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

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