标题： 具有时滞的反应扩散神经网络在狄利克雷零边界值下的非平凡平稳解和常数平衡点的稳定性分析 显示英文标题

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

摘要： 在本文中，采用Lyapunov-Razumikhin技巧、状态依赖切换定律的设计、不动点定理和变分方法，推导出在Dirichlet零边界值条件下延迟反应扩散细胞神经网络的全局指数稳定（正）平稳解的存在性和唯一性结果，包括经典意义下的全局稳定性准则。 接下来，提出了充分条件以保证在扩散影响下常微分系统全局稳定性不变。 新的定理表明，扩散在判断扩散系统的稳定性时是一把双刃剑。 此外，构造了一个例子来阐明在Dirichlet零边界值下，任何非零常数平衡点都不可能存在于动态系统的相平面上，或者它必然导致矛盾。 接下来，在激活函数满足Lipschitz条件的情况下，设计了另一个例子，通过鞍点定理证明小的扩散效应会导致延迟神经网络动态行为相平面结构的本质变化。 最后，一个数值例子说明了所提出方法的可行性。 值得一提的是，在最后一章中最初提出了一些有趣的数学问题。 显示英文摘要

摘要： 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.