For such a system it was shown that, if the angle between the stable and unstable directions is uniformly bounded away from 0, then a noisy trajectory can be shadowed for all time. For non-hyperbolic systems, it appears that shadows may exist only for finite time. Refinement is an iterative process that perturbs each point of a noisy orbit in an attempt to produce a nearby orbit with less noise. A refinement iteration is successful if the trajectory before the iteration has noise and the trajectory after the iteration has noise , and , for some reasonable . Otherwise the iteration is unsuccessful .
Containment is a rigorous method to prove the existence of a shadow orbit.
See GHYS for details. Although this is a good area for further work, containment is beyond the scope of this thesis. QT make the distinction between dynamical noise and observational noise . Observational noise does not effect the future evolution of the system. In contrast, dynamical noise does effect the future evolution of the system.