Shadowing lemma

In the theory of dynamical systems, the shadowing lemma is a lemma describing the behaviour of pseudo-orbits near a hyperbolic invariant set. Informally, the theory states that every pseudo-orbit (which one can think of as a numerically computed trajectory with rounding errors on every step) stays uniformly close to some true trajectory (with slightly altered initial position)—in other words, a pseudo-trajectory is "shadowed" by a true one. This suggests that numerical solutions can be trusted to represent the orbits of the dynamical system. However, caution should be exercised as some shadowing trajectories may not always be physically realizable.