Multistability of Quaternion-Valued Recurrent Neural Networks with Discontinuous Nonmonotonic Piecewise Nonlinear Activation Functions

In this article, the coexistence and dynamical behaviors of multiple equilibrium points for quaternion-valued neural networks (QVNNs) are investigated, whose activation functions are discontinuous and nonmonotonic piecewise nonlinear. According to the Hamilton rules, the QVNNs can be divided into four real-valued parts. By utilizing the Brouwer’s Fixed Point Theorem and property of strictly diagonally dominant matrices, some sufficient conditions are derived to ensure that the QVNNs have at least 54n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$5^{4n}$$\end{document} equilibrium points, 34n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3^{4n}$$\end{document} of them are locally exponentially stable, and the others are unstable. It is shown that the number of stable equilibria in QVNNs is more than that in the real-valued ones. Finally, a numerical simulation is presented to clarify the theoretical analysis is valid.