# Time series-based bifurcation analysis

Re-posting this one from my old university website. A brief introduction of my graduate research topic.

Studies in nonlinear time series analysis have provided reliable techniques for the evaluation of signals from dynamical systems. Some of these techniques are used to gain insights into the unknown physical processes, to do prediction, as well as to determine invariants associated to the dynamics of the system. Others are employed to determine whether irregularities in signals are due to the intrinsic nonlinearity of the system or are caused by extrinsic random processes impinging on the system. Several others are applied to build models capturing the dynamics of the system from the observed data. In this study, new methods to uncover the underlying mechanisms of dynamical systems using time series are introduced. This set of algorithms can be used to evaluate how sensitive the system is to the values of its parameters and how the system’s behavior changes as the parameters are varied. Since these issues are best explored by means of **bifurcation diagrams** (BDs), this work describes the algorithms in constructing BDs from time series.

Extracting physically interesting and useful information from the observed data is the primary goal of time series analysis. Nonlinear time series analysis has particularly provided additional tools for the characterization of irregular, broadband signals that are products of nonlinear dynamical systems. Without these tools, these signals are incomprehensible observations rather than vital sources of physically interesting information. Since these signals are prevalent in nature, their evaluation is very important and relevant. A few examples include the electrical activities within the brain, the beating of the heart, the spiking of neurons, the spread of epidemics, the swings in animal populations, and the changes in global climate. The analyses of these signals can be loosely grouped into: 1) the reconstruction and quantification of attractors, and 2) model-building and prediction, which includes parameter estimation using time series.

Dissipative systems are typically characterized by the presence of attracting sets or **attractors **in the phase space. An attractor is a bounded subregion of the phase space of a dynamical system to where regions of initial conditions of nonzero volumes eventually converge with increasing time. It can be a point in the phase space, of dimension equal to 0, or a closed curve, of dimension equal to 1. Some other attracting sets can be very irregular and in fact, can have a dimension which is not an integer value. Such sets are called **fractals**, and when they are attractors they are referred to as strange attractors. Strange attractors can be characterized by a spectrum of dimension values such as **box-counting dimension**, **information dimension**, and **correlation dimension**. The motion on a strange attractor can also display **sensitivity to initial conditions** such that the distance between neighboring points on the attractor can grow exponentially with time. This motion is referred to as being **chaotic**. The existence of chaos means that small errors grow exponentially in time that long term prediction becomes impossible. A quantitative description of the sensitivity to initial condition is provided by the **Lyapunov exponents**, quantities characterizing the stretching of infinitesimal displacements in a strange attractor.

The reconstruction of attractors by **delay embedding** and their quantification in terms of dimensions, Lyapunov exponents, among other attractor invariants have yielded means of revealing intrinsic nonlinear behavior of a dynamical system from time series. For example, the presence of a positive Lyapunov exponent or a fractional value of the attractor’s dimension affirms the nonlinear nature of the system. These invariants can also be used to identify systems in a manner similar to natural frequencies of some physical systems. These quantities, however, do not give a complete description of the dynamics *per se* and thus, a different set of methods is required.

On the other hand, the goal of model-building is to construct a template of the dynamics using the observed data. Loosely speaking, this can be done by obtaining an appropriate set of coefficients in a predetermined class of functional forms such that the resulting function captures the dynamics of the system under study. The model is used either to represent the global behavior of the observed data or to describe the local dynamics in the reconstructed attractor or a mixture of both. The use of nonlinear autoregressive models, the measure-based functional reconstruction of Giona, radial basis functions, and neural networks are among the many functions that represent global models describing the dynamics in the whole phase-space. On the other hand, local linear maps using neighboring points, local averaging, and the use of higher-order polynomials whose coefficients are fitted using near neighbors are just a few of the many local modeling approaches. The effectiveness of the model is measured by its **prediction performance**, defined as the ability of the model to give accurate values at steps forward in time.

The description of dynamical systems in terms of invariants or sets of coefficients in a predetermined class of functions is already sufficient to solve many significant problems. However, these invariants, though they remain constant with coordinate transformation, are not robust to changes in system parameters. Model coefficients also vary from one observation to another when observations are taken at different parameter values. Thus, problems that involve changes in parameters require a broader framework than the above description. This framework is provided by the study of **bifurcations**. A bifurcation is a qualitative change in the dynamics, for example from a stable behavior to an unstable behavior, which occurs as a system parameter varies. The knowledge of the bifurcation structure of a dynamical system is therefore important in order to understand the system’s response to the changes in parameter values. This is particularly necessary for the case of nonlinear systems where small perturbations, for parameter values near critical points, can cause dramatic changes in the system’s output. The study of bifurcations from time series, however, has received less attention in the past years. It is only recently that this problem is addressed rigorously. The problem is that the analysis requires *a priori* knowledge of the dynamics in the form of differential or difference equations that can prove difficult to construct even for simple systems. This makes the problem of reconstructing bifurcation structures from time series a formidable task.

In this research, a new and equally important theme in nonlinear time series analysis, **the study of bifurcations**, is introduced. The tools developed can be used to analyze time series measured at different parameter values. The motivation of the study is to unveil the bifurcation structure of the system using the observed data. In particular, the study aims to: 1) know the sequence of bifurcations that the system undergoes as the parameters are changed; and 2) uncover behaviors of the system, which may be present but not readily observed. To achieve these goals, the problem of reconstructing bifurcation diagrams is systematically investigated. Methods to obtain qualitatively the same BD as that of the given system using time series at a finite number of parameter values are presented. The reconstruction does not assume any knowledge of the explicit form of the dynamical system (differential/difference equation). Instead, time series at different parameter values are used to obtain a suitable family of predictor functions, which exhibits qualitatively similar bifurcations as the given system. The BD of this family of predictor functions on some parameter region, termed as **projection region**, is then regarded as the reconstructed BD. In other words, the projection region is the region in the parameter space of the model with similar bifurcation structure as the system. For parameter values within this region, the dynamics of the model is therefore the same as that of the given system. Thus, one can take the BD of the model in this region as the reconstructed BD. The problem therefore is to determine the projection region using parameter values computed from the available time series.