Patrick McSharry, University of Oxford and London School of Economics

Abstract: With the discovery of chaos came the hope of finding simple models that would be capable of explaining complex phenomena. Numerous papers claimed to find low-dimensional chaos in a number of areas ranging from the weather to the stock market. Years later, many of these claims have been disproved and the fantastic hopes pinned on chaos have been toned down as research with more realistic objectives follows. The difficulty in calculating reliable estimates of the correlation dimension and the maximal Lyapunov exponent, two of the hallmarks of chaos, are explored. Given that nonlinear dynamics is a relatively new and growing field of science, the need for statistical testing is greater than ever. Surrogate data provides one possible approach but great care is needed in generating relevant surrogates and in interpreting the results. Examples of misleading applications and challenges for the future of research in nonlinear dynamics are discussed.

Keywords: chaos, nonlinear dynamics, correlation dimension, Lyapunov exponent, surrogates