An instability framework of Hopf-Turing-Turing singularity in 2-component reaction-diffusion systems

This paper investigates pattern formation in 2-component reaction-diffusion systems with linear diffusion and local reaction terms. We propose a novel instability framework characterized by 0-mode Hopf instability, m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textit{m}$$\end{document} and m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textit{m}$$\end{document} + 1 mode Turing instabilities in 2-component reaction-diffusion systems. A normal form for the codimension 3 bifurcation is derived via the center manifold reduction, representing one of the main results in this paper. Additionally, we present numerical results on the bifurcation of certain reaction-diffusion systems and on the chaotic behavior of the normal form.