Mathematical and Computational Foundations of Recurrence Quantifications

Abstract

Real-world systems possess deterministic trajectories, phase singularities and noise. Dynamic trajectories have been studied in temporal and frequency domains, but these are linear approaches. Basic to the field of nonlinear dynamics is the representation of trajectories in phase space. A variety of nonlinear tools such as the Lyapunov exponent, Kolmogorov–Sinai entropy, correlation dimension, etc. have successfully characterized trajectories in phase space, provided the systems studied were stationary in time. Ubiquitous in nature, however, are systems that are nonlinear and nonstationary, existing in noisy environments all of which are assumption breaking to otherwise powerful linear tools. What has been unfolding over the last quarter of a century, however, is the timely discovery and practical demonstration that the recurrences of system trajectories in phase space can provide important clues to the system designs from which they derive. In this chapter we will introduce the basics of recurrence plots (RP) and their quantification analysis (RQA). We will begin by summarizing the concept of phase space reconstructions. Then we will provide the mathematical underpinnings of recurrence plots followed by the details of recurrence quantifications. Finally, we will discuss computational approaches that have been implemented to make recurrence strategies feasible and useful. As computers become faster and computer languages advance, younger generations of researchers will be stimulated and encouraged to capture nonlinear recurrence patterns and quantification in even better formats. This particular branch of nonlinear dynamics remains wide open for the definition of new recurrence variables and new applications untouched to date.

Appendix: Mathematical Models

To illustrate our statements we are using some prototypical model systems, which are listed below:

1.1 Auto-Regressive Process of 1st Order

$$\displaystyle{ x_{i} = 0.95x_{i-1} + 0.05x_{i-2} + 0.9\xi \quad \text{where }\ \xi \ \mathrm{is\ white\ Gaussian\ noise.}}$$

(1.33)

1.2 Lorenz System [17]

$$\displaystyle\begin{array}{rcl} \dot{x}& =& -\sigma (x - y), \\ \dot{y}& =& -xz + rx - y, \\ \dot{z}& =& xy - bz. {}\end{array}$$

(1.34)

1.3 Rössler System [54]

$$\displaystyle\begin{array}{rcl} \dot{x}& =& -y - z, \\ \dot{y}& =& x + a\,y, \\ \dot{z}& =& b + z\,(x - c).{}\end{array}$$

(1.35)

1.4 Mutually Coupled Rössler Systems

$$\displaystyle\begin{array}{rcl} \dot{x}_{1}& =& -(1+\nu )x_{2} - x_{3} +\mu (y_{1} - x_{1}), \\ \dot{x}_{2}& =& (1+\nu )x_{1} + a\,x_{2}, \\ \dot{x}_{3}& =& b + x_{3}\,(x_{1} - c), {}\end{array}$$

(1.36)

$$\displaystyle\begin{array}{rcl} \dot{y}_{1}& =& -(1-\nu )y_{2} - y_{3} +\mu (x_{1} - y_{1}), \\ \dot{y}_{2}& =& (1-\nu )y_{1} + a\,y_{2}, \\ \dot{y}_{3}& =& b + y_{3}\,(y_{1} - c). {}\end{array}$$

(1.37)