Nonlinear measures and dynamics from time series data

Abstract

Our attempts to understand the dynamics of natural systems often rely on the observations or average responses of the system. The science of extracting information from this time series of observations forms the area of time series analysis. The underlying dynamics of natural systems is nonlinear and therefore complex. This nonlinearity leaves signatures in the dynamics of the system, captured by the time series of its variables. Nonlinear time series analysis strives to seek and interpret these signatures in order to capture the nature of the underlying dynamics. This exercise is complicated by the presence of various shortcomings of real world data. Some of the primary problems we encounter in this respect are noise, data gaps and finite size of data. This thesis attempts to address these problems in the context of nonlinear time series analysis.

The origins of dynamical systems theory can be traced back to the celestial. Many of the ideas of this theory were developed while trying to address the three body problem. Subsequently, nonlinear dynamics has found applications in many fields of astrophysics. These include stellar pulsations, accretion disc physics, galactic models and so on. The use of time series analysis in astrophysics has been limited in the past due to the absence of long, continuous and high quality datasets. The advent of space telescopes has begun to overcome some of these problems. Subsequently period doubling, chaotic and strange non chaotic behavior etc have been observed in a large number of stars. This thesis aims to explore the dynamics of variable stars in more detail, seeking out signatures of nonlinearity and chaos in various scenarios.

After exploring the basics of nonlinear time series analysis and the physics of variable stars, we proceed to address the issues commonly encountered in the nonlinear time series analysis of real world data. We identify three main issues in this context, namely the presence of datagaps, noise and finite sizes of data.

We start by analyzing the effect that data gaps have on the estimation of the correlation dimension and multifractal spectrum from time series data. This is implemented by introducing gaps into long evenly sampled datasets of standard nonlinear dynamical systems. The frequency and size of gaps introduced are drawn from two Gaussian distributions, with varying means. After the introduction of gaps, the time series is merged, ignoring the gaps and the quantifier of interest is calculated for this gap-affected or unevenly sampled time series. The variation of the value of the quantifier from the evenly sampled value then enables us to identify a region where reliable conclusions can be drawn about the nature of the underlying dynamics of the system. We use the analysis to calculate the D2 of pulsating variable stars, from their light curves. We also use the results of the analysis to calculate the multifractal spectrum of multiple ecological and meteorological time-series. We use the method of surrogate data testing to establish that the multifractality arises as a result of deterministic nonlinearity.

We then proceed to address the problem of noise while identifying the underlying dynamics of a nonlinear dynamical system. We point out the difficulty in differentiating between limit cycle and chaotic dynamics, in a dynamical system evolving in the presence of noise. We show that the bicoherence, which is a higher order spectrum proves to be a useful tool in differentiating between the two dynamical states.

Identifying strange non chaotic behavior from data is a major challenge. Spectral scaling of peaks in the strobed power spectrum is one of the most accepted methods to identify strange non chaotic behavior. We show that noise contaminated quasiperiodicity shows the exact same scaling behavior as a strange non chaotic time series. We show that the use of a bicoherence based filter while identifying peaks during scaling can differentiate between the two dynamical states. We further use the bicoherence to analyse the underlying dynamics of RR Lyrae stars. We show that RRab Lyrae stars may be exhibiting chaotic behavior. We show that RRc Lyrae stars may be grouped into two subclasses that exhibit strange non chaotic and quasiperiodic dynamics.

Recurrence networks have been shown to be successful in addressing the problem of small data sizes. We utilize the power of recurrence networks to distinguish between the different classes of RRc Lyrae stars identified using the bicoherence analysis. We also show that the recurrence networks quantifiers can help classify close binary stars into semi detached, over-contact and ellipsoidal binaries.

We also use the methods of time series analysis to study the Kepler light curves of over contact binary stars. Using the correlation dimension, multifractal spectrum and the bicoherence, we see that most overcontact binaries show chaotic dynamics. We also find that the extend of contact between the component stars in these cases is significantly correlated with its nonlinear properties, specifically the correlation dimension and bicoherence. Hence the computation of these nonlinear measures gives us an estimate of the extend of contact between these stars.

The thesis is arranged as follows. In Chapter 1 and 2 we explore the basics of nonlinear time series analysis and the physics of variable stars respectively.

In Chapter 3 we describe the effect that data gaps have on estimating the correlation dimension and multifractal spectrum from time series data and use the results to analyze variable stars light curves and multiple ecological and meteorological time-series.

In Chapter 4 we use the bicoherence function to distinguish between periodic states contaminated with noise and chaotic and strange non chaotic states.

In Chapter 5 we use the correlation dimension, multifractal spectrum and the bicoherence to study the dynamics of overcontact binary stars.

In Chapter 6 we use recurrence networks to classify RRc Lyrae stars and contact binary stars into various sub categories.

In the concluding chapter we summarize the importance of the results presented in the thesis and provide future directions to the work carried out. Identifying noisy quasiperiodicity and distinguishing it from strange non chaotic behavior is a difficult task, which we could successfully address in our study. We could relate measures from nonlinear time series analysis to astrophysical properties of the stars studied. This could lead to classification of different types of astrophysical systems.