Coping with Chaos deals with the evolution of the techniques for reconstructing chaotic attractors, computing their invariant statistical properties (principally dimension and Lyapunov exponents), and the use of these techniques for forcasting and control in dynamical systems. It contains an introductory section of five chapters which review succintly the theory of dynamics, dimension, symbolic dynamics, Lyapunov exponents and entropy, and the theory of embedding. The remainder of the book contains reprints of many the classic papers that contributed to this development. While the book is written by and for scientists in the physical sciences, and its mathematics is quite sophisticated, it is very clearly written and comprehensible to those of us in the social, psychological, and life sciences. It is a great book.
It is important for several reasons. First of all, it should help legitimize the word "chaos" for those who might have been concerned with its respectability. Secondly, the design and analysis of experiments in our fields has been insufficiently developed (Abraham, 1997). This book and the recent one by Abarbanel (1996) provide some very sophisticated techniques that should do much to fill this void and promote development of this embryonic field. Thirdly, for many of us whose mathematics are strained dealing with dynamics, this book can help us considerably. Its exposition is so clear that we can follow the basic concepts and improve our mathematics at the same time.
Part I, "Background" is a concise introduction to some basics of dynamics. The first chapter is a brief review of dynamical systems, attractors, and chaos. It gives an excellent summary of these basics, and clear destinctions between an invertible map (Hénon) and a noninvertible map (logistic), and between attractors from conservative Hamiltonian systems and nonconservative systems. The box-counting definition of fractal dimensions is given, and then a definition of chaos. It also shows how second-order nonautonomous equations can be converted into autonomous first order differential equations, and ends by providing a definition of Lyapunov exponents. Some terms may seem a bit formidable at first, but readily become quite clear and tractable.
Part II, "Analysis of Data from Chaotic Systems" is comprised of four chapters of 3-6 reprints each (pp. 65-203). Each chapter starts with a review of basic principles by the editors, and a description of what each paper contributes to the subject. These papers include many that we have wished were in our own libraries. This book conveniently collects them for us. For example, in chapter 6, The Practice of Embedding, the paper by Packard, Crutchfield, Farmer, & Shaw (1980) 'Geometry from a time series', is the foremost introduction to the time-lagged (delay-coordinate) technique of reconstructing an attractor from a single time series. Broomhead & King's (1986) 'Extracting qualitative dynamics from experimental data', added filtering using singular value decomposition to the technique. Kennel, Brown, & Abarbanel's (1992) 'Determining embedding dimension for phase-space reconstruction using a geometrical construction', added the false-neighbors improvements to diminish the number of dimensions needed for emedding and assisted in distinguishing deterministic from stochastic components. Kaplan & Glass' (1992) 'Direct test for determinism in a time series' adds another technique for evaluating deterministic and stochastic contributions by comparing an averaged vectorfield with that from a randomized vectorfield. I liked the editors pointing out that these techniques, as they also did in their chapter 5 (from Part I) on the theory of embedding, were appropriate for analyzing simultaneous multiple variables as well as time-delayed variables (Abraham, 1997; Sauer, Yorke, & Casdagli, 1991; Stewart, 1996).
Chapter 7, Dimension Calculations, includes the papers by Albano et al., Eckmann & Ruelle, Ding et al., Theiler et al., Brandstater & Swinney, and Guckenheimer & Buzyna that apply the techniques, calibrating them with synthetic data from known dynamical systems and applying them to experimental data. For example, the paper by Theiler et al (1992), 'Testing for nonlinearity in time series may be best known to our fields because testing their method of surrogate data used EEG and so it is frequently quoted in our disciplines. Econometric and epidemiological data have also been so tested. They introduced Monte Carlo techniques for establishing a null hypothesis and statistical testing of attractor invariants. Guckenheimer & Buzyna's (1983) 'Dimension measurements for geostrophic turbulence', a study of bifurcation in turbulence, used multiple simultaneous time series. The editors note, "It is our feeling that simultaneous measurements will often give superior results [to the delay-coordinate procedure], and should be used, if available (p. 106).
Chapter 8, Calculation of Lyapunov Exponents, contains three papers which take up the evolution of these computations since Wolf, Swift, Swinney, & Vastano's (1985) heuristic method for estimating the largest exponents. The idea is to convolute an m-frame of orthogonal vectors over a trajectory to get at local linear dynamics. The first paper by Eckmann et al., an application of this method, is one of the first to provide computation of several characteristic exponents. The next paper ( Bryant et al.) adds the use of hypothetical higher degree polynomial models during the convolution (see Brown, Bryant, & Abarbanel, 1991, for more details of this technique which provides more reliable estimates of the entire spectrum, including negative Lyapuov exponents). The bad news is that the process is seriously degraded by even small amounts of noise. The last paper in this chapter by Parlitz addresses the issue of getting spurious Lyapunov's from these techniques and suggests using time reversals which should reverse the sign of true Lyapunov exponents. The good news is that regularization and smoothing data contaminated with noise can overcome some of the degredative effects of noise.
