TY  - JOUR
A1  - Romano, Maria Carmen
A1  - Thiel, Marco
A1  - Kurths, Jürgen
A1  - von Bloh, Werner
T1  - Multivariate recurrence plots
N2  - We propose a new approach to calculate recurrence plots of multivariate time series, based on joint recurrences in phase space. This new method allows to estimate dynamical invariants of the whole system, like the joint Renyi entropy of second order. We use this entropy measure to quantitatively study in detail the phase synchronization of two bidirectionally coupled chaotic systems and identify different types of transitions to chaotic phase synchronization in dependence on the coupling strength and the frequency mismatch. By means of this analysis we find several new phenomena, such a chaos-period-chaos transition to phase synchronization for rather large coupling strengths. (C) 2004 Elsevier B.V. All rights reserved
Y1  - 2004
SN  - 0375-9601
ER  - 
TY  - JOUR
A1  - Thiel, M.
A1  - Romano, Maria Carmen
A1  - Schwarz, Udo
A1  - Kurths, Jürgen
A1  - Timmer, Jens
T1  - Surrogate-based hypothesis test without surrogates
N2  - Fourier surrogate data are artificially generated time series, that - based on a resampling scheme - share the linear properties with an observed time series. In this paper we study a statistical surrogate hypothesis test to detect deviations from a linear Gaussian process with respect to asymmetry in time (Q-statistic). We apply this test to a Fourier representable function and obtain a representation of the asymmetry in time of the sample data, a characteristic for nonlinear processes, and the significance in terms of the Fourier coefficients. The main outcome is that we calculate the expected value of the mean and the standard deviation of the asymmetries of the surrogate data analytically and hence, no surrogates have to be generated. To illustrate the results we apply our method to the saw tooth function, the Lorenz system and to measured X-ray data of Cygnus X-1
Y1  - 2004
SN  - 0218-1274
ER  - 
TY  - JOUR
A1  - Thiel, M.
A1  - Romano, Maria Carmen
A1  - Kurths, Jürgen
T1  - How much information is contained in a recurrence plot?
N2  - Recurrence plots have recently been recognized as a powerful tool for the analysis of data. Not only the visualization of structures of the time series but also the possibility to estimate invariants from them and the possibility to analyze non-stationary data sets are remarkable. However, the question of how much information is encoded in such a two-dimensional and binary representation has not been discussed so far. In this Letter we show that-under some conditions-it is possible to reconstruct an attractor from the recurrence plot, at least topologically. This means that all relevant dynamical information is contained in the plot. (C) 2004 Elsevier B.V. All rights reserved
Y1  - 2004
SN  - 0375-9601
ER  - 
TY  - JOUR
A1  - Thiel, M.
A1  - Romano, Maria Carmen
A1  - Read, P. L.
A1  - Kurths, Jürgen
T1  - Estimation of dynamical invariants without embedding by recurrence plots
N2  - In this paper we show that two dynamical invariants, the second order Renyi entropy and the correlation dimension, can be estimated from recurrence plots (RPs) with arbitrary embedding dimension and delay. This fact is interesting as these quantities are even invariant if no embedding is used. This is an important advantage of RPs compared to other techniques of nonlinear data analysis. These estimates for the correlation dimension and entropy are robust and, moreover, can be obtained at a low numerical cost. We exemplify our results for the Rossler system, the funnel attractor and the Mackey-Glass system. In the last part of the paper we estimate dynamical invariants for data from some fluid dynamical experiments and confirm previous evidence for low dimensional chaos in this experimental system. (C) 2004 American Institute of Physics
Y1  - 2004
SN  - 1054-1500
ER  -