Timely detection of unusual and/or unexpected events in natural and man-made systems has deep scientific and practical relevance. We show that the recently proposed conceptually simple and easily calculated measure of permutation entropy can be effectively used to detect qualitative and quantitative dynamical changes. We illustrate our results on two model systems as well as on clinically characterized brain wave data from epileptic patients.

Due to the ubiquity of time series with long-range correlation in many areas of science and engineering, analysis and modeling of such data is an important problem. While the field seems to be mature, three major issues have not been satisfactorily resolved. ͑i͒ Many methods have been proposed to assess long-range correlation in time series. Under what circumstances do they yield consistent results? ͑ii͒ The mathematical theory of long-range correlation concerns the behavior of the correlation of the time series for very large times. A measured time series is finite, however. How can we relate the fractal scaling break at a specific time scale to important parameters of the data? ͑iii͒ An important technique in assessing long-range correlation in a time series is to construct a random walk process from the data, under the assumption that the data are like a stationary noise process. Due to the difficulty in determining whether a time series is stationary or not, however, one cannot be 100% sure whether the data should be treated as a noise or a random walk process. Is there any penalty if the data are interpreted as a noise process while in fact they are a random walk process, and vice versa? In this paper, we seek to gain important insights into these issues by examining three model systems, the autoregressive process of order 1, on-off intermittency, and Lévy motions, and considering an important engineering problem, target detection within sea-clutter radar returns. We also provide a few rules of thumb to safeguard against misinterpretations of long-range correlation in a time series, and discuss relevance of this study to pattern recognition.

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By studying recurrence time statistics for chaotic systems, we identify two different types of recurrences and develop scaling laws relating the mean recurrence time to the information dimension of the chaotic attractor. We then design two novel and simple ways of using the recurrence time statistics for analyzing transient as well as nonstationary time series. We show that the methods are capable of detecting nonstationarity due to drift of parameters, locating bifurcations, telling the periodicity of major transient periodic motions, and other types of changes in the dynamics. PACS numbers: 05.45.Tp, 02.50.Fz Since the time of Boltzmann and Poincare, the study of the recurrence of states of a dynamical system (i.e., how often a small region in phase space is visited) has been recognized to be fundamental to classical statistical mechanics [1]. It is also of central importance in the study of chaos [2]. However, recurrence time statistics for chaotic systems has only been considered for 1D discrete maps [3] and for a model stationary flow [4]. Thus, an important and challenging question is whether a scaling law relating the mean recurrence time to the attractor dimensions and/ or Lyapunov exponents can be developed for multivariate chaotic systems.Almost all existing linear and nonlinear time series analysis techniques assume that the time series is stationary. However, many time series occurring in geophysics, physiology, finance, etc., are nonstationary. Sometimes the changes in the dynamics can be the most interesting feature of a phenomenon under study. Thus, simple and efficient methods capable of detecting the nonstationarity of a time series and studying transient dynamics would be valuable to researchers from diverse fields. This subject has attracted much attention recently [5][6][7][8][9][10]. In this Letter, we first consider the recurrence time statistics for multivariate chaotic systems. We shall identify recurrence times of two different types and derive scaling laws relating the mean recurrence times of these two types to the information dimension of the attractor. We then design two simple and novel algorithms that use the recurrence time statistics for analyzing transient as well as nonstationary dynamics.Given a chaotic dynamical system described either by a discrete map or by a set of ordinary differential equations (ODEs), we first iterate the map or integrate the ODEs (with sampling time t) until the dynamics is free of transience. We then arbitrarily choose a reference point X 0 on the attractor and consider recurrence to its neighborhood of radius r: B r ͑X 0 ͒ ͕X : kX 2 X 0 k # r͖. If the underlying dynamics on the chaotic attractor is ergodic, then X 0 can be chosen arbitrarily. Next, we consider a trajectory of length N and denote the subset of the trajectory that belongs to B r ͑X 0 ͒ by S 1 ͕X t 1 , X t 2 , . . . , X t i , . . .͖. These are the Poincare recurrence points, and the Poincare re-

Crutchfield et al. observed that noise truncates period-doubling cascade and induces chaos. To date, however, very little low-dimensional chaos has been unambiguously identified experimentally. This discrepancy stimulates us to reexamine the noisy logistic map. We find that noise can indeed induce chaos. However, this is not associated with the main 2 n cascade. We identify three basic conditions for noise to induce chaos. We also show that when noise induces chaos the complete period-doubling cascade is inhibited, otherwise the cascade is simply masked by noise. [S0031-9007(99)

Multifractal theory provides an elegant statistical characterization of many complex dynamical variations in Nature and engineering. It is conceivable that it may enrich characterization of the sun's magnetic activity and its dynamical modeling. Recently, on applying Fourier truncation to remove the 11-year cycle and carrying out multifractal detrended fluctuation analysis of the filtered sunspot time series, Movahed et al reported that sunspot data are characterized by multifractal scaling laws with the exponent for the second-order moment, h(2), being 1.12. Moreover, they think the filtered sunspot data are like a fractional Brownian motion process with anti-persistent long-range correlations characterized by the Hurst parameter H = h(2) − 1 = 0.12. By designing an adaptive detrending algorithm and critically assessing the effectiveness of Fourier truncation in eliminating the 11-year cycle, we show that the values of the fractal scaling exponents obtained by Movahed et al are artifacts of the filtering algorithm that they used. Instead, sunspot data with the 11-year cycle properly filtered are characterized by a different type of multifractal with persistent longrange correlations characterized by H ≈ 0.74.

Time series from complex systems with interacting nonlinear and stochastic subsystems and hierarchical regulations are often multiscaled. In devising measures characterizing such complex time series, it is most desirable to incorporate explicitly the concept of scale in the measures. While excellent scale-dependent measures such as ⑀ entropy and the finite size Lyapunov exponent ͑FSLE͒ have been proposed, simple algorithms have not been developed to reliably compute them from short noisy time series. To promote widespread application of these concepts, we propose an efficient algorithm to compute a variant of the FSLE, the scale-dependent Lyapunov exponent ͑SDLE͒. We show that with our algorithm, the SDLE can be accurately computed from short noisy time series and readily classify various types of motions, including truly low-dimensional chaos, noisy chaos, noise-induced chaos, random 1 / f ␣ and ␣-stable Levy processes, stochastic oscillations, and complex motions with chaotic behavior on small scales but diffusive behavior on large scales. To our knowledge, no other measures are able to accurately characterize all these different types of motions. Based on the distinctive behaviors of the SDLE for different types of motions, we propose a scheme to distinguish chaos from noise.

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