Multiple Lagrange stability and Lyapunov asymptotical stability of delayed fractional-order Cohen-Grossberg neural networks

This paper addresses the coexistence and local stability of multiple equilibrium points for fractional-order Cohen-Grossberg neural networks (FOCGNNs) with time delays. Based on Brouwer's fixed point theorem, sufficient conditions are established to ensure the existence of Pi(n)(i=1) (2K(i) + 1) equilibrium points for FOCGNNs. Through the use of Hardy inequality, fractional Halanay inequality, and Lyapunov theory, some criteria are established to ensure the local Lagrange stability and the local Lyapunov asymptotical stability of Pi(n)(i=1)(K-i + 1) equilibrium points for FOCGNNs. The obtained results encompass those of integer-order Hopfield neural networks with or without delay as special cases. The activation functions are nonlinear and nonmonotonic. There could be many corner points in this general class of activation functions. The structure of activation functions makes FOCGNNs could have a lot of stable equilibrium points. Coexistence of multiple stable equilibrium points is necessary when neural networks come to pattern recognition and associative memories. Finally, two numerical examples are provided to illustrate the effectiveness of the obtained results.

本文研究了具有时滞的分数阶科恩 - 格罗斯伯格神经网络（FOCGNNs）多个平衡点的共存性和局部稳定性。基于布劳威尔不动点定理，建立了充分条件以确保FOCGNNs存在\(\prod_{i = 1}^{n}(2k_{i}+ 1)\)个平衡点。通过利用哈代不等式、分数阶哈拉奈不等式以及李雅普诺夫理论，建立了一些准则以确保FOCGNNs的\(\prod_{i = 1}^{n}(k_{i}+ 1)\)个平衡点的局部拉格朗日稳定性和局部李雅普诺夫渐近稳定性。所得到的结果包含了有或无时滞的整数阶霍普菲尔德神经网络的结果作为特殊情形。激活函数是非线性且非单调的。在这类一般的激活函数中可能存在许多转折点。激活函数的结构使得FOCGNNs可能具有大量稳定平衡点。当神经网络用于模式识别和联想记忆时，多个稳定平衡点的共存是必要的。最后，提供了两个数值例子以说明所得到结果的有效性。