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Phase relationships between two or more interacting processes from one-dimensional time series. I. Basic theory.

N B Janson1, A G Balanov, V S Anishchenko

  • 1Department of Physics, Lancaster University, Lancaster, LA1 4YB, United Kingdom.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|March 23, 2002
PubMed
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This study introduces a novel method to detect phase relationships in interacting oscillatory systems using single time series. The approach utilizes return time maps to analyze angle dynamics, offering a new tool for complex system analysis.

Area of Science:

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

Background:

  • Understanding phase relationships is crucial for analyzing coupled oscillatory systems.
  • Existing methods often require multi-dimensional data or have limitations under non-stationary conditions.

Purpose of the Study:

  • To develop a general approach for detecting phase relationships between multiple interacting oscillatory processes.
  • To utilize one-dimensional time series for phase relationship analysis.
  • To establish a unique relationship between a novel angle measure and conventional phase difference.

Main Methods:

  • Introduction of angles and radii of return times maps.
  • Analysis of the dynamics of these angles.
  • Derivation of a unique relationship between angles and phase difference under weak forcing.

Related Experiment Videos

  • Numerical confirmation using a non-stationary forced Van der Pol system.

    • Main Results:

    • A general method for phase relationship detection from single time series is established.
    • An explicit, unique relationship between the new angle measure and conventional phase difference is derived for weakly forced systems.
    • The method's validity is confirmed numerically in a non-stationary forced Van der Pol system.
    • A model for angle dynamics under weak quasiperiodic forcing with multiple frequencies is developed.

    Conclusions:

    • The developed approach provides a powerful tool for analyzing phase relationships in complex oscillatory systems using minimal data.
    • This method extends the analysis of coupled oscillators to scenarios previously challenging due to data limitations or system non-stationarity.