Estimation of the Kolmogorov entropy from a chaotic signal

While there has been recently a dramatic growth in new mathematical concepts related to chaotic systems, ' the detailed comparison between models and experimental data has lagged somewhat. After observing a seemingly chaotic signal in the laboratory, the researcher is faced with the question of how to characterize the signal, how to be sure that it is chaotic (rather than multiperiodic or random), and how to quantify "how" chaotic the signal is. In this Rapid Communication we propose a method to estimate the Kolmogorov entropy K directly from a time signal. If one obtains a finite, positive E entropy one can answer some of the above questions with a degree of confidence. Although we have dissipative systems in mind, the idea presented below should prove useful for conservative systems as well. The Kolmogorov entropy is defined as follows: Consider a dynamical system with F degrees of freedom. Suppose that the F-dimensional phase space is partitioned to boxes of size eF. Suppose that there is an attractor in phase space and that the trajectory x(t) is in the basin of attraction. The state of the system is now measured at intervals of time Let p(ii, i2, . . . , id) be the joint probability that x(t=r) is in box ii, x(t=2r) is in box i2, . . . , and x(t = dr) is in box iq The Kolm. ogorov entropy is then'