Abstract
We show that the phase portrait of a dynamical system near a stationary hyperbolic point is reproduced correctly by numerical methods such as one-step methods or multi-step methods satisfying a strong root condition. This means that any continuous trajectory can be approximated by an appropriate discrete trajectory, and vice versa, to the correct order of convergence and uniformly on arbitrarily large time intervals. In particular, the stable and unstable manifolds of the discretization converge to their continuous counterparts.
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