dynamics of an evolutionary differential equation system possess a low-dimensional, attracting, slow manifold, there are many advantages to using this manifold to perform computations for long term dynamics, locating features such as stationary points, limit cycles, or bifurcations. Approximating the slow manifold, however, may be computationally as challenging as the original problem. If the system is defined by a legacy simulation code or a microscopic simulator, it may be impossible to perform the manipulations needed to directly approximate the slow manifold. In this paper we demonstrate that with the knowledge only of a set of ``slow'' variables that can be used to {\it parameterize} the slow manifold, we can conveniently compute, using a legacy simulator, on a nearby manifold. Forward and reverse integration, as well as the location of fixed points are illustrated for a discretization of the Chafee-Infante PDE for parameter values for which an Inertial Manifold is known to exist, and can be used to validate the computational results.