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# Recurrence plot

Natural processes can have a distinct recurrent behaviour, e.g. periodicities (as seasonal or Milankovich cycles ), but also irregular cyclicities (as El Niño Southern Oscillation). Moreover, the recurrence of states, in the meaning that states are arbitrarily close after some time, is a fundamental property of deterministic dynamical systems and is typical for nonlinear or chaotic systems. The recurrence of states in nature has been known for a long time and has also been discussed in early publications (e.g. recurrence phenomena in cosmic-ray intensity, Monk 1939).

Eckmann et al. (1987) [No reference to a paper of Eckmann et al published in 1987 appears below. Could someone add that?] have introduced a tool which can visualize the recurrence of states in a phase space. Usually, a phase space does not have a dimension (two or three) which allows it to be pictured. Higher dimensional phase spaces can only be visualized by projection into the two or three dimensional sub-spaces. However, Eckmann's tool enables us to investigate the m-dimensional phase space trajectory through a two-dimensional representation of its recurrences. Such recurrence of a state at time i at a different time j is pictured within a two-dimensional squared matrix with black and white dots, where black dots mark a recurrence, and both axes are time axes. This representation is called recurrence plot (RP). Such an RP can be mathematically expressed as

$R(i,j) = \Theta(\varepsilon - || \vec{x}(i) - \vec{x}(j)||), \quad \vec{x}(i) \in \mathbf{R}^m, \quad i, j=1 \dots N,$

where N is the number of considered states $\vec{x}(i)$, $\varepsilon$ is a threshold distance, $|| \cdot ||$ a norm (e.g. Euclidean norm) and $\Theta( \cdot )$ the Heaviside step function.

Close returns plots are similar to recurrence plots. The difference is that the relative time between recurrences is used for the y-axis (instead of absolute time).

Recent developments allow the quantification of recurrence plots (Zbilut and Weber, 1992) and, thus, the study of transitions or interesting nonlinear parameters in the data.

[An explanation of this graph is needed here.]

## References

[Eckmann et al and Zlibut and Webber are mentioned above, so they need to get listed here.]