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Qiang Song, Wenwu Yu, Jinde Cao

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    This study presents a distributed control algorithm for synchronizing harmonic oscillators using delayed position states. It establishes conditions for synchronization based on coupling strength and time delays, crucial for networked systems.

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    Area of Science:

    • Control Theory
    • Networked Systems
    • Nonlinear Dynamics

    Background:

    • Synchronization is a key phenomenon in many physical and biological systems.
    • Networked harmonic oscillators present complex dynamics influenced by coupling and time delays.

    Purpose of the Study:

    • To develop a distributed control algorithm for achieving synchronization in coupled harmonic oscillators.
    • To analyze the impact of delayed state information on synchronization.
    • To derive conditions for synchronization considering positive and negative coupling strengths.

    Main Methods:

    • A distributed control algorithm utilizing delayed position states.
    • Design of coupling strength based on network Laplacian eigenvalues.
    • Analysis of stability switches with respect to time delay variations.

    Main Results:

    • Necessary and sufficient conditions for synchronization derived.
    • Synchronization is achievable within specific time-delay intervals.
    • The method is effective for both positive and negative coupling strengths.

    Conclusions:

    • The proposed distributed control algorithm enables synchronization in networked harmonic oscillators.
    • Time delay plays a critical role, with synchronization possible only within bounded intervals.
    • The findings provide valuable insights for designing stable and synchronized complex networks.