@article{ThielRomanoSchwarzetal.2004,
  author    = {Thiel, M. and Romano, Maria Carmen and Schwarz, Udo and Kurths, J{\"u}rgen and Timmer, Jens},
  title     = {Surrogate-based hypothesis test without surrogates},
  issn      = {0218-1274},
  year      = {2004},
  abstract  = {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},
  language  = {en}
}
@article{ThielRomanoKurths2004,
  author    = {Thiel, M. and Romano, Maria Carmen and Kurths, J{\"u}rgen},
  title     = {How much information is contained in a recurrence plot?},
  issn      = {0375-9601},
  year      = {2004},
  abstract  = {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},
  language  = {en}
}
@article{ThielRomanoReadetal.2004,
  author    = {Thiel, M. and Romano, Maria Carmen and Read, P. L. and Kurths, J{\"u}rgen},
  title     = {Estimation of dynamical invariants without embedding by recurrence plots},
  issn      = {1054-1500},
  year      = {2004},
  abstract  = {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},
  language  = {en}
}
@article{RomanoThielKurthsetal.2005,
  author    = {Romano, Maria Carmen and Thiel, M. and Kurths, J{\"u}rgen and Kiss, Istvan Z. and Hudson, J. L.},
  title     = {Detection of synchronization for non-phase-coherent and non-stationary data},
  issn      = {0295-5075},
  year      = {2005},
  abstract  = {We present a new method to detect phase as well as generalized synchronization in a wide class of complex systems. It is based on the recurrences of the system's trajectory to the neighborhood of a former state in phase space. We illustrate the applicability of the algorithm for the paradigmatic chaotic Rossler system in the funnel regime and for noisy data, where other methods to detect phase synchronization fail. Furthermore, we demonstrate for electrochemical experiments that the method can easily detect phase and generalized synchronization in non-phase- coherent and even non-stationary time series},
  language  = {en}
}
@article{ZouThielRomanoetal.2006,
  author    = {Zou, Yong and Thiel, M. and Romano, Maria Carmen and Kurths, J{\"u}rgen and Bi, Q.},
  title     = {Shrimp structure and associated dynamics in parametrically excited oscillators},
  series = {International journal of bifurcation and chaos : in applied sciences and engineering},
  volume    = {16},
  journal   = {International journal of bifurcation and chaos : in applied sciences and engineering},
  number    = {12},
  publisher = {World Scientific Publ. Co},
  address   = {Singapore},
  issn      = {0218-1274},
  doi       = {10.1142/S0218127406016987},
  pages     = {3567 -- 3579},
  year      = {2006},
  abstract  = {We investigate the bifurcation structures in a two-dimensional parameter space (PS) of a parametrically excited system with two degrees of freedom both analytically and numerically. By means of the Renyi entropy of second order K-2, which is estimated from recurrence plots, we uncover that regions of chaotic behavior are intermingled with many complex periodic windows, such as shrimp structures in the PS. A detailed numerical analysis shows that, the stable solutions lose stability either via period doubling, or via intermittency when the parameters leave these shrimps in different directions, indicating different bifurcation properties of the boundaries. The shrimps of different sizes offer promising ways to control the dynamics of such a complex system.},
  language  = {en}
}