We explored the phenomenon of finite-time synchronization for coupled complex-valued neural networks that were subject to mixed-variable delays. To address this challenge, an impulsive pinning control strategy was proposed. The method presented entailed the dynamic adjustment of specific nodes at distinct impulsive intervals, contingent upon the magnitude of the errors observed in those nodes. Furthermore, an enhanced technique utilizing the sign function was employed to simultaneously manage the real and imaginary components of the complex-valued neural networks. By applying finite-time stability theorems and utilizing complex-valued inequalities, sufficient conditions for achieving finite-time synchronization and determining stability time under the influence of delayed impulsive effects were established. A comprehensive discussion on the interaction between impulsive effects and pinning strategies was also included. It was noted that integrating impulsive effects with pinning ratios enabled precise control over nodes exhibiting significant errors, thereby promoting rapid convergence within finite time frames. Our findings highlight the effectiveness of impulsive pinning control in enhancing synchronization stability, providing significant insights into the practical applications of complex-valued neural networks, particularly in image processing.

We investigate the synchronizability of multilayer star-ring networks. Two types of multilayer networks, including aggregated coupling and divergent coupling, are established based on the connections between the hub node and the leaf nodes in the subnetwork. The eigenvalue spectrum of the two types of multilayer networks is strictly derived, and the correlation between topological parameters and synchronizability is analyzed by the master stability function framework. Moreover, the variable coupling strength has been investigated, revealing that it is significantly related to the synchronizability of the aggregated coupling while having no influence on the divergent coupling. Furthermore, the validity of the synchronizability analysis is obtained by implementing adaptive control on the multilayer star-ring networks previously mentioned. Calculations and comparisons show that the differences caused by the sizes of multilayer networks and interlayer coupling strength are not negligible. Finally, numerical examples are also provided to validate the effectiveness of the theoretical analysis.