Numerical approximations of strong (un)stable manifolds

Abstract

The method of computing global one-dimensional stable or unstable manifolds of a hyperbolic equilibrium of a smooth vector field is well known. Such manifolds consist only of two trajectories and arbitrarily large pieces can, for example, be generated using an initial point close to the equilibrium on the linear approximation of the manifold. The attraction properties (in forward or backward time) of the local manifolds ensure that the computational error, which depends on the arclength of the computed piece, remains bounded. This paper discusses how these error bounds change as the equilibrium loses its hyperbolicity, or when the one-dimensional, say, unstable manifold is, in fact, a ‘strong’ unstable manifold that is contained in a higher-dimensional unstable manifold. For these cases, the local manifolds are not locally attracting either in forward or in backward time and the standard error bound does not work. We illustrate the theoretical analysis with numerical computations, using an example for which the global manifolds can be found explicitly, as well as more general vector fields where the true manifolds are not known.