Multistability and Multiperiodicity for a Class of Cohen–Grossberg BAM Neural Networks with Discontinuous Activation Functions and Time Delays

In this paper, a general class of Cohen–Grossberg bidirectional associative memory neural networks (CGBAMNNs) with time-varying delays, distributed delays and discontinuous activation functions is investigated. By partitioning the state space, employing analysis approach and Cauchy convergence principle, sufficient conditions are established for the existence and local exponential stability of multiple equilibrium points, which ensure that 2n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2n$$\end{document}-dimensional CGBAMNNs with k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-level discontinuous activation functions can have kn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k^n$$\end{document} equilibrium points. As an extension of multistability, sufficient conditions are obtained to ensure the existence of kn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k^n$$\end{document} locally exponentially stable periodic orbits of the system when time-varying delays and external inputs are periodic. Finally, three examples are given to illustrate the feasibility and application of the obtained results.