Recurrence plot: Difference between revisions

Revision as of 11:28, 1 March 2023

Revision as of 22:03, 4 January 2024

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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). 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.

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 such recurrences, i.e., the systemset returnsof to<math>(i,j)</math> awith <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 thatof itthe hassystem beforeat time <math>i</math> (___location of the trajectory at time <math>i</math>).

The recurrence plot represents the collection of pairs of times 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 can be expressed by the binary recurrence matrix