Recurrence Plots Analysis

• Recurrence plots are two-dimensional binary representations of phase space recurrences that clearly illustrate periodic, chaotic, and intermittent dynamics.
• They utilize threshold-based matrices and time-delay embedding to reveal structural patterns and support quantitative analysis through recurrence quantification measures.
• Recent methods combine deep learning and spatial statistics to enhance regime detection and parameter inference in complex dynamical systems.

A recurrence plot (RP) is a two-dimensional, binary representation of all pairwise recurrences of a trajectory in phase space, constructed by marking pairs of time points at which the system’s state returns within a specified distance in the phase space. RPs provide a visual and quantitative framework to elucidate periodicity, chaos, laminarity, and regime shifts in both low- and high-dimensional and spatially extended dynamical systems. In combination with recurrence quantification analysis (RQA) and, more recently, deep learning and novel spatial statistics methods, RPs serve as non-parametric, model-independent tools for the analysis of nonlinear dynamical behavior across scientific disciplines including physics, neuroscience, geosciences, economics, and machine learning.

1. Mathematical Formulation and Construction

Given a trajectory $\{\mathbf{x}_i\}_{i=1}^N$ in $\mathbb{R}^m$ (directly measured or reconstructed via time-delay embedding), the binary recurrence matrix $R$ is defined by

$$R_{ij} = \Theta(\varepsilon - \|\mathbf{x}_i - \mathbf{x}_j\|)$$

for $i, j = 1, \dots, N$, where $\|\cdot\|$ is a chosen norm (commonly Euclidean or maximum norm), $\varepsilon > 0$ is the recurrence threshold, and $\Theta(\cdot)$ is the Heaviside step-function. The RP is visualized as a black-and-white image, where a black dot at $(i, j)$ indicates that the system’s states at times $i$ and $j$ are “close” in phase space.

Phase space trajectories are often reconstructed using Takens’ time-delay embedding: for a scalar series $\{s_i\}$,

$$\mathbf{x}_i = (s_i, s_{i+\tau}, \ldots, s_{i+(m-1)\tau}) \in \mathbb{R}^m$$

with embedding dimension $m$ and delay $\tau$ set via false-nearest-neighbors and the first minimum of mutual information, respectively.

RP Interpretation

• Long diagonal lines: Indicate deterministic or periodic dynamics; repeated evolution.
• Short, broken diagonals: Characterize chaotic behavior due to sensitive dependence and divergence.
• Vertical or horizontal lines: Signal laminar (intermittent, slowly evolving) phases.
• Isolated points: Indicate stochastic, uncorrelated, or high-dimensional noise.

2. Recurrence Quantification Analysis (RQA)

RQA translates the geometric patterns in the RP into quantitative measures that characterize different dynamical regimes. Standard RQA metrics include:

Metric	Mathematical Definition	Interpretation
RR	$RR = \frac{1}{N^2}\sum_{i,j} R_{i,j}$	Recurrence rate (density of black points)
DET	$$DET = \frac{\sum_{\ell=\ell_{\min}}^N \ell\,P(\ell)}{\sum_{\ell=1}^N \ell\,P(\ell)}$$	Degree of determinism (fraction in diagonals)
$L_{\max}$	$L_{\max} = \max\{\ell : P(\ell) > 0\}$	Longest diagonal line, related to stability
DIV	$DIV = 1 / L_{\max}$	Divergence (proxy for Lyapunov exponent)
LAM	$$LAM = \frac{\sum_{v=v_{\min}}^N v\,V(v)}{\sum_{v=1}^N v\,V(v)}$$	Laminarity (vertical lines, intermittency)
ENTR	$$ENTR = -\sum_{\ell=\ell_{\min}}^N p(\ell)\ln p(\ell)$$, $p(\ell) = P(\ell)/\sum P(\ell)$	Entropy of diagonal line distribution
TT	$$TT = \frac{\sum_{v=v_{\min}}^N v\,V(v)}{\sum_{v=v_{\min}}^N V(v)}$$	Trapping time (average vertical line length)

$P(\ell)$ and $V(v)$ denote the histograms of diagonal and vertical line lengths, respectively.

RQA measures such as $L_{\max}$ have been shown to correlate strongly (anti-correlation, e.g. Pearson $r \approx -0.7$) with the maximal local Lyapunov exponent $\lambda$, supporting their utility in diagnosing transitions to instability and chaos (Bianchi et al., 2016, Sales et al., 2022).

3. Extensions, Novel Quantifiers, and Spatial Statistics

3.1 Recurrence Pattern Correlation (RPC)

RPC generalizes RQA by quantifying the correlation of the RP with user-defined spatial motifs or patterns $w_{\Delta i, \Delta j}$ (Marghoti et al., 15 Aug 2025). The RPC statistic is

$$\mathrm{RPC} = \frac{1}{W\,rr(1-rr)}\sum_{i\ne j}\sum_{i'\ne j'} w_{i'-i, j'-j} (R_{i,j} - rr)(R_{i',j'} - rr)$$

where $rr$ is the global recurrence rate and $W$ normalizes the weights. By choosing $w$ to represent diagonals, verticals, or arbitrary motifs, RPC recovers DET, LAM, or captures new nonlocal structures (e.g., unstable manifolds, higher-order periodicities). RPC enables the detection of long-range correlations and complex phase-space topologies beyond the local structures captured by RQA.

3.2 Entropy-Based Measures

Novel entropy quantifiers have been developed to address deficiencies of classical entropy (ENTR) based on diagonal lines (Corso et al., 2017, Sales et al., 2022). Corso et al. (Corso et al., 2017) define the microstate entropy $S(N^*)$ over $N\times N$ binary submatrices (“microstates”) of the RP:

$S = -\sum_{i=1}^{N^*} p_i \ln p_i$

where $p_i$ is the empirical probability of microstate $i$. This measure is monotonic in system complexity and robust to noise and parameter choices.