The final chapter of this section contains four papers on Periodic Orbits and Symbolic Dynamics. These papers provide attractor reconstructions, return and Poincaré maps, and parameter maps from the analysis of various physical systems.Part III (pp. 206-395) deals with the practical problems of "Prediction, Filtering, Control and Communication in Chaotic Systems". The editors note that after the initial development of the concept of reconstruction of the attractor (Taken's embedding theorem, 1981), estimating the correlation dimension (Grassberger & Procaccia, 1983), reconstruction of the local linear dynamics (Eckmann & Ruelle, 1985), and the approximation of Lyapunov exponents (Sano & Sawada, 1985), the way was paved for the development of ideas of prediction and nonlinear forecasting. The first chapter (10), Prediction, contains papers iby Farmer & Sidorowich who "discuss the scaling of prediction error for several artificial and experimental time series", Casdagli who "compares local linear approximations to the dynamics with global approximations", Sugihara & May who use local linear short term techniques and discuss difficult issues of measurement error, and Sauer "using singlular value decomposition to restrict the dynamics to the tangent plane of the attractor, and using Fourier interpolation to counteract undersampling difficulties." Most of these use known systems to calibrate the techniques in addition to testing them on experimental data.
Chapter 11 on Noise Reduction is important because distinguishing noise from deterministic chaos is difficult due to the continuous nature of chaotic power spectra.
Chapter 12, Control: Theory of Stabilization of Unstable Orbits deals with the theory of stabilizing periodic orbits in order to control undesirable chaotic orbits which depend on them.
Chapter 13, Control: Experimental Stabilization of Unstable Orbits contains 6 papers which provide examples of such control in several physical and physiological systems. The paper by Garfinkel et al. states that chaotic systems are "highly susceptible to control, provided that the developing chaos can be analyzed in real time and that analysis is then used to make small control interventions. . . . By administering electrical stimuli to the heart at irregular times determined by chaos theory, the arrhythmia was converted to periodic beating." Besides physiological issues of health and disease, control theory could be considered for use in psychological therapy, and in the control of organizations and social systems. Can we borrow some of these techniques in our fields where noise becomes more prominant? Time will tell.
Chapter 14, Control: Targeting and Goal Dynamics contains papers which go beyond the reduction of undesirable chaotic behavior and tries to restrict the orbit to "a small region about some specified point on the chaotic attractor" or to a specific dynamical goal.
Chapter 15, the final chapter, on Synchronism and Communication, has obvious applications for precise control in living systems, but at first blush might seem, as topics generated from fairly esoteric physical systems with low noise conditions, to have the little applicability to psychological and social systems. On the other hand, these topics involve the aspects of psychological and social systems, the self-organizational features, that we find most important. Thus these techniques might help in dealing with communication and harmony within and between individuals or within and between components of social organization. Becoming more familiar with some of these technical considerations may yield some more explicit formulations to our metaphorical fancies as well as some suggestions for psychological, social, and biological research.
For any of us interested in doing or reading research on chaotic systems, this book should become a cornerstone of our personal libraries. The clarity and elegance of the exposition is exceptional; the importance of the reprints undeniable.
Abarbanel, H. D. I. (1996). Analysis of observed chaotic data. New York: Springer-Verlag.
Abraham, F. D. (1997). Nonlinear coherence in multivariate research: Invariants and the reconstruction of attractors. Nonlinear Dynamics, Psychology, and Life Sciences, 1, 7-33.
Brown, R., Bryant, P., & Abarbanel, H. D.. I. (1991). Computing the Lyapunov spectrum of a dynamical system from observed time series. Physical Review A, 43, 27-87.
Eckmann, J. -P., & Ruelle, D. (1985) Ergodic theory of chaos and strange attractors. Reviews of Modern Physics, 57, 617-619.
Grassberger, P., & Procaccia, I. (1983). Characterisation of strange attractors. Physical Review Letters, 50, 346-369.
Sano, M., & Sawada. Y. (1985). Measurement of the Lyapunov spectrum from chaotic time series. Physical Review Letters, 55, 1082-1085.
Sauer, T., Yorke, J. A., & Casdagli, M. (1991). Embedology. Journal of Statistical Physics, 65, 579-616.
Stewart. H. B. (1996). Chaos, dynamical structure and climate variability. In D. Herbert (Ed.), Chaos and the changing nature of science and medicine: An Introduction. Conference Proceedings, 376 (pp. 80-115). New York: American Institue of Physics.
Takens, F. (1981). Detecting strange attractors in turbulence. In D. A. Rand & L. -S. Young (Eds), Dynamical systems and turbulence, Warwick, 1980. Lecture Notes in Mathematics, 898 (pp. 366-381). Berlin: Springer-Verlag.
Wolf, A., Swift, J. B., Swinney, H., & Vastano, J. A. (1985). Determining Lyapunov exponents from a time series. Physica D, 16, 285--317.