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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
We present different tests for phase synchronization which improve the procedures currently used in the literature. This is accomplished by using a two-samples test setup and by utilizing insights and methods from directional statistics and bootstrap theory. The tests differ in the generality of the situation in which they can be applied as well as in their complexity, including computational cost. A modification of the resampling technique of the bootstrap is introduced, making it possible to fully utilize data from time series.
A method for the multivariate analysis of statistical phase synchronization phenomena in empirical data is presented. A first statistical approach is complemented by a stochastic dynamic model, to result in a data analysis algorithm which can in a specific sense be shown to be a generic multivariate statistical phase synchronization analysis. The method is applied to EEG data from a psychological experiment, obtaining results which indicate the relevance of this method in the context of cognitive science as well as in other fields.
In this article we review the application of the synchronization theory to the analysis of multivariate biological signals. We address the problem of phase estimation from data and detection and quantification of weak interaction, as well as quantification of the direction of coupling. We discuss the potentials as well as limitations and misinterpretations of the approach
We study the effects of mutual and external chaotic phase synchronization in ensembles of bursting oscillators. These oscillators (used for modeling neuronal dynamics) are essentially multiple time scale systems. We show that a transition to mutual phase synchronization takes place on the bursting time scale of globally coupled oscillators, while on the spiking time scale, they behave asynchronously. We also demonstrate the effect of the onset of external chaotic phase synchronization of the bursting behavior in the studied ensemble by a periodic driving applied to one arbitrarily taken neuron. We also propose an explanation of the mechanism behind this effect. We infer that the demonstrated phenomenon can be used efficiently for controlling bursting activity in neural ensembles
We study phase synchronization effects of chaotic oscillators with a type-I intermittency behavior. The external and mutual locking of the average length of the laminar stage for coupled discrete and continuous in time systems is shown and the mechanism of this synchronization is explained. We demonstrate that this phenomenon can be described by using results of the parametric resonance theory and that this correspondence enables one to predict and derive all zones of synchronization
Being one of the fundamental phenomena in nonlinear science, synchronization of oscillations has permanently remained an object of intensive research. Development of many asymptotic methods and numerical simulations has allowed an understanding and explanation of various phenomena of self-synchronization. But even in the classical case of coupled van der Pol oscillators a full description of all possible dynamical regimes, their mutual transitions and characteristics is still lacking. We present here a study of the phenomenon of mutual synchronization for two non-scalar- coupled non-identical limit-cycle oscillators and analyze phase, frequency and amplitude characteristics of synchronization regimes. A series of bifurcation diagrams that we obtain exhibit various regions of qualitatively different behavior. Among them we find mono-, bi- and multistability regions, beating and "oscillation death" ones; also a region, where one of the oscillators dominates the other one is observed. The frequency characteristics that we obtain reveal three qualitatively different types of synchronization: (i) on the mean frequency (the in-phase synchronization), (ii) with a shift from the mean frequency caused by a conservative coupling term (the anti-phase synchronization), and (iii) on the frequency of one of the oscillators (when one oscillator dominates the other). (C) 2003 Elsevier B.V. All rights reserved
Charged dust grains in circumplanetary environments experience, beyond various deterministic forces, also stochastic perturbations caused, by fluctuations of the magnetic field, the charge of the grains, by chaotic rotation of aspherical grains, etc. Here we investigate the dynamics of a dust population in a circular orbit around a planet which is perturbed by a stochastic planetary magnetic field B', modeled by an isotropically Gaussian white noise. The resulting perturbation equations give rise to a modified diffusion of the inclinations i and eccentricities e. The diffusion coefficient is found to be D proportional to w^2 O /n^2 , where the gyrofrequency, the Kepler frequency, and the synodic frequency are denoted by w , O, and n, respectively. This behavior has been checked against numerical simulations. We have chosen dust grains (1 m in radius) ejected from Jupiter's satellite Europa in circular equatorial orbits around Jupiter and integrated numerically their trajectories over their typical lifetimes (100 years). The particles were exposed to a Gaussian fluctuating magnetic field B' with the same statistical properties as in the analytical treatment. These simulations have confirmed the analytical results. The theoretical studies showed the statistical properties of B' to be of decisive importance. To estimate them, we analyzed the magnetic field data obtained by the Galileo spacecraft magnetometer at Jupiter and found almost Gaussian fluctuations of about 5% of the mean field and exponentially decaying correlations. This results in a diffusion of orbital inclinations and eccentricities of the dust grains of about ten percent over the lifetime of the particles. For smaller dusty motes or for close-in particles (e.g., in Jovian gossamer rings) stochastics might well dominate the dynamics.
We study numerically the behavior of the autocorrelation function (ACF) and the power spectrum of spiral attractors without and in the presence of noise. It is shown that the ACF decays exponentially and has two different time scales. The rate of the ACF decrease is defined by the amplitude fluctuations on small time intervals, i.e., when tau < tau(cor), and by the effective diffusion coefficient of the instantantaneous phase on large time intervals. it is also demonstrated that the ACF in the Poincare map also decreases according to the exponential law exp(-lambda(+)k), where lambda(+) is the positive Lyapunov exponent. The obtained results are compared with the theory of fluctuations for the Van der Pol oscillator
This paper treats a problem of reconstructing ordinary differential equation from a single analytic time series with observational noise. We suppose that the noise is Gaussian (white). The investigation is presented in terms of classical theory of dynamical systems and modern time series analysis. We restrict our considerations on time series obtained as a numerical analytic solution of autonomous ordinary differential equation, solved with respect to the highest derivative and with polynomial right-hand side. In case of an approximate numerical solution with a rather small error, we propose a geometrical basis and a mathematical algorithm to reconstruct a low-order and low-power polynomial differential equation. To reduce the noise the given time series is smoothed at every point by moving polynomial averages using the least-squares method. Then a specific form of the least-squares method is applied to reconstruct the polynomial right-hand side of the unknown equation. We demonstrate for monotonous, periodic and chaotic solutions that this technique is very efficient