“Recurrence time entropy” (RTE) (Sales et al., 2022, Marwan et al., 2014) is constructed from the distribution of vertical line lengths (gaps), reflecting uncertainty in return intervals and correlating strongly (Pearson $r > 0.9$) with $\lambda_{\max}$. Multimodality in finite-time RTE can reveal hierarchical stickiness (islands-around-islands) in quasi-integrable systems.

3.3 Network and Symbolic Extensions

RPs have been interpreted as adjacency matrices for undirected graphs, leading to network-theoretic measures such as transitivity and the transitivity dimension $D_\mathcal{T}$ (Marwan et al., 2014). Symbolic recurrence plots, constructed via rewriting grammars on the RP and selection of entropy-maximizing partitions, segment time indices into “recurrence domains,” revealing functional components in high-dimensional signals (e.g. event-related EEG) (Graben et al., 2012).

4. High-Dimensional, Spatially Extended, and Non-Stationary Systems

RP analysis extends to high-dimensional and spatially extended systems by directly constructing distance matrices in the full state space (no embedding required if the entire system state is measured) (Marwan et al., 2014, Riedl et al., 18 Jan 2024). For spatial data such as satellite imagery or vegetation indices, image frames are treated as state vectors and compared pairwise or via spatial similarity (e.g., mapograms, weighted Bhattacharyya coefficients).

A novel approach replaces the global threshold $\varepsilon$ with temporal smoothing of the distance (or similarity) matrix and the extraction of column-wise local minima or maxima (Riedl et al., 18 Jan 2024). This “local-minima RP” method emphasizes slow regime shifts and external drivers (e.g., ENSO in climate data) that produce horizontal bands in the plot, providing a sensitive tool for detecting environmental regime transitions.

In high dimensions, careful normalization and adaptive thresholding (e.g., by fixing RR) are critical due to the curse of dimensionality and concentration of measure. Best practices include Theiler windows to avoid trivial recurrences and the use of surrogate data to assess statistical significance.

5. Deep Learning and Data-Driven Approaches

Recent work has exploited the image-like structure of RPs to train convolutional neural networks (CNNs) for parameter inference and dynamical regime classification (Lober et al., 30 Oct 2024, Mohan et al., 20 Jun 2025). RPs, when used as input to CNNs, yield more robust and accurate parameter estimation for nonlinear maps than direct regression on raw time series; for the logistic and standard maps, RP-based models achieve regression RMSE $<0.04$ and per-class $f_1$ scores $>0.98$ on held-out data (Lober et al., 30 Oct 2024). Classification architectures (e.g., dual-branch models combining standard and directional features) can distinguish periodic, quasi-periodic, chaotic, hyperchaotic, white and red noise, and generalize to experimental and astrophysical datasets (Mohan et al., 20 Jun 2025). This approach eliminates the need for handcrafted RQA features, automating dynamical state identification.

6. Regime Detection, Confidence Assessments, and Practical Considerations

Sliding-window RQA with bootstrapped confidence intervals permits statistically rigorous detection of dynamical transitions in noisy, nonstationary series (Marwan et al., 2013). The moving-window approach partitions time, computes local RQA histograms (e.g. $P_t(l)$), and uses bootstrapping over the empirical distribution of line structures to set $95\%$ or $99\%$ confidence bands. Excursions of RQA measures outside these bounds signify regime shifts (e.g. climate transitions in paleo SST records).

Proper selection of RP parameters (embedding, norm, threshold, window length) is critical. For large datasets ($N\gg 10^4$), computational cost and memory ($O(N^2)$) become limiting; approximate nearest-neighbor search or windowed analysis can mitigate this. Practitioners are urged to fix RR by adaptive $\varepsilon$ and to interpret RQA fluctuations in light of statistical significance.

7. Applications and Impact Across Disciplines

RPs and their quantitative analysis have been applied to a diverse array of systems:

• Echo state networks (Bianchi et al., 2016): Visualization and quantification of high-dimensional reservoir dynamics, with RQA identifying regimes of maximal computational capacity.
• Exchange rates and geophysical time series (Sparavigna, 2014, Sparavigna, 2014): Discerning periodicity, stochastic drift, and responses to external shocks (e.g., monetary policy, hydrological coupling).
• Neuroscience and brain signals (Graben et al., 2012): Segmentation of recurrence domains underlying functional signal components.
• High-dimensional spatial and climate data (Marwan et al., 2014, Riedl et al., 18 Jan 2024): Characterizing periodic vs. irregular spatiotemporal dynamics, detecting regime shifts and external drivers.
• Dynamical systems: Detection of stickiness and invariant set hierarchies in Hamiltonian maps (Sales et al., 2022), visualization of unstable periodic orbits via motif-based RPC (Marghoti et al., 15 Aug 2025).
• Automated parameter inference and classification via deep learning models operating on RPs (Lober et al., 30 Oct 2024, Mohan et al., 20 Jun 2025).

RPs offer a flexible, transparent window into dynamical systems, providing both visual and metric-based evidence for transitions, nonlinear behavior, and complex temporal organization.

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In summary, recurrence plots and associated measures have evolved into a comprehensive toolkit for dynamical systems analysis. They bridge pure visualization, quantitative analysis, and modern machine learning, enabling the detection and characterization of periodic, chaotic, laminar, and transitional regimes in diverse and high-dimensional datasets. Their adaptability and integration with spatial statistics and deep learning extend their reach across the physical, life, climate, and data sciences.