Neimark–Sacker Bifurcation with Zn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_n$$\end{document}-Symmetry and a Neural Application

Effects of Zn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_n$$\end{document}-symmetry, (n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 2$$\end{document}), on normal form of Neimark–Sacker bifurcation in discrete time dynamical systems are investigated. As an application, we consider three dimensional discrete Hopfield neural network with Z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_2$$\end{document}-symmetry. We drive analytical conditions for stability and bifurcations of the trivial fixed point of the system and compute analytically the normal form coefficients for the codimension 1 and codimension 2 bifurcation points including pitchfork, period-doubling, Neimark–Sacker, Z2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}_2$$\end{document}-symmetric Neimark–Sacker and resonance 1:4. By using numerical continuation in numerical software matcontm, we compute bifurcation curves of trivial fixed point and cycle with period 4 under variation of one and two parameters, and all codimension 1 and codimension 2 bifurcations supported by matcontm, on the corresponding curves are computed.