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Can we trust in numerical computations of chaotic solutions of dynamical systems ?

2013; World Scientific; Linguagem: Inglês

10.1142/9789814434867_0004

1793-1010

René Lozi,

Chaos-based Image/Signal Encryption

World Scientific Series on Nonlinear Science Series ATopology and Dynamics of Chaos, pp. 63-98 (2013) No AccessCan we trust in numerical computations of chaotic solutions of dynamical systems ?René LoziRené LoziLaboratoire J.A. Dieudonné — UMR CNRS 7351, Université de Nice Sophia-Antipolis Parc Valrose, 06108 Nice Cedex 02, Francehttps://doi.org/10.1142/9789814434867_0004Cited by:32 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail Abstract: Since the famous paper of Edward Lorenz in 1963 numerical computations using computers play a central role in order to display and analyze solutions of nonlinear dynamical systems. By these means new structures have been emphasized like hyperbolic and/or strange attractors. However theoretical proofs of their existences are very difficult and limited to very special linear cases. Computer aided proofs are also complex and require special interval arithmetic analysis. Nevertheless, numerous researchers in several fields linked to chaotic dynamical systems are confident in the numerical solutions they found using popular software and publish without checking carefully the reliability of their results. In the simple case of discrete dynamical systems (e.g. Hénon map) there are concerns about the nature of what a computer find out: long unstable pseudo-orbits or strange attractors? The shadowing property and its generalizations which ensure that pseudo-orbits of a homeomorphism can be traceable by actual orbits even if rounding errors are not inevitable are not of great help in order to validate the numerical results. Continuous dynamical systems (e.g. Chua, Lorenz, Rössler) are even more difficult to handle in this scope and researchers have to be very cautious to back up theory with numerical computations. 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