Numerical continuation of normally hyperbolic invariant manifolds

Abstract

This paper deals with the numerical continuation of invariant manifolds regardless of the restricted dynamics. Common examples of such manifolds include limit sets, codimension 1 manifolds separating basins of attraction (separatrices), stable/unstable/centre 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 graph transform and linear graph transform compute the perturbed manifold with its hyperbolic splitting. To globally discretize manifolds, a discrete tubular neighbourhood is used, induced by a transverse bundle composed of discrete stable and unstable bundles. This approach allows the development of the discrete graph transform/linear graph transform analogous to the usual smooth case. Convergence results are given. The discrete vector bundle construction and associated local k-plane interpolation may be of independent interest. A practical numerical implementation for solving the global equations underlying the graph transform is proposed. Relevant numerical techniques are discussed and computational tests included. An additional application is the computation of the ‘slow-transient’ surface of an enzyme reaction.