NUMERICAL STUDY OF MANIFOLD COMPUTATIONS

Abstract

In this paper we study the numerical accuracy of computing one-dimensional manifolds of (non-hyperbolic) equilibria in a planar vector field for which the manifolds are known explicitly. We consider the (strong) stable manifolds of a saddle, a sink, and a centre-stable equilibrium. Introduction The numerical approximation of stable or unstable manifolds is often used as a tool to investigate the global behaviour of a dynamical system. Most methods are designed for manifolds of hyperbolic equilibria and utilize the fact that integration can be done such that the manifold is (locally) an attractor. This means that a bounded approximation of the manifold can be computed as close to the true manifold as one likes. In this paper, we consider the cases where a one-dimensional (strong) stable manifold of a vector field is computed. This necessarily implies that it is the manifold of an equilibrium. We consider both hyperbolic and non-hyperbolic equilibria. The standard method to compute the one-dimensional stable manifold of an equilibrium in a vector field is to choose a point on the local manifold close to the equilibrium and integrate it backward in time; for example, see [3]. Most methods use the linear manifold as an approximation for the local manifold, because the (strong) stable manifold is tangent at the equilibrium to the corresponding (strong) stable eigenspace [2]. Our investigations use this method as well. If the equilibrium is hyperbolic and has only a one-dimensional stable manifold, then this manifold is an attractor in backward time, at least locally near the equilibrium. This means that initially the distance of points on the computed trajectory to the true manifold decays. This property is utilized to show that the computational error decays to zero as δ → 0, even though the integration time that is needed to reach a specific finite arclength evidently goes to infinity in the process; see [4]. If one considers a vector field with an equilibrium that is not hyperbolic, or if one is interested in finding the strong stable manifold of an equilibrium with more than one stable eigenvalue, then standard error bounds no longer apply. In this paper we investigate how the maximum error varies with δ using a two-dimensional vector field for which the manifolds are known explicitly. We consider the hyperbolic saddle, the hyperbolic sink and the case where the equilibrium has one zero and one stable eigenvalue.