Stability and exponential lag-synchronization of a class of neural network with state dependent and distributed delays over a time scale

This article tackles the stability and synchronization challenges of neural networks with distributed and state-dependent delays on a temporal domain, leveraging the powerful framework of time scales theory. By formulating the problem within this unified framework, we enable applications to both uniform and non-uniform time domains. Our investigation begins with a thorough analysis of the local exponential stability of the zero solution, employing a combination of pure analysis method, reduction to absurdity technique, and time scale theory. We derive a set of sufficient conditions that guarantee local exponential stability of neural networks with distributed and state-dependent delays. Furthermore, we examine exponential lag synchronization results, utilizing a range of analytical tools, including time scale theory, matrix norm theory, unified matrix-measure theory, and the Halanay inequality. To demonstrate the efficacy and broad applicability of our findings, we present a simulated example on random time scales. Specifically, the time scales theory allows us to effectively handle time scales by providing a unified framework that can seamlessly integrate both continuous and discrete time domains, thereby enabling the analysis of complex systems with varying time scales. Moreover, our approach leverages the flexibility of time scales theory to accommodate non-uniform time domains, making it an ideal tool for tackling real-world problems with intricate temporal dynamics.