Recurrence plot

A Rekurrenzplot ( " return " from the Latin recurrere ) is a modern method of non-linear data analysis. The return of property is typical of deterministic dynamical systems ( chaos, nonlinear dynamics) and is also reflected in many natural processes, such as El Niño, Milankovitch cycles or sunspot cycles. Return not means that exactly the original condition occurs again, but that he was only made arbitrarily close again. Already Poincaré had postulated the infinite recurrence of states.

The method of Rekurrenzplots was introduced in 1987 by Eckmann. It is used for the representation of higher-dimensional phase space trajectories.

Description

The Rekurrenzplot is a square matrix with two time axes. In this matrix, those time pairs are represented by black dots whose states are nearly equal, ie when the corresponding condition is returned. The return is usually determined from the distance between all pairs of data:

Here is the Heaviside function, the maximum distance and a norm, such as the Euclidean norm.

The appearance of the Rekurrenzplots is determined by the behavior of the phase space trajectory. There is a distinction between the small structures, such as single points, diagonal lines or vertical lines, and the overall impression of the plot (texture).

Recent developments allow further investigation of data by a quantitative evaluation of Rekurrenzplots ( Zbilut and Webber, 1992; Marwan et al, 2002. ).

A Close returns plot is a Rekurrenzplot with a slightly different mode of application of recurrence times. Here, the y- axis is not equal to the absolute time, but the time difference (that is, the time after which the state returns ).