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Recurrence patterns correlation.

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Summary
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We developed recurrence pattern correlation (RPC), a new method to analyze complex time-series data. RPC offers a more flexible way to study localized structures in dynamical systems than traditional recurrence plots (RPs).

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Area of Science:

  • Nonlinear dynamics
  • Complex systems analysis
  • Time-series analysis

Background:

  • Recurrence plots (RPs) are valuable for visualizing time-series dynamics.
  • Traditional recurrence quantification analysis often uses global metrics, missing localized structures.
  • A gap exists between qualitative RP inspection and quantitative analysis.

Purpose of the Study:

  • Introduce recurrence pattern correlation (RPC) to bridge the gap in recurrence analysis.
  • Develop a flexible tool for analyzing pattern formation in recurrent dynamical systems.
  • Measure the correlation degree of RPs to patterns of arbitrary shape and scale.

Main Methods:

  • Introduce recurrence pattern correlation (RPC), inspired by spatial statistics.
  • Apply RPC to visualize unstable manifolds in the Logistic map.
  • Dissect the mixed phase space of the Standard map using RPC.
  • Track unstable periodic orbits in the Lorenz '63 system.

Main Results:

  • RPC successfully visualizes localized structures missed by traditional methods.
  • The method reveals correlations between recurrence patterns and underlying dynamical properties.
  • RPC demonstrates flexibility in analyzing diverse nonlinear systems.

Conclusions:

  • Recurrence pattern correlation (RPC) provides a more nuanced quantitative analysis of time-series data.
  • Long-range correlations in recurrence patterns encode crucial information about nonlinear dynamics.
  • RPC offers a flexible framework for studying pattern formation in complex systems.