Stability analysis of nontrivial stationary solution and constant equilibrium point of reaction-diffusion neural networks with time delays under Dirichlet zero boundary value

TL;DR

This paper investigates the stability of stationary solutions in reaction-diffusion neural networks with delays, employing Lyapunov-Razumikhin techniques and variational methods to establish stability criteria and analyze the effects of diffusion.

Contribution

It introduces new theorems on the stability of delayed reaction-diffusion neural networks, highlighting the dual role of diffusion and providing conditions for global stability and phase plane changes.

Findings

Diffusion can both stabilize and destabilize neural network solutions.

Small diffusion effects significantly alter the phase plane structure.

Numerical examples confirm the theoretical stability criteria.

Abstract

In this paper, Lyapunov-Razumikhin technique, design of state-dependent switching laws, a fixed point theorem and variational methods are employed to derive the existence and the unique existence results of globally exponentially stable (positive) stationary solution of delayed reaction-diffusion cell neural networks under Dirichlet zero boundary value, including the global stability criteria in the classical meaning. Next, sufficient conditions are proposed to guarantee the global stability invariance of ordinary differential systems under the influence of diffusions. New theorems show that the diffusion is a double-edged sword in judging the stability of diffusion systems. Besides, an example is constructed to illuminate that any non-zero constant equilibrium point must be not in the phase plane of dynamic system under Dirichlet zero boundary value, or it must lead to a contradiction. Next, under Lipschitz assumptions on active function, another example is designed to prove that the small diffusion effect will cause the essential change of the phase plane structure of the dynamic behavior of the delayed neural networks via a Saddle point theorem. Finally, a numerical example illustrates the feasibility of the proposed methods. It is worth mentioning that some interesting mathematical problems are originally put forward in the last chapter.