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Aimefournier (talk | contribs) m →Detailed description: Clarifying that linear trend is separate from logistic map |
Aimefournier (talk | contribs) m →Detailed description: Remedied that ε had two meanings |
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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>. 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</math> and counting as a recurrence any time the trajectory gets sufficiently close (say, within
Operationally the plot is drawn as follows:
(a) A certain time window <math>\vec{w} = <t_1, t_2, ..., t_T></math> is chosen where any two successive time steps are separated by the time interval <math>\
(b) A 2D plot is created where the x-axis and y-axis both report <math>\vec{w}</math>, forming a <math>T \times T</math> lattice of little squares each with side measuring <math>\
(c) The data <math>\mathbf{X}</math> are used to compute a matrix <math>\mathbf{R}</math> formed by binary elements recording the recurrence/non-recurrence of values <math>\vec{x}</math> through the binary function:
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