We present the first algorithms that allow the estimation of non-negative Lyapunov exponents from an experimental time series. Lyapunov exponents, which provide a qualitative and quantitative characterization of dynamical behavior, are related to the exponentially fast divergence or convergence of nearby orbits in phase space. A system with one or more positive Lyapunov exponents is defined to be chaotic. Our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems: we monitor the long-term growth rate of small volume elements in an attractor.The method is tested on model systems with known Lyapunov spectra, and applied to data for the Belousov-Zhabotinskii reaction and Couette-Taylor flow.

Chaos, orbital divergence and the loss of predictabilityChaos has been discovered both in the laboratory and in the mathematical models that describe a wide variety of systems [1,3], In common usage chaos is taken to mean a state in which chance prevails. To the nonlinear dynamicist the word chaos has a more precise and rather different mean ing. A chaotic system is one in which long-term prediction of the system's state is impossible because the omnipresent uncertainty in determining its initial state grows exponentially fast in time. The rapid loss of predictive power is due to the property that orbits (trajectories) that arise from nearby initial conditions diverge exponentially fast on the average. Nearby orbits correspond to almost identically prepared systems, so that systems whose differences we may not be able to resolve initially soon behave quite differently. In non-chaotic systems, nearby orbits either converge exponentially fast, or at worst exhibit a slower than exponential diverg ence: long-term prediction is at least theoretically possible.Rates of orbital divergence or convergence, called Lyapunov exponents [2,9,13,16], are clearly of fundamental importance in studying chaos. Positive Lyapunov exponents indicate orbital divergence and chaos, and set the time scale on which state prediction is possible. Negative Lyapunov exponents set the time scale on which transients or perturbations of the system's state will decay. In this chapter we define the spectrum of Lyapunov exponents, describe the well-known technique for computing a system's spectrum from its defining equations of motion, and outline a new technique for estimating non-negative exponents from experimental data.

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Experiments on the Belousov-Zhabotinskii reaction in a stirred flow reactor reveal behavior that is strikingly similar to that generated by one-dimensional maps with a single extremum. In particular, a period-doubling sequence is observed that leads to a regime containing both chaotic and periodic states. Within the experimental resolution the ordering of the periodic states is in accord with the theory of one-dimensional maps.PACS numbers: 05.70. Ln, 47.70.Fw, We have conducted experiments on a complex chemically reacting system (with about 25 chemical species) which exhibits, as a function of the flow rate of the chemicals through the reactor, a sequence of periodic and chaotic states that is in good agreement with that exhibited by unimodal (single-extremum) one-dimensional (ID) maps. From the data we have constructed ID maps that correspond to the different periodic and chaotic states. A decade ago Metropolis, Stein, and Stein 1 showed that unimodal maps, x n+1 = Xf(x n ) f exhibit universal (map-independent) dynamics as a function of the bifurcation parameter X. Analysis of higher-dimensional systems has led to the conjecture that, if such a system were to exhibit a period-doubling sequence, then the dynamics of the system would be similar to that of a ID map. 2 * 3 Indeed, period-doubling sequences have been discovered in recent experiments 4 on a variety of physical systems, and the observed behavior for at least the first few doublings has been in accord with the theory for ID maps. However, ID maps were not obtained in any of those experiments, and the rich dynamical structure that ID maps exhibit beyond the period-doubling sequence has been observed only in the experiments of Testa, P£rez, and Jeffries 4 on the simplest nonlinear physical system that has been studied, an electrical oscillator with three degrees of freedom. We will now review the properties of ID maps and then present the results of our experiments.ID maps. 5 -A method called symbolic dynamics can be used to show that the dynamics of unimodal ID maps of the interval [0, 1] is exhausted by the periodic states of the "U (universal) sequence" of Metropolis, Stein, and Stein 1 and the chaotic states of the "reverse bifurcation sequence" of Lorenz. 6 The theory uses only the unimodal property of the map to deduce the nature of the states and the order in which they appear as a function of the bifurcation parameter X. Feigenbaum and others, making the additional assumption that the map has a quadratic extremum, have obtained detailed predictions for the scaling of various dynamical quantities. 2 ' 5 We will confine our discussion to the results of symbolic dynamics theory since it is the ordering and nature of the states and not their scaling properties that have been determined in our experiments. We begin with the mechanics of map iteration. 1 ' 5 For a given value of X one picks any initial condition (except for a set of measure zero) and iter-

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