In this paper, we study the dynamic phase transition for one dimensional Brusselator model. By the linear stability analysis, we define two critical numbers $ {\lambda}_0 $ and $ {\lambda}_1 $ for the control parameter $ {\lambda} $ in the equation. Motivated by [9], we assume that $ {\lambda}_0< {\lambda}_1 $ and the linearized operator at the trivial solution has multiple critical eigenvalues $ \beta_N^+ $ and $ \beta_{N+1}^+ $. Then, we show that as $ {\lambda} $ passes through $ {\lambda}_0 $, the trivial solution bifurcates to an $ S^1 $-attractor $ {\mathcal A}_N $. We verify that $ {\mathcal A}_N $ 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.

中文翻译：

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具有多个关键特征值的Brusselator模型的图灵不稳定性和动态相变

在本文中，我们研究了一维Brusselator模型的动态相变。通过线性稳定性分析，我们为方程式中的控制参数$ {\ lambda} $定义了两个临界数$ {\ lambda} _0 $和$ {\ lambda} _1 $。由[9]，我们假设$ {\ lambda} _0 <{\ lambda} _1 $和平凡解的线性化算符具有多个关键特征值$ \ beta_N ^ + $和$ \ beta_ {N + 1} ^ + $。然后，我们证明，当$ {\ lambda} $通过$ {\ lambda} _0 $时，平凡解分叉成一个$ S ^ 1 $吸引子$ {\ mathcal A} _N $。我们验证$ {\ mathcal A} _N $由八个稳态解和连接它们的轨道组成。我们通过中心流形分析计算每个稳态解的前导系数。我们还给出数值结果来解释主定理。