We propose a new method to compute Lyapunov exponents from limited experimental data. The method is tested on a variety of known model systems, and it is found that the algorithm can be used to obtain a reasonable Lyapunov-exponent spectrum from only 5000 data points with a precision of 10 -lor 10 -2 in three-or four-dimensional phase space, or 10000 data points in five-dimensional phase space. We also apply our algorithm to the daily-averaged data of surface temperature observed at two locations in the United States to quantitatively evaluate atmospheric predictability. PACS numbers: 05.45.+b, 02.60.+y, 47.20.Tg, 92.60.Wc Nonlinear phenomena occur in nature in a wide range of apparently different contexts, yet they often display common features, or can be understood using similar concepts. Deterministic chaos and fractal structure in dissipative dynamical systems are among the most important nonlinear paradigms. The spectrum of Lyapunov exponents provides a quantitative measure of the sensitivity to initial conditions (i.e., the divergence of neighboring trajectories exponentially in time) and is the most useful dynamical diagnostic for chaotic systems. In fact, any system containing at least one positive Lyapunov exponent is defined to be chaotic, with the magnitude of the exponent determining the time scale for predictability. In any well-behaved dissipative dynamical system, one of the Lyapunov exponents must be strictly negative. I If the Lyapunov-exponent spectrum can be determined, the Kolmogorov entropy2 can be computed by summing all of the positive exponents, and the fractal dimension may be estimated using the Kaplan-Yorke conjecture. 3 The Lyapunov-exponent spectrum can be computed relatively easily for known model systems.4 However, it is difficult to estimate Lyapunov exponents from experimental data for a complex system (e.g., the atmosphere). Wolf et al.5 proposed a method to estimate one or two positive exponents. Sano and Sawada 6 and Eckmann et al.7 developed similar procedures to determine several of the Lyapunov exponents (including positive, zero, and even negative values). This is now a very active research area, and several authors8 have introduced further improvements. However, all of these methods require rela-@ 1991 The Americ tively long time series and/or data of high precision (for example, Eckmann et at. used 64000 data points with a precision of 10-4 for the Lorenz equations9), but such high-quality data cannot be obtained in many real-world situations. The infinitesimal length scales used to define Lyapunov exponents are inaccessible in experimental data. 5 The presence of noise or limited precision leads to a length scale Ln below which the structure of the underlying strange attractor is obscured. Also, for a finite data set of N points, there is a minimum length scale Lo-L/N1/D, where L is the horizontal extent of the attractor and D is its information dimension,lo below which structure cannot be resolved. When Lo~ Ln, increasing N is not likely to provide any further information on the structure of the attractor, so that a relatively small data set can be sufficient for computing Lyapunov exponents. Furthermore, if the length scales Lo and Ln are small enough for the chaotic dynamics to be the same as at infinitesimal length scales, then the computation of Lyapunov exponents using these length scales should yield reasonable results. Abraham et at. II have demonstrated that it is possible to calculate the dimensions of attractors from small, noisy data sets. The purpose of this paper is to develop a procedure by which one can evaluate the Lyapunovexponent spectrum from relatively small data sets of low precision. We test the method on a variety of known model systems, and we also use the method to study the predictability of the atmosphere from observational meteorological data. It should be noted, as pointed ou
