This paper explores the multistability issue for fractional-order Hopfield neural networks with Gaussian activation function and multiple time delays. First, several sufficient criteria are presented for ensuring the exact coexistence of $3^n$ equilibria, based on the geometric characteristics of Gaussian function, the fixed point theorem and the contraction mapping principle. Then, different from the existing methods used in the multistability analysis of fractional-order neural networks without time delays, it is shown that $2^n$ of $3^n$ total equilibria are locally asymptotically stable, by applying the theory of fractional-order linear delayed system and constructing suitable Lyapunov function. The obtained results improve and extend some existing multistability works for classical integer-order neural networks and fractional-order neural networks without time delays. Finally, an illustrative example with comprehensive computer simulations is given to demonstrate the theoretical results.

本文探讨了具有高斯激活函数和多个时间延迟的分数阶 Hopfield 神经网络的多稳定性问题。首先，提出了几个充分的标准来确保准确共存$3^n$平衡，基于高斯函数的几何特性、不动点定理和收缩映射原理。然后，与现有的用于无时滞分数阶神经网络多稳定性分析的方法不同，它表明：$2^n$ 的 $3^n$通过应用分数阶线性延迟系统的理论并构造合适的Lyapunov函数，总平衡是局部渐近稳定的。所获得的结果改进和扩展了一些现有的无时延经典整数阶神经网络和分数阶神经网络的多稳定性工作。最后，给出了一个具有综合计算机模拟的说明性例子来证明理论结果。