It is well-known how to compute global one-dimensional stable or unstable manifolds of a hyperbolic equilibrium of a smooth vector field. Such manifolds consist only of two trajectories and arbitrarily large pieces can 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 computation 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, stable manifold is in fact a strong stable manifold that is contained in a higher-dimensional stable 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. The theoretical analysis is illustrated with numerical computations.
|Publication status||Published - 2002|