Recurrence analysis of strange nonchaotic dynamics in driven excitable systems

Complex network based techniques to identify extreme events and (sudden) transitions in spatio-temporal systems

We present here two promising techniques for the application of the complex network approach to continuous spatio-temporal systems that have been developed in the last decade and show large potential for future application and development of complex systems analysis. First, we discuss the transforming of a time series from such systems to a complex network. The natural approach is to calculate the recurrence matrix and interpret such as the adjacency matrix of an associated complex network, called recurrence network. Using complex network measures, such as transitivity coefficient, we demonstrate that this approach is very efficient for identifying qualitative transitions in observational data, e.g., when analyzing paleoclimate regime transitions. Second, we demonstrate the use of directed spatial networks constructed from spatio-temporal measurements of such systems that can be derived from the synchronized-in-time occurrence of extreme events in different spatial regions. Although there are many possibilities to investigate such spatial networks, we present here the new measure of network divergence and how it can be used to develop a prediction scheme of extreme rainfall events.

Distinguishing dynamics using recurrence-time statistics

The probability densities of the mean recurrence time, which is the average time needed for a system to recur to a previously visited neighborhood, are investigated in various dynamical regimes and are found to be in agreement with those of the finite-time Lyapunov exponents. The important advantages of the former ones are that they are easy to estimate and that comparable short time series are sufficient. Asymmetric distributions with exponential tails are observed for intermittency and crisis-induced intermittency, while for typical chaos, the distribution has a Gaussian shape. Further, the shape of the distribution distinguishes intermittent strange nonchaotic attractors from those appearing through fractalization and tori collision mechanisms. Furthermore, statistics performed on the peaks in the frequency distribution of recurrence times unveil scaling behavior in agreement with that obtained from the spectral distribution function defined as the number of peaks in the Fourier spectrum greater than a predefined value. The results of the present recurrence statistics are of relevance in classifying different dynamics and providing important insights into the dynamics of a system when only one realization of this system is available. The practical use of this approach for experimental data is shown on experimental electrochemical time series.

Remote synchronization in star networks

We study phase synchronization in a network motif with a starlike structure in which the central node's (the hub's) frequency is strongly detuned against the other peripheral nodes. We find numerically and experimentally a regime of remote synchronization (RS), where the peripheral nodes form a phase synchronized cluster, while the hub remains free with its own dynamics and serves just as a transmitter for the other nodes. We explain the mechanism for this RS by the existence of a free amplitude and also show that systems with a fixed or constant amplitude, such as the classic Kuramoto phase oscillator, are not able to generate this phenomenon. Further, we derive an analytic expression which supports our explanation of the mechanism.

Identifying complex periodic windows in continuous-time dynamical systems using recurrence-based methods

The identification of complex periodic windows in the two-dimensional parameter space of certain dynamical systems has recently attracted considerable interest. While for discrete systems, a discrimination between periodic and chaotic windows can be easily made based on the maximum Lyapunov exponent of the system, this remains a challenging task for continuous systems, especially if only short time series are available (e.g., in case of experimental data). In this work, we demonstrate that nonlinear measures based on recurrence plots obtained from such trajectories provide a practicable alternative for numerically detecting shrimps. Traditional diagonal line-based measures of recurrence quantification analysis as well as measures from complex network theory are shown to allow an excellent classification of periodic and chaotic behavior in parameter space. Using the well-studied Rössler system as a benchmark example, we find that the average path length and the clustering coefficient of the resulting recurrence networks are particularly powerful discriminatory statistics for the identification of complex periodic windows.

Recurrence-based detection of the hyperchaos-chaos transition in an electronic circuit

Some complex measures based on recurrence plots give evidence about hyperchaos-chaos transitions in coupled nonlinear systems [E. G. Souza et al., "Using recurrences to characterize the hyperchaos-chaos transition," Phys. Rev. E 78, 066206 (2008)]. In this paper, these measures are combined with a significance test based on twin surrogates to identify such a transition in a fourth-order Lorenz-like system, which is able to pass from a hyperchaotic to a chaotic behavior for increasing values of a single parameter. A circuit analog of the mathematical model has been designed and implemented and the robustness of the recurrence-based method on experimental data has been tested. In both the numerical and experimental cases, the combination of the recurrence measures and the significance test allows to clearly identify the hyperchaos-chaos transition.