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    Area of Science:

    • Complex Networks
    • Nonlinear Systems
    • Control Theory

    Background:

    • Cluster synchronization is crucial for complex systems.
    • Finite-time and fixed-time synchronization offer enhanced performance over traditional methods.
    • Controlling synchronization in networks with or without pinning control is a significant challenge.

    Purpose of the Study:

    • To investigate finite-time and fixed-time cluster synchronization in complex networks.
    • To design distributed control protocols for achieving these synchronization types.
    • To analyze the impact of network topology and pinning control on synchronization performance.

    Main Methods:

    • Design of novel distributed protocols for finite-time and fixed-time cluster synchronization.
    • Rigorous mathematical proofs to validate protocol effectiveness.
    • Analysis of synchronization criteria, including the role of coupling terms.
    • Investigation of special cases: complete synchronization and master-slave systems.

    Main Results:

    • Development of effective distributed protocols for finite-time and fixed-time cluster synchronization.
    • Sufficient conditions derived to understand the influence of coupling on synchronization.
    • Demonstration of the method's applicability to complete synchronization and master-slave systems.
    • Validation of theoretical findings through numerical simulations.

    Conclusions:

    • The proposed protocols successfully achieve finite-time and fixed-time cluster synchronization.
    • The study provides a theoretical framework and practical methods for controlling synchronization in complex networks.
    • The results have implications for understanding and designing coordinated behaviors in various complex systems.