A versatile algorithm for computing invariant manifolds

Abstract

This paper deals with the numerical computation of invariant manifolds using a method of discretizing global manifolds. It provides a geometrically natural algorithm that converges regardless of the restricted dynamics. Common examples of such manifolds include limit sets, co-dimension 1 manifolds separating basins of attraction (separatrices), stable/unstable/center manifolds, nested hierarchies of attracting manifolds in dissipative systems and manifolds appearing in bifurcations. The approach is based on the general principle of normal hyperbolicity, where the graph transform leads to the numerical algorithms. This gives a highly multiple purpose method. The algorithm fits into a continuation context, where the graph transform computes the perturbed manifold. Similarly, the linear graph transform computes the perturbed hyperbolic splitting. To discretize the graph transform, a discrete tubular neighborhood and discrete sections of the associated vector bundle are constructed. To discretize the linear graph transform, a discrete (un)stable bundle is constructed. Convergence and contractivity of these discrete graph transforms are discussed, along with numerical issues. A specific numerical implementation is proposed. An application to the computation of the 'slow-transient' surface of an enzyme reaction is demonstrated. © 2006 Springer-Verlag Berlin Heidelberg.