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When are synchronization errors small?

Lucas Illing1, Jochen Bröcker, Ljupco Kocarev

  • 1Institute for Nonlinear Science, University of California, San Diego, La Jolla 93093-0402, USA. lilling@ucsd.edu

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|October 9, 2002
PubMed
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Bounds on synchronization error in nonlinear systems are explored. Negative Lyapunov exponents alone are insufficient; manifold deformation bounds are crucial for small synchronization errors in coupled systems.

Area of Science:

  • Nonlinear dynamics
  • Systems theory
  • Control theory

Background:

  • Synchronization of nonlinear systems is vital in many fields.
  • Assessing synchronization error bounds is complex for nearly identical systems.
  • Existing methods may not fully capture error dynamics under perturbations.

Purpose of the Study:

  • To investigate bounds on synchronization error for nearly identical nonlinear systems.
  • To demonstrate the insufficiency of negative largest conditional Lyapunov exponents for guaranteeing small synchronization error.
  • To provide analytic bounds for synchronization error deformation.

Main Methods:

  • Analysis of synchronization manifolds in nonlinear systems.
  • Calculation of bounds for manifold deformation due to perturbations.

Related Experiment Videos

  • Derivation of analytic bounds for specific system subclasses, including Lur'e systems.
  • Case study using the Lorenz system.

    • Main Results:

    • Negative largest conditional Lyapunov exponents are not sufficient to guarantee small synchronization error.
    • Bounds for manifold deformation grow with the largest singular value of the linearized system.
    • The derived bounds are applicable beyond Lur'e systems, as shown with the Lorenz system.

    Conclusions:

    • Accurate synchronization error bounds require considering manifold deformation.
    • The largest singular value of the linearized system impacts synchronization error bounds.
    • The findings are generalizable to various nonlinear systems, including the Lorenz system.