Granger Causality and Cross Recurrence Plots in Rheochaos

Our stress relaxation measurements on wormlike micelles using a Rheo-SALS (rheology + small angle light scattering) apparatus allow simultaneous measurements of the stress and the scattered depolarised intensity. The latter is sensitive to orientational ordering of the micelles. To determine the presence of causal influences between the stress and the depolarised intensity time series, we have used the technique of linear and nonlinear Granger causality. We find there exists a feedback mechanism between the two time series and that the orientational order has a stronger causal effect on the stress than vice versa. We have also studied the phase space dynamics of the stress and the depolarised intensity time series using the recently developed technique of cross recurrence plots (CRPs). The presence of diagonal line structures in the CRPs unambiguously proves that the two time series share similar phase space dynamics.Comment: 10 pages, 7 figure

Brownian motion and diffusion: from stochastic processes to chaos and beyond

One century after Einstein's work, Brownian Motion still remains both a fundamental open issue and a continous source of inspiration for many areas of natural sciences. We first present a discussion about stochastic and deterministic approaches proposed in the literature to model the Brownian Motion and more general diffusive behaviours. Then, we focus on the problems concerning the determination of the microscopic nature of diffusion by means of data analysis. Finally, we discuss the general conditions required for the onset of large scale diffusive motion.Comment: RevTeX-4, 11 pages, 5 ps-figures. Chaos special issue "100 Years of Brownian Motion

Extended Recurrence Plot Analysis and its Application to ERP Data

We present new measures of complexity and their application to event related potential data. The new measures base on structures of recurrence plots and makes the identification of chaos-chaos transitions possible. The application of these measures to data from single-trials of the Oddball experiment can identify laminar states therein. This offers a new way of analyzing event-related activity on a single-trial basis.Comment: 21 pages, 8 figures; article for the workshop ''Analyzing and Modelling Event-Related Brain Potentials: Cognitive and Neural Approaches`` at November 29 - December 01, 2001 in Potsdam, German

The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian

We study the spectrum of the Fibonacci Hamiltonian and prove upper and lower bounds for its fractal dimension in the large coupling regime. These bounds show that as $\lambda \to \infty$, $\dim (\sigma(H_\lambda)) \cdot \log \lambda$ converges to an explicit constant ($\approx 0.88137$). We also discuss consequences of these results for the rate of propagation of a wavepacket that evolves according to Schr\"odinger dynamics generated by the Fibonacci Hamiltonian.Comment: 23 page

Detecting Determinism in High Dimensional Chaotic Systems

A method based upon the statistical evaluation of the differentiability of the measure along the trajectory is used to identify in high dimensional systems. The results show that the method is suitable for discriminating stochastic from deterministic systems even if the dimension of the latter is as high as 13. The method is shown to succeed in identifying determinism in electro-encephalogram signals simulated by means of a high dimensional system.Comment: 8 pages (RevTeX 3 style), 5 EPS figures, submitted to Phys. Rev. E (25 apr 2001

Better Nonlinear Models from Noisy Data: Attractors with Maximum Likelihood

A new approach to nonlinear modelling is presented which, by incorporating the global behaviour of the model, lifts shortcomings of both least squares and total least squares parameter estimates. Although ubiquitous in practice, a least squares approach is fundamentally flawed in that it assumes independent, normally distributed (IND) forecast errors: nonlinear models will not yield IND errors even if the noise is IND. A new cost function is obtained via the maximum likelihood principle; superior results are illustrated both for small data sets and infinitely long data streams.Comment: RevTex, 11 pages, 4 figure

Asymptotically stable phase synchronization revealed by autoregressive circle maps

A new type of nonlinear time series analysis is introduced, based on phases, which are defined as polar angles in spaces spanned by a finite number of delayed coordinates. A canonical choice of the polar axis and a related implicit estimation scheme for the potentially underlying auto-regressive circle map (next phase map) guarantee the invertibility of reconstructed phase space trajectories to the original coordinates. The resulting Fourier approximated, Invertibility enforcing Phase Space map (FIPS map) is well suited to detect conditional asymptotic stability of coupled phases. This rather general synchronization criterion unites two existing generalisations of the old concept and can successfully be applied e.g. to phases obtained from ECG and airflow recordings characterizing cardio-respiratory interaction.Comment: PDF file, 232 KB, 24 pages, 3 figures; cheduled for Phys. Rev. E (Nov) 200

H\"older Continuity of the Integrated Density of States for the Fibonacci Hamiltonian

We prove H\"older continuity of the integrated density of states for the Fibonacci Hamiltonian for any positive coupling, and obtain the asymptotics of the H\"older exponents for large and small couplings.Comment: 18 page

Remarks on the Spectral Properties of Tight Binding and Kronig-Penney Models with Substitution Sequences

We comment on some recent investigations on the electronic properties of models associated to the Thue-Morse chain and point out that their conclusions are in contradiction with rigorously proven theorems and indicate some of the sources of these misinterpretations. We briefly review and explain the current status of mathematical results in this field and discuss some conjectures and open problems.Comment: 15,CPT-94/P.3003,tex,

Enlarged scaling ranges for the KS-entropy and the information dimension

Numerical estimates of the Kolmogorov-Sinai entropy based on a finite amount of data decay towards zero in the relevant limits. Rewriting differences of block entropies as averages over decay rates, and ignoring all parts of the sample where these rates are uncomputable because of the lack of neighbours, yields improved entropy estimates. In the same way, the scaling range for estimates of the information dimension can be extended considerably. The improvement is demonstrated for experimental data.Comment: 5 pages, 6 figure