Matrix Measure-Based Projective Synchronization on Coupled Neural Networks With Clustering Trees.

Area of Science:

• Complex Systems and Networks
• Computational Neuroscience
• Control Theory

Background:

• Neural networks with heterogeneous coupling and time-varying delays present synchronization challenges.
• Cluster-tree topologies introduce complexities due to inter-cluster influences and mismatched parameters.
• Achieving synchronization within a bounded error is crucial for practical applications.

Purpose of the Study:

• To investigate projective quasisynchronization for nonlinear heterogeneous-coupled neural networks with mixed time-varying delays and a cluster-tree topology.
• To design pinning impulsive controllers for selected nodes within different clusters.
• To derive conditions ensuring synchronization within a prescribed error bound and estimate convergence rates.

Main Methods:

• Design of pinning impulsive controllers applied to nodes with largest error norms.
• Utilizing average impulsive interval, matrix measure method, and Lyapunov stability theorem.
• Employing the formula of variation of parameters and comparison principle for impulsive systems.

Main Results:

• Sufficient conditions for cluster projective quasisynchronization are derived.
• Convergence rate and synchronization error bound are precisely estimated.
• Synchronization error bound is optimized based on impulsive effects.

Conclusions:

• The proposed pinning impulsive control strategy effectively achieves projective quasisynchronization in complex neural network arrays.
• The theoretical findings are validated through numerical experiments, demonstrating the efficacy of the derived conditions and estimations.
• The study provides a robust framework for controlling synchronization in distributed nonlinear systems with intricate network structures.