Analyses of transient chaotic time series

Time-series-analysis-based detection of critical transitions in real-world non-autonomous systems

Real-world non-autonomous systems are open, out-of-equilibrium systems that evolve in and are driven by temporally varying environments. Such systems can show multiple timescale and transient dynamics together with transitions to very different and, at times, even disastrous dynamical regimes. Since such critical transitions disrupt the systems' intended or desired functionality, it is crucial to understand the underlying mechanisms, to identify precursors of such transitions, and to reliably detect them in time series of suitable system observables to enable forecasts. This review critically assesses the various steps of investigation involved in time-series-analysis-based detection of critical transitions in real-world non-autonomous systems: from the data recording to evaluating the reliability of offline and online detections. It will highlight pros and cons to stimulate further developments, which would be necessary to advance understanding and forecasting nonlinear behavior such as critical transitions in complex systems.

Hybridization of the swarming and interior point algorithms to solve the Rabinovich-Fabrikant system

In this study, a trustworthy swarming computing procedure is demonstrated for solving the nonlinear dynamics of the Rabinovich-Fabrikant system. The nonlinear system's dynamic depends upon the three differential equations. The computational stochastic structure based on the artificial neural networks (ANNs) along with the optimization of global search swarming particle swarm optimization (PSO) and local interior point (IP) algorithms, i.e., ANNs-PSOIP is presented to solve the Rabinovich-Fabrikant system. An objective function based on the differential form of the model is optimized through the local and global search methods. The correctness of the ANNs-PSOIP scheme is observed through the performances of achieved and source solutions, while the negligible absolute error that is around 10^-05-10^-07 also represent the worth of the ANNs-PSOIP algorithm. Furthermore, the consistency of the ANNs-PSOIP scheme is examined by applying different statistical procedures to solve the Rabinovich-Fabrikant system.

Finding nonlinear system equations and complex network structures from data: A sparse optimization approach

In applications of nonlinear and complex dynamical systems, a common situation is that the system can be measured, but its structure and the detailed rules of dynamical evolution are unknown. The inverse problem is to determine the system equations and structure from time series. The principle of exploiting sparse optimization to find the equations of dynamical systems from data was first articulated in 2011 by the ASU group. The basic idea is to expand the system equations into a power series or a Fourier series of a finite number of terms and then to determine the vector of the expansion coefficients based solely on data through sparse optimization. This Tutorial presents a brief review of the recent progress in this area. Issues discussed include discovering the equations of stationary or nonstationary chaotic systems to enable the prediction of critical transition and system collapse, inferring the full topology of complex oscillator networks and social networks hosting evolutionary game dynamics, and identifying partial differential equations for spatiotemporal dynamical systems. Situations where sparse optimization works or fails are pointed out. The relation with the traditional delay-coordinate embedding method is discussed, and the recent development of a model-free, data-driven prediction framework based on machine learning is mentioned.

Inferring directional interactions from transient signals with symbolic transfer entropy

We extend the concept of symbolic transfer entropy to enable the time-resolved investigation of directional relationships between coupled dynamical systems from short and transient noisy time series. For our approach, we consider an observed ensemble of a sufficiently large number of time series as multiple realizations of a process. We derive an index that quantifies the preferred direction of transient interactions and assess its significance using a surrogate-based testing scheme. Analyzing time series from noisy chaotic systems, we demonstrate numerically the applicability and limitations of our approach. Our findings obtained from an analysis of event-related brain activities underline the importance of our method to improve understanding of gross neural interactions underlying cognitive processes.

An automated algorithm for the generation of dynamically reconstructed trajectories

The lack of long enough data sets is a major problem in the study of many real world systems. As it has been recently shown [C. Komalapriya, M. Thiel, M. C. Romano, N. Marwan, U. Schwarz, and J. Kurths, Phys. Rev. E 78, 066217 (2008)], this problem can be overcome in the case of ergodic systems if an ensemble of short trajectories is available, from which dynamically reconstructed trajectories can be generated. However, this method has some disadvantages which hinder its applicability, such as the need for estimation of optimal parameters. Here, we propose a substantially improved algorithm that overcomes the problems encountered by the former one, allowing its automatic application. Furthermore, we show that the new algorithm not only reproduces the short term but also the long term dynamics of the system under study, in contrast to the former algorithm. To exemplify the potential of the new algorithm, we apply it to experimental data from electrochemical oscillators and also to analyze the well-known problem of transient chaotic trajectories.

Exploring partial control of chaotic systems

In this paper we make a thorough exploration of the technique of partial control of chaotic systems. This control technique allows one to keep the trajectories of a dynamical system close to a chaotic saddle even if the control applied is smaller than the effects of environmental noise in the system, provided that the chaotic saddle is due to the existence of a horseshoelike mapping in phase space. We state this here in a mathematically precise way using the Conley-Moser conditions, and we prove that they imply that our partial control strategy can be applied. We also give an upper bound of the control-noise ratio needed to achieve this goal, and we describe how this technique can be applied for large noise values. Finally, we study in detail the effect of imperfect targeting in our control technique. All these results are illustrated numerically with the paradigmatic Hénon map.

Reconstruction of a system's dynamics from short trajectories

Long data sets are one of the prime requirements of time series analysis techniques to unravel the dynamics of an underlying system. However, acquiring long data sets is often not possible. In this paper, we address the question of whether it is still possible to understand the complete dynamics of a system if only short (but many) time series are observed. The key idea is to generate a single long time series from these short segments using the concept of recurrences in phase space. This long time series is constructed so as to exhibit a dynamics similar to that of a long time series obtained from the corresponding underlying system.

Detection of seizure rhythmicity by recurrences

Epileptic seizures show a certain degree of rhythmicity, a feature of heuristic and practical interest. In this paper, we introduce a simple model of this type of behavior, and suggest a measure for detecting and quantifying it. To evaluate our method, we develop a set of test segments that incorporate rhythmicity features, and present results from the application of this measure to test segments. We then analyze electrocorticogram segments containing seizures, and present two examples. Finally, we discuss the similarity of our method to techniques for detecting unstable periodic orbits in chaotic time series.

Partial control of chaotic systems

In a region in phase space where there is a chaotic saddle, all initial conditions will escape from it after a transient with the exception of a set of points of zero Lebesgue measure. The action of an external noise makes all trajectories escape faster. Attempting to avoid those escapes by applying a control smaller than noise seems to be an impossible task. Here we show, however, that this goal is indeed possible, based on a geometrical property found typically in this situation: the existence of a horseshoe. The horseshoe implies that there exist what we call safe sets, which assures that there is a general strategy that allows one to keep trajectories inside that region with control smaller than noise. We call this type of control partial control of chaos.

Measurements of brain activity complexity for varying mental loads

Using functional magnetic resonance imaging, we investigate the variation in dynamical complexity of human brain activity for different mental loads. Our experiments measured the activity of ten subjects under three experimental conditions: a rest condition, a periodic task of finger opposition, and a task of finger opposition alternated with mathematical serial calculation. We used the correlation dimension to gauge the spatiotemporal complexity of brain activity. The experiments show a direct relationship between this complexity and the difficulty of the task. A natural interpretation is that higher levels of mental load recruit a larger number of independent neural processes that contribute to complex brain dynamics. These results suggest the possibility that the relative change in correlation dimension can be a useful global measure of brain dynamics, e.g., in determining the levels of mental activity, even if little is known about the underlying neurological processes.