Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues

Abstract

<p style='text-indent:20px;'>In this paper, we study the dynamic phase transition for one dimensional Brusselator model. By the linear stability analysis, we define two critical numbers <inline-formula><tex-math id="M1">\begin{document}$ {\lambda}_0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ {\lambda}_1 $\end{document}</tex-math></inline-formula> for the control parameter <inline-formula><tex-math id="M3">\begin{document}$ {\lambda} $\end{document}</tex-math></inline-formula> in the equation. Motivated by [<xref ref-type="bibr" rid="b9">9</xref>], we assume that <inline-formula><tex-math id="M4">\begin{document}$ {\lambda}_0&lt; {\lambda}_1 $\end{document}</tex-math></inline-formula> and the linearized operator at the trivial solution has multiple critical eigenvalues <inline-formula><tex-math id="M5">\begin{document}$ \beta_N^+ $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ \beta_{N+1}^+ $\end{document}</tex-math></inline-formula>. Then, we show that as <inline-formula><tex-math id="M7">\begin{document}$ {\lambda} $\end{document}</tex-math></inline-formula> passes through <inline-formula><tex-math id="M8">\begin{document}$ {\lambda}_0 $\end{document}</tex-math></inline-formula>, the trivial solution bifurcates to an <inline-formula><tex-math id="M9">\begin{document}$ S^1 $\end{document}</tex-math></inline-formula>-attractor <inline-formula><tex-math id="M10">\begin{document}$ {\mathcal A}_N $\end{document}</tex-math></inline-formula>. We verify that <inline-formula><tex-math id="M11">\begin{document}$ {\mathcal A}_N $\end{document}</tex-math></inline-formula> consists of eight steady state solutions and orbits connecting them. We compute the leading coefficients of each steady state solution via the center manifold analysis. We also give numerical results to explain the main theorem.