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==Detailed description==
One way to visualize the recurring nature of states by their trajectory through a [[phase space]] is the recurrence plot, introduced by Eckmann et al. (1987)
| author=J. P. Eckmann, S. O. Kamphorst, [[David Ruelle|D. Ruelle]]
| title=Recurrence Plots of Dynamical Systems
| journal=Europhysics Letters
| volume=5
| issue=9
| pages=973–977
| year=1987
| doi=10.1209/0295-5075/4/9/004
| bibcode=1987EL......4..973E
| s2cid=250847435
}}</ref>. 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. One frequently used tool to study the behaviour of such phase space trajectories is then the [[Poincaré map]]. Another tool, is the recurrence plot, which enables us to investigate many aspects of the ''m''-dimensional phase space trajectory through a two-dimensional representation.
At a '''recurrence''' the trajectory returns to a ___location (state) in phase space it has visited before up to a small error <math>\varepsilon</math> . The recurrence plot represents the collection of pairs of times of such recurrences, i.e., the set of <math>(i,j)</math> with <math>\vec{x}(i) \approx \vec{x}(j)</math>, with <math>i</math> and <math>j</math> discrete points of time and <math>\vec{x}(i)</math> the state of the system at time <math>i</math> (___location of the trajectory at time <math>i</math>).
Mathematically, this
:<math>R(i,j) = \begin{cases} 1 &\text{if} \quad \| \vec{x}(i) - \vec{x}(j)\| \le \varepsilon \\ 0 & \text{otherwise}, \end{cases}</math>
where <math>\| \cdot \|</math> is a norm and <math>\varepsilon</math> the recurrence threshold.
:<math>R(i,j)=\Theta(\varepsilon - \| \vec{x}(i)- \vec{x}(j) \|).</math>
If only a [[time series]] is available, the phase space can be reconstructed, e.g., by using a time delay embedding (see [[Takens' theorem]]):▼
The recurrence plot visualises <math>\mathbf{R}</math> with coloured (mostly black) dot at coordinates <math>(i,j)</math> if <math>R(i,j)=1</math>, with time at the <math>x</math>- and <math>y</math>-axes.
▲If only a [[time series]] <math>u(i)</math> is available, the phase space can be reconstructed, e.g., by using a time delay embedding (see [[Takens' theorem]]):
:<math>\vec{x}(i) = (u(i), u(i+\tau), \ldots, u(i+\tau(m-1)),</math>
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[[Image:Rp examples740.gif|thumb|center|740px|Typical examples of recurrence plots (top row: [[time series]] (plotted over time); bottom row: corresponding recurrence plots). From left to right: uncorrelated stochastic data ([[white noise]]), [[harmonic oscillation]] with two frequencies, chaotic data ([[logistic map]]) with linear trend, and data from an [[autoregressive process|auto-regressive process]].]]
The small-scale structures in RPs are used by the [[recurrence quantification analysis]]<ref>{{cite
| journal=Physics Reports | volume=438 | issue=5–6 | year=2007 | doi=10.1016/j.physrep.2006.11.001 | pages=237 |bibcode = 2007PhR...438..237M Close returns plots are similar to recurrence plots. The difference is that the relative time between recurrences is used for the <math>y</math>-axis (instead of absolute time).
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