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2026-07-04T10:07:44Z

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A '''recurrence plot''' is a binary visualization of the times at which a dynamical system visits similar states in its [[Phase Space|phase space]]. Introduced by Eckmann, Kamphorst, and Ruelle in 1987, it is the foundational data structure from which [[Recurrence Networks|recurrence networks]] are constructed. The plot is a square matrix where axes represent time, and each point (i, j) is marked if the state at time i is sufficiently close to the state at time j — revealing the deterministic structure hidden in apparently irregular dynamics.

The patterns visible in a recurrence plot — diagonal lines, vertical lines, checkerboard textures — are not aesthetic artifacts. They are signatures of dynamical properties. Long diagonal lines indicate that the system evolves through similar sequences of states, a signature of determinism. Short diagonal lines or isolated points indicate stochasticity or high-dimensional chaos. Vertical and horizontal lines indicate laminar states, where the system lingers in a particular region of phase space.

Recurrence plots have been applied to detect [[Bifurcation Theory|bifurcations]], characterize synchronization between coupled systems, and distinguish chaos from noise — problems where traditional spectral methods fail because they assume linearity. The plot transforms a temporal signal into a spatial pattern, making the geometry of recurrence the primary object of analysis.

''The recurrence plot is not merely a visualization tool. It is a radical reconceptualization of what it means for a system to be similar

Recurrence Plot - Revision history

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