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    A new time-scale-dependent coupling scheme enhances synchronization in complex networks. This method allows for faster synchronization by adapting inner coupling matrices, removing prior stability assumptions for subsystems.

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    Area of Science:

    • Complex Networks
    • Nonlinear Dynamics
    • Control Theory

    Background:

    • Two-time-scale nonlinear complex networks exhibit modular architectures.
    • Existing coupling schemes often assume time-scale independence and require fast subsystem stability.

    Purpose of the Study:

    • To propose a novel time-scale-dependent coupling scheme for two-time-scale nonlinear complex networks.
    • To remove the restrictive assumption of exponentially stable fast dynamics in subsystems.

    Main Methods:

    • Developing time-scale-dependent inner coupling matrices linked to fast subsystem dynamics.
    • Employing time-scale separation and weighted ε-dependent Lyapunov functions for stability analysis.
    • Establishing criteria for local and global exponential synchronization.

    Main Results:

    • The proposed scheme guarantees an exponential decay rate for synchronization error, independent of the singular perturbation parameter ε.
    • Criteria for local and global exponential synchronization are derived.
    • The allowable upper bound for ε is determined by solving matrix inequalities.

    Conclusions:

    • The novel time-scale-dependent coupling scheme effectively achieves synchronization in two-time-scale complex networks.
    • The method relaxes previous stability constraints on subsystems.
    • Numerical simulations confirm the efficiency of the proposed strategy.