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A unifying definition of synchronization for dynamical systems.

Reggie Brown1, Ljupco Kocarev

  • 1Department of Physics and Department of Applied Science, College of William and Mary, Williamsburg, Virginia 23187-8795.

Chaos (Woodbury, N.Y.)
|June 5, 2003
PubMed
Summary

We introduce a unified definition for synchronization in deterministic dynamical systems. This framework unifies existing synchronization phenomena and addresses limitations of prior definitions, paving the way for broader applications.

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Area of Science:

  • Dynamical Systems Theory
  • Control Theory
  • Nonlinear Dynamics

Background:

  • Synchronization is a key phenomenon in complex systems.
  • Existing definitions of synchronization are often system-specific.
  • A unified approach is needed to encompass diverse synchronization behaviors.

Purpose of the Study:

  • To propose a unifying definition for synchronization in stationary finite-dimensional deterministic dynamical systems.
  • To demonstrate how existing synchronization phenomena fit within the new framework.
  • To identify and discuss issues with previous synchronization definitions.

Main Methods:

  • Developing a formal, unifying mathematical definition for synchronization.
  • Applying the definition to various examples from dynamical systems literature.

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  • Analyzing the limitations of prior synchronization definitions.

    • Main Results:

    • A novel, unifying definition for synchronization is proposed.
    • The definition successfully encompasses diverse synchronization examples.
    • Key problems and limitations of previous synchronization definitions are highlighted.

    Conclusions:

    • The proposed definition provides a consistent framework for understanding synchronization.
    • This work lays the groundwork for extending synchronization concepts.
    • Future extensions may include infinite-dimensional systems and noisy environments.