Recurrence Patterns Correlation

TL;DR

This paper introduces Recurrence Pattern Correlation (RPC), a new method for analyzing time series dynamics that captures localized structures and patterns in recurrence plots, enhancing the understanding of nonlinear systems.

Contribution

The paper presents RPC, a novel quantifier inspired by spatial statistics, enabling flexible analysis of recurrence patterns of arbitrary shape and scale in dynamical systems.

Findings

RPC visualizes unstable manifolds in bifurcation diagrams

Dissects mixed phase space of the Standard map

Tracks unstable periodic orbits in Lorenz system

Abstract

Recurrence plots (RPs) are powerful tools for visualizing time series dynamics; however, traditional Recurrence Quantification Analysis (RQA) often relies on global metrics, such as line counting, that can overlook system-specific, localized structures. To address this, we introduce Recurrence Pattern Correlation (RPC), a quantifier inspired by spatial statistics that bridges the gap between qualitative RP inspection and quantitative analysis. RPC is designed to measure the correlation degree of an RP to patterns of arbitrary shape and scale. By choosing patterns with specific time lags, we visualize the unstable manifolds of periodic orbits within the Logistic map bifurcation diagram, dissect the mixed phase space of the Standard map, and track the unstable periodic orbits of the Lorenz '63 system's 3-dimensional phase space. This framework reveals how long-range correlations in recurrence patterns encode the underlying properties of nonlinear dynamics and provides a more flexible tool to analyze pattern formation in recurrent dynamical systems.