On the Approximation of Stable and Unstable Fiber Bundles of (Non)Autonomous ODEs — A Contour Algorithm

Abstract

We propose an algorithm for the approximation of stable and unstable fibers that applies to autonomous as well as to nonautonomous ODEs. The algorithm is based on computing the zero-contour of a specific operator; an idea that was introduced in [Hüls, 2006] for discrete time systems. We present precise error estimates for the resulting contour algorithm and demonstrate its efficiency by computing stable and unstable fibers for a (non)autonomous pendulum equation in two space dimensions. Our second example is the famous three-dimensional Lorenz system for which several approximations of the two-dimensional Lorenz manifold are calculated. In both examples, we observe equally well performance for autonomously and nonautonomously chosen parameters.