Recurrence plot: Difference between revisions

Revision as of 23:52, 2 February 2018

Revision as of 17:38, 9 February 2019

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Eckmann et al. (1987) introduced recurrence plots, which provide a way to visualize the periodic nature of a trajectory through a [[phase space]]. Often, the phase space does not have a low enough dimension (two or three) to be pictured, since higher-dimensional phase spaces can only be visualized by projection into the two or three-dimensional sub-spaces. However, making a recurrence plot enables us to investigate certain aspects of the ''m''-dimensional phase space trajectory through a two-dimensional representation.

A '''recurrence''' is a time the trajectory returns to a ___location it has visited before. The recurrence plot depicts the collection of pairs of times at which the trajectory is at the same place, i.e. the set of <math>(i,j)</math> with <math>\vec{x}(i) = \vec{x}(j)</math>. This can show many things,: for instance, if the trajectory is strictly periodic with period <math>T</math>, then all such pairs of times will be separated by a multiple of <math>T</math> and visible as diagonal lines. To make the plot, continuous time and continuous phase space are discretized, taking e.g. <math>\vec{x}(i)</math> as the ___location of the trajectory at time <math>i \tau</math> and counting as a recurrence any time the trajectory gets sufficiently close (say, within &epsilon;) to a point it has been previously. Concretely then, recurrence/non-recurrence can be recorded by the binary function

:<math>R(i,j) = \begin{cases} 1 \quad &\text{if} \quad \| \vec{x}(i) - \vec{x}(j)\| \le \varepsilon \\ 0 \quad & \text{otherwise}, \end{cases}</math>