Dynamical systems are usually modelled as a set of coupled differential equations. Information regarding the temporal evolution of real world systems is most often available in the form of a scalar time series in a single variable (example includes light curve from a star, data from Electrocardiogram etc.). For the past few decades, the conventional methods like estimation of fractal or correlation dimension, Lyapunov exponents and mutual information have been widely used for studying the underlying dynamics of a real world system using its time series. However, most of these methods require long time series data. Modern data analysis techniques have been achieving more popularity in many scientific disciplines for understanding the underlying complex dynamics of the system. In the last two decades, method of recurrence plot has been developed as a new technique for the analysis of time series of complex dynamical systems. Here, the fundamental property of all dynamical systems, namely the recurrence of trajectory points, is utilised for the construction of the recurrence plot (RP). With this new method, a large and diverse amount of information regarding the dynamics of the underlying system can be extracted and statistically quantified using the measures developed for the analysis of RP. The methods of RP have been further developed by constructing complex networks from time series using properties of recurrence. Such networks are called recurrence networks (RN) and they have become one of the major tools in modern nonlinear time series analysis. Over the last few years, we have developed, in collaboration with R. Misra of IUCAA, several methods and measures for characterising time series using RN.
The details of our work done during the last three years are listed below.
1. Uniform framework for the RN analysis of time series
In this work, we try to generalise the methods for the construction and analysis of un-weighted recurrence network. For that, a new scheme is proposed for the selection of critical threshold that provides a uniform framework for the non-subjective comparison of the statistical measures of the recurrence networks constructed from various chaotic attractors. Also shown that the degree distributions of the recurrence network constructed by our scheme is characteristic to the structure of the attractor. Two practical applications of our scheme is also demonstrated : (a) detection of transition between two dynamical regimes in a time – delayed system (b) identification of the dimensionality of the underlying system from real world data with a limited number of points through recurrence network measures. Advantage of Recurrence network measures over conventional methods is that they can be applied on short and non stationary time series data.
Ref: Rinku Jacob, K. P. Harikrishnan, R. Misra and G. Ambika, Phys. Rev. E, Vol.93, 012202, 2016.
2. Characterization of chaotic attractors under noise
In this work, a detailed numerical investigation has been done to understand the effect of white and colored noise addition (additive noise) to a chaotic time series and how it changes the topology and structure of the underlying attractor. We made an observation that, the structure of the attractor is found to be robust even up to high noise levels of 50%. We also analysed the light curves from a dominant black hole system (GRS 1915+105) and shown that the recurrence network measures are capable of identifying the nature of noise contamination in a time series.
Ref: Rinku Jacob, K. P. Harikrishnan, R. Misra and G. Ambika, Comm. Nonlinear Sci. Num Simulations, Vol. 41, 32, 2016.
3. Surrogate analysis using network measures
In this work, surrogate analysis is done on the time series using the network measures by converting the time series into recurrence network using our scheme and we show that, the characteristic path length (a measure in network analysis) is especially efficient as a discriminating measure even if the time series is short. As an application to real world data, we use the light curves from GRS 1915+105 and show that a combined analysis using three primary network measures can provide vital information regarding the nature of the temporal variability of light curves from different spectroscopic classes.
Ref: Rinku Jacob, K. P. Harikrishnan, R. Misra and G. Ambika, Comm. Nonlinear Sci. Num Simulations, Vol. 54, 84, 2018 .
Future goals :
Our future goal is to study the effect of additive noise on a standard model (say Lorenz system) and to compare the results with the time series (here it is light curve) from a dominant black hole system GRS 1915+105, as an attempt to understand the reason for the existence of different classes of light curves categorized as a result of switching between three temporal states by Beloni et al.