Chen Xiaofeng, Song Qiankun, Li Zhongshan, Zhao Zhenjiang, Liu Yurong
IEEE Trans Neural Netw Learn Syst. 2018 Jul;29(7):2769-2781. doi: 10.1109/TNNLS.2017.2704286. Epub 2017 Jun 5.
This paper addresses the problem of stability for continuous-time and discrete-time quaternion-valued neural networks (QVNNs) with linear threshold neurons. Applying the semidiscretization technique to the continuous-time QVNNs, the discrete-time analogs are obtained, which preserve the dynamical characteristics of their continuous-time counterparts. Via the plural decomposition method of quaternion, homeomorphic mapping theorem, as well as Lyapunov theorem, some sufficient conditions on the existence, uniqueness, and global asymptotical stability of the equilibrium point are derived for the continuous-time QVNNs and their discrete-time analogs, respectively. Furthermore, a uniform sufficient condition on the existence, uniqueness, and global asymptotical stability of the equilibrium point is obtained for both continuous-time QVNNs and their discrete-time version. Finally, two numerical examples are provided to substantiate the effectiveness of the proposed results.
本文研究了具有线性阈值神经元的连续时间和离散时间四元数值神经网络（QVNNs）的稳定性问题。将半离散化技术应用于连续时间QVNNs，得到了离散时间类似物，它们保留了连续时间对应物的动力学特性。通过四元数的复数分解方法、同胚映射定理以及李雅普诺夫定理，分别为连续时间QVNNs及其离散时间类似物推导了关于平衡点存在性、唯一性和全局渐近稳定性的一些充分条件。此外，还得到了连续时间QVNNs及其离散时间版本关于平衡点存在性、唯一性和全局渐近稳定性的统一充分条件。最后，给出了两个数值例子以证实所提结果的有效性。
