G. Ambika

Synchronized states in chaotic systems coupled indirectly through a dynamic environment.

We consider synchronization of chaotic systems coupled indirectly through a common environmnet where the environment has an intrinsic dynmics of its own modulated via feedback from the systems. We find that a rich vareity of synchronization behavior, such as in-phase, anti-phase,complete and anti- synchronization is possible. We present an approximate stability analysis for the different synchronization behaviors. The transitions to different states of synchronous behaviour are analyzed in the parameter plane of coupling strengths by numerical studies for specific cases such as Rossler and Lorenz systems and are characterized using various indices such as correlation, average phase difference and Lyapunov exponents. The threshold condition obtained from numerical analysis is found to agree with that from the stability analysis.

A non subjective approach to the GP algorithm for analysing noisy time series

We present an adaptation of the standard Grassberger-Proccacia (GP) algorithm for estimating the correlation dimension of a time series in a non-subjective manner. The validity and accuracy of this approach are tested using different types of time series, such as those from standard chaotic systems, pure white and colored noise and chaotic systems with added noise. The effectiveness of the scheme in analysing noisy time series, particularly those involving colored noise, is investigated. One interesting result we have obtained is that, for the same percentage of noise addition, data with colored noise is more distinguishable from the corresponding surrogates than data with white noise. As examples of real life applications, analyses of data from an astrophysical X-ray object and a human brain EEG are presented.

The Nonlinear Behavior of the Black Hole System GRS 1915+105

Using nonlinear time series analysis, along with surrogate data analysis, it is shown that the various types of long-term variability exhibited by the black hole system GRS 1915+105 can be explained in terms of a deterministic nonlinear system with some inherent stochastic noise. Evidence is provided for a nonlinear limit cycle origin of one of the low-frequency QPOs detected in the source, while some other types of variability could be due to an underlying low-dimensional chaotic system. These results imply that the partial differential equations that govern the magnetohydrodynamic flow of the inner accretion disk can be approximated by a small number (≈3-5) of nonlinear but ordinary differential equations. While this analysis does not reveal the exact nature of these approximate equations, they may be obtained in the future, after results of magnetohydrodynamic simulation of realistic accretion disks become available.

THE CHAOTIC BEHAVIOR OF THE BLACK HOLE SYSTEM GRS 1915+105

A modified nonlinear time series analysis technique, which computes the correlation dimension D2, is used to analyze the X-ray light curves of the black hole system GRS 1915+105 in all 12 temporal classes. For four of these temporal classes, D2 saturates to ≈4-5, which indicates that the underlying dynamical mechanism is a low-dimensional chaotic system. Of the other eight classes, three show stochastic behavior, while five show deviation from randomness. The light curves for four classes that depict chaotic behavior have the smallest ratios of the expected Poisson noise to the variability ( 0.2). This suggests that the temporal behavior of the black hole system is governed by a low-dimensional chaotic system whose nature is detectable only when the Poisson fluctuations are much smaller than the variability.

On interrelations of recurrences and connectivity trends between stock indices

In this paper, we present the results of Monte Carlo simulations for two popular techniques of long-range correlation detection — classical and modified rescaled range analyses. A focus is put on an effect of different distributional properties on an ability of the methods to efficiently distinguish between short-term memory and long-term memory. To do so, we analyze the behavior of the estimators for independent, short-range dependent, and long-range dependent processes with innovations from eight different distributions. We find that apart from a combination of very high levels of kurtosis and skewness, both estimators are quite robust to distributional properties. Importantly, we show that R/S is biased upwards (yet not strongly) for short-range dependent processes, while M-R/S is strongly biased downwards for long-range dependent processes regardless of the distribution of innovations.Financial data has been extensively studied for correlations using Pearson’s crosscorrelation coefficient ρ as the point of departure. We employ an estimator based on recurrence plots — the correlation of probability of recurrence (CPR) — to analyze connections between nine stock indices spread worldwide. We suggest a slight modification of the CPR approach in order to get more robust results. We examine trends in CPR for an approximately 19-month window moved along the time series and compare them to trends in ρ. Binning CPR into three levels of connectedness (strong, moderate, and weak), we extract the trends in number of connections in each bin over time. We also look at the behavior of CPR during the dot-com bubble by shifting the time series to align their peaks. CPR mainly uncovers that the markets move in and out of periods of strong connectivity erratically, instead of moving monotonically towards increasing global connectivity. This is in contrast to ρ, which gives a picture of ever-increasing correlation. CPR also exhibits that time-shifted markets have high connectivity around the dot-com bubble of 2000.We use significance tests using twin surrogates to interpret all the measures estimated in the study.

