A modified non-linear time series analysis technique, which computes the correlation dimension D 2 , is used to analyze the X-ray light curves of the black hole system GRS 1915+105 in all twelve temporal classes. For four of these temporal classes D 2 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 which depict chaotic behavior have the smallest ratio of the expected Poisson noise to the variability (< 0.05) while those for the three classes which depict stochastic behavior is the highest (> 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.

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 is tested using different types of time series, such as, those from standard chaotic systems, pure white and colored noise and chaotic systems added with noise. The effectiveness of the scheme in analysing noisy time series, particularly those involving colored noise, is investigated. An 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 for real life applications, analysis of data from an astrophysical X-ray object and human brain EEG, are presented.

Using non-linear 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 non-linear system with some inherent stochastic noise. Evidence is provided for a non-linear limit cycle origin of one of the low frequency QPO 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 which govern the magneto-hydrodynamic flow of the inner accretion disk, can be approximated by a small number (≈ 3 − 5) of non-linear 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.

GRS 1915+105 is a prominent black hole system exhibiting variability over a wide range of time scales and its observed light curves have been classified into 12 temporal states. Here we undertake a complete analysis of these light curves from all the states using various quantifiers from nonlinear time series analysis, such as, the correlation dimension (D 2 ), the correlation entropy (K 2 ), singular value decomposition (SVD)and the multifractal spectrum (f (α) spectrum). An important aspect of our analysis is that, for estimating these quantifiers, we use algorithmic schemes which we have proposed recently and tested successfully on synthetic as well as practical time series from various fields. Though the schemes are based on the conventional delay embedding technique, they are automated so that the above quantitative measures can be computed using conditions prescribed by the algorithm and without any intermediate subjective analysis. We show that nearly half of the 12 temporal states exhibit deviation from randomness and their complex temporal behavior could be approximated by a few (3 or 4) coupled ordinary nonlinear differential equations. These results could be important for a better understanding of the processes that generate the light curves and hence for modelling the temporal behavior of such complex systems. To our knowledge, this is the first complete analysis of an astrophysical object (let alone a black hole system) using various techniques from nonlinear dynamics.

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.