Do numerical orbits of chaotic dynamical processes represent true orbits?

• Published in: Journal of Complexity Vol. 3; no. 2; pp. 136 - 145
• Main Authors: Hammel, Stephen M, Yorke, James A, Grebogi, Celso
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 1987
• ISSN: 0885-064X; 1090-2708
• DOI: 10.1016/0885-064X(87)90024-0

Chaotic processes have the property that relatively small numerical errors tend to grow exponentially fast. In an iterated process, if errors double each iterate and numerical calculations have 50-bit (or 15-digit) accuracy, a true orbit through a point can be expected to have no correlation with a numerical orbit after 50 iterates. On the other hand, numerical studies often involve hundreds of thousands of iterates. One may therefore question the validity of such studies. A relevant result in this regard is that of Anosov and Bowen who showed that systems which are uniformly hyperbolic will have the shadowing property: a numerical (or noisy) orbit will stay close to (shadow) a true orbit for all time. Unfortunately, chaotic processes typically studied do not have the requisite uniform hyperbolicity, and the Anosov-Bowen result does not apply. We report rigorous results for nonhyperbolic systems: numerical orbits typically can be shadowed by true orbits for long time periods.