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This study introduces recurrence plots and network analysis to identify periodic behaviors in continuous dynamical systems, overcoming limitations with short time series data.

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Area of Science:

  • Dynamical Systems and Chaos Theory
  • Nonlinear Dynamics
  • Time Series Analysis

Background:

  • Identifying periodic windows in continuous dynamical systems is challenging, especially with limited experimental data.
  • Traditional methods like Lyapunov exponents are less effective for continuous systems and short time series.
  • Periodic and chaotic behaviors in parameter spaces require robust discrimination methods.

Purpose of the Study:

  • To present recurrence plots and nonlinear measures as a practical method for detecting periodic windows in continuous dynamical systems.
  • To demonstrate the effectiveness of recurrence quantification analysis and complex network theory for classifying periodic versus chaotic behavior.
  • To identify specific network measures that excel at distinguishing complex periodic windows.

Main Methods:

  • Utilizing recurrence plots generated from system trajectories.
  • Applying traditional recurrence quantification analysis (RQA) measures, specifically diagonal line-based metrics.
  • Employing complex network theory measures, including average path length and clustering coefficient.
  • Benchmarking the methods using the Rössler system.

Main Results:

  • Recurrence plot-based nonlinear measures effectively classify periodic and chaotic behaviors.
  • Both traditional RQA and network measures provide excellent discrimination.
  • Average path length and clustering coefficient of recurrence networks are powerful indicators for identifying periodic windows.
  • The Rössler system analysis confirmed the efficacy of these methods.

Conclusions:

  • Recurrence plots and associated nonlinear measures offer a viable alternative for analyzing continuous dynamical systems.
  • Network measures derived from recurrence plots are particularly effective for distinguishing complex periodic windows from chaotic behavior.
  • This approach enhances the analysis of experimental data with short time series.