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

Yuncherl Choi; Taeyoung Ha; Jongmin Han; Sewoong Kim; Doo Seok Lee

doi:10.3934/dcds.2021035

Article Contents

2021, Volume 41, Issue 9: 4255-4281. Doi: 10.3934/dcds.2021035

This issue Previous Article Local well-posedness for the derivative nonlinear Schrödinger equation with

$ L^2 $

-subcritical data Next Article Singular double-phase systems with variable growth for the Baouendi-Grushin operator

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

1.
Ingenium College of Liberal Arts, Kwangwoon University, Seoul 01891, Korea
2.
Division of Medical Mathematics, National Institute for Mathematical Sciences, Daejeon 34047, Korea
3.
Department of Mathematics, Kyung Hee University, Seoul 02447, Korea
4.
School of Mathematics, Korea Institute for Advanced Study, Seoul 02455, Korea
5.
Samsung Fire & Marine Insurance, Seoul 04523, Korea
6.
Department of Undergraduate Studies, Daegu Gyeongbuk Institute of Science and Technology, Daegu 42988, Korea

^* Corresponding author: Jongmin Han
^* Corresponding author: Jongmin Han

Received: September 2020

Early access: February 2021

Published: September 2021

Abstract / Introduction Full Text(HTML) Figure(8) / Table(4) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 35B32, 35B41; Secondary: 35K40.

Citation:

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Figure 1. Examples of Structure of $ {\mathcal A}_N $ in Table 1, 2 and 3

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Figure 2. Examples of Structure of $ {\mathcal A}_N $ in Table 4

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Figure 3. Case (ⅰ) of (4.3) and $ N = 4 $. With $ w (x,0) = w_0(x) $, (a) $ u_h(x,t) \to u_1^+(x) $ and (b) $ v_h(x,t) \to v_1^+(x) $. With $ w (x,0) = w_1(x) $, (c) $ u_h(x,t) \to u_1^+(x) $ and (d) $ v_h(x,t) \to v_1^+(x) $

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Figure 4. Case (ⅱ) of (4.3) and $ N = 4 $. With $ w (x,0) = w_0(x) $, (a) $ u_h(x,t) \to u_2^+(x) $ and (b) $ v_h(x,t) \to v_2^+(x) $. With $ w (x,0) = w_1(x) $, (c) $ u_h(x,t) \to u_1^+(x) $ and (d) $ v_h(x,t) \to v_1^+(x) $

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Figure 5. Case (ⅲ) of (4.3) and $ N = 4 $. With $ w (x,0) = w_0(x) $, (a) $ u_h(x,t) \to u_1^+(x) $ and (b) $ v_h(x,t) \to v_1^+(x) $. With $ w (x,0) = w_1(x) $, (c) $ u_h(x,t) \to u_1^+(x) $ and (d) $ v_h(x,t) \to v_1^+(x) $

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Figure 6. Case (ⅰ) of (4.3) and $ N = 8 $. With $ w (x,0) = w_0(x) $, (a) $ u_h(x,t) \to u_1^-(x) $ and (b) $ v_h(x,t) \to v_1^-(x) $. With $ w (x,0) = w_1(x) $, (c) $ u_h(x,t) \to u_1^+(x) $ and (d) $ v_h(x,t) \to v_1^+(x) $

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Figure 7. Case (ⅱ) of (4.3) and $ N = 8 $. With $ w (x,0) = w_0(x) $, (a) $ u_h(x,t) \to u_1^+(x) $ and (b) $ v_h(x,t) \to v_2^+(x) $. With $ w (x,0) = w_2(x) $, (c) $ u_h(x,t) \to u_1^-(x) $ and (d) $ v_h(x,t) \to v_1^-(x) $

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Figure 8. Case (ⅲ) of (4.3) and $ N = 8 $. With $ w (x,0) = w_0(x) $, (a) $ u_h(x,t) \to u_2^+(x) $ and (b) $ v_h(x,t) \to v_2^+(x) $. With $ w (x,0) = w_1(x) $, (c) $ u_h(x,t) \to u_1^-(x) $ and (d) $ v_h(x,t) \to v_1^-(x) $

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Table 1. Stability for $ k = 2 $

subcases	$ w_1^+ $	$ w_1^- $	$ w_2^+ $	$ w_2^- $
(ⅰ-1)	stable	saddle	$ \times $	$ \times $
(ⅰ-2)	saddle	stable	$ \times $	$ \times $
(ⅰ-3)	$ \times $	$ \times $	stable	saddle
(ⅰ-4)	$ \times $	$ \times $	saddle	stable

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Table 2. Stability for $ k = 4 $

subcases	$ w_1^\pm $	$ w_2^\pm $	$ w_3^\pm $	$ w_4^\pm $
(ⅱ-1)	stable	saddle	$ \times $	$ \times $
(ⅱ-2)	saddle	stable	$ \times $	$ \times $
(ⅱ-3)	$ \times $	$ \times $	stable	saddle
(ⅱ-4)	$ \times $	$ \times $	saddle	stable

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Table 3. Stability for $ k = 6 $

subcases	$ w_1^+ $	$ w_1^- $	$ w_2^+ $	$ w_2^- $	$ w_3^\pm $	$ w_4^\pm $
(ⅲ-1)	stable	saddle	$ \times $	$ \times $	saddle	stable
(ⅲ-2)	saddle	stable	$ \times $	$ \times $	stable	saddle
(ⅲ-3)	$ \times $	$ \times $	stable	saddle	saddle	stable
(ⅲ-4)	$ \times $	$ \times $	saddle	stable	stable	saddle

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Table 4. Stability for $ k = 8 $

subcases	$ w_1^\pm $	$ w_2^\pm $	$ w_3^\pm $	$ w_4^\pm $
(ⅳ-1)	stable	stable	saddle	saddle
(ⅳ-2)	saddle	saddle	stable	stable

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