Chapter 26

As we saw at the beginning of this course, the exponential growth of errors iterating a chaotic dynamical system implies that a computer generated trajectory from some initial condition will rapidly diverge from the true orbit due to truncation errors or approximations in the numerical integration scheme, so that after a relatively short time the computer generated orbit will have no correlation with the true orbit. This means, for example, that the relationship of numerical modeling of a physical system to the actual behavior is unclear. Our confidence in the modeling is partially rescued, at least in chaotic systems of moderate dimension, by the phenomenon of shadowing Shadowing is the existence of a true trajectory that remains close to the numerically produced trajectory (called the pseudo-trajectory) for very long times, This true trajectory will not in general be the one with the same initial condition as the numerical trajectory. The phenomenon of shadowing has been known for a long time for hyperbolic systems [1]. Recent work has addressed the question for nonhyperbolic physical attractors.