Editorial

Abstract

Researchers in different subject areas are tackling similar questions that require a complex systems approach: Can we develop indicators that serve as warning signals for impending regime shifts or critical thresholds in the system behavior? What is the characteristic observation scale that allows for an optimal description of the system dynamics in space and time? How can the resilience (return time to equilibrium) and stability (resistance to external forcing) of a system subjected to disturbance regimes be quantified? Can we derive generalities on how natural and man-made systems develop in time? Amongst interacting units of the system, which ones are the keystones for its global functioning? In this context, recurrence plots are useful tools as they provide a common language for the study of complex systems [Webber, Jr. et al., 2009]. Furthermore, since the method does not require fitting a specific model to data, recurrence plots logically preserve a maximum of the information that is embedded in the sequence of observations. Recurrence plot based methods are also powerful visual and quantitative tools for pattern detection and, as such, continue to birth novel research hypotheses as well as challenge established ones. The ability of these methods to cope with nonlinear and transient behaviors, as well as observational noise is of particular interest to experimentalists and empiricists.The recurrence plot (RP) was introduced more than twenty years ago in order to visualize the dynamics of complex systems by their recurrences. The first natural extension was the quantification of small-scale structures in an RP by means of recurrence quantification analysis (RQA). RQA has turned out to be a powerful tool for distinguishing different types of dynamics, detecting dynamical transitions, or describing changes in the complexity of spatial objects [Marwan et al., 2007;Marwan, 2008].

Strong Laws for Recurrence Quantification Analysis

“…Thus the first equality is an immediate consequence of the definition of diagonal lines and the second one follows from (10). Further, (6) gives…”

“…Proof. By (4), d m k (x ∞ i , x ∞ j ) ≤ r if and only if (x i+l , x j+l ) ≤ r for every 0 ≤ l < h. Thus the first equality is an immediate consequence of the definition of diagonal lines and the second one follows from (10). Further, (6) gives…”

“…As a corollary of Proposition 1 we can immediately obtain a formula for the determinism in terms of correlation sums. Since (10) gives…”

“…Since the seminal paper [18], new RQA tools, quantities and modifications were introduced and recurrence quantification has been applied in many areas of science, cf. [12] and [10], among others.…”