Eckmann et al. (1987) introduced recurrence plots, which provide aOne way to visualize the periodicrecurring nature of a trajectory through a [[phase space]] is the recurrence plot, introduced by Eckmann et al. (1987). 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.
AAt a '''recurrence''' is a time the trajectory returns to a ___location in phase space it has visited before (i.e., the system returns to a state that it had before). The recurrence plot depictsrepresents the collection of pairs of times at which the trajectory is at the samesuch placerecurrences, i.e., the set of <math>(i,j)</math> with <math>\vec{x}(i) =\approx \vec{x}(j)</math>. To make the plot, continuouswith time<math>i</math> and continuous<math>j</math> phasediscrete spacepoints areof discretized,time taking e.g.and <math>\vec{x}(i)</math> as the ___locationstate of the trajectorysystem at time <math>i</math> and(___location counting as a recurrence any timeof the trajectory getsat sufficiently close (say, withintime <math>\varepsiloni</math>) to a point it has been previously.