Anticipatory synchronization with variable time delay and reset

A method to synchronize two chaotic systems with anticipation or lag, coupled in the drive response mode, is proposed. The coupling involves variable delay with three time scales. The method has the advantage that synchronization is realized with intermittent information about the driving system at intervals fixed by a reset time. The stability of the synchronization manifold is analyzed with the resulting discrete error dynamics. The numerical calculations in standard systems such as the Rössler and Lorenz systems are used to demonstrate the method and the results of the analysis.

Computing the multifractal spectrum from time series: an algorithmic approach.

We show that the existing methods for computing the f(alpha) spectrum from a time series can be improved by using a new algorithmic scheme. The scheme relies on the basic idea that the smooth convex profile of a typical f(alpha) spectrum can be fitted with an analytic function involving a set of four independent parameters. While the standard existing schemes [P. Grassberger et al., J. Stat. Phys. 51, 135 (1988); A. Chhabra and R. V. Jensen, Phys. Rev. Lett. 62, 1327 (1989)] generally compute only an incomplete f(alpha) spectrum (usually the top portion), we show that this can be overcome by an algorithmic approach, which is automated to compute the D(q) and f(alpha) spectra from a time series for any embedding dimension. The scheme is first tested with the logistic attractor with known f(alpha) curve and subsequently applied to higher-dimensional cases. We also show that the scheme can be effectively adapted for analyzing practical time series involving noise, with examples from two widely different real world systems. Moreover, some preliminary results indicating that the set of four independent parameters may be used as diagnostic measures are also included.

Bubbling and bistability in two parameter discrete systems

We present a graphical analysis of the mechanisms underlying the occurrences of bubbling sequences and bistability regions in the bifurcation scenario of a special class of one dimensional two parameter maps. The main result of the analysis is that whether it is bubbling or bistability is decided by the sign of the third derivative at the inflection point of the map function.

Uniform framework for the recurrence-network analysis of chaotic time series.

We propose a general method for the construction and analysis of unweighted ε-recurrence networks from chaotic time series. The selection of the critical threshold ε_{c} in our scheme is done empirically and we show that its value is closely linked to the embedding dimension M. In fact, we are able to identify a small critical range Δε numerically that is approximately the same for the random and several standard chaotic time series for a fixed M. This provides us a uniform framework for the nonsubjective comparison of the statistical measures of the recurrence networks constructed from various chaotic attractors. We explicitly show that the degree distribution of the recurrence network constructed by our scheme is characteristic to the structure of the attractor and display statistical scale invariance with respect to increase in the number of nodes N. We also present two practical applications of the scheme, detection of transition between two dynamical regimes in a time-delayed system and identification of the dimensionality of the underlying system from real-world data with a limited number of points through recurrence network measures. The merits, limitations, and the potential applications of the proposed method are also highlighted.

Transition to chaos in a driven pendulum with nonlinear dissipation

The Melnikov-Holmes method is used to study the onset of chaos in a driven pendulum with nonlinear dissipation. Detailed numerical studies reveal many interesting features like a chaotic attractor at low frequencies, band formation near escape from the potential well and a sequence of subharmonic bifurcations inside the band that accumulates at the homoclinic bifurcation point.