• Previous Article
    Singular double-phase systems with variable growth for the Baouendi-Grushin operator
  • DCDS Home
  • This Issue
  • Next Article
    Local well-posedness for the derivative nonlinear Schrödinger equation with $ L^2 $-subcritical data
September  2021, 41(9): 4255-4281. doi: 10.3934/dcds.2021035

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

Received  September 2020 Published  September 2021 Early access  February 2021

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.

Citation: Yuncherl Choi, Taeyoung Ha, Jongmin Han, Sewoong Kim, Doo Seok Lee. Turing instability and dynamic phase transition for the Brusselator model with multiple critical eigenvalues. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4255-4281. doi: 10.3934/dcds.2021035
References:
[1]

R. Anguelov and S. M. Stoltz, Stationary and oscillatory patterns in a coupled Brusselator model, Math. Computers Simul., 133 (2017), 39-46.  doi: 10.1016/j.matcom.2015.06.002.  Google Scholar

[2]

K. J. Brown and F. A. Davidson, Global bifurcation in the Brusselator system, Nonlin. Anal., 12 (1995), 1713-1725.  doi: 10.1016/0362-546X(94)00218-7.  Google Scholar

[3] I. R. Epstein and J. A. Pojman, An Introduction to Nonlinear Chemical Dynamics,, Oxford Univ. Press, 1998.   Google Scholar
[4]

M. Ghergu, Non-constant steady-state solutions for Brusselator type systems, Nonlinearity, 21 (2008), 2331-2345.  doi: 10.1088/0951-7715/21/10/007.  Google Scholar

[5]

B. Guo and Y. Han, Attractor and spatial chaos for the Brusselator in $ \mathbb{R}^N$, Nonlin. Anal., 70 (2009), 3917-3931.  doi: 10.1016/j.na.2008.08.002.  Google Scholar

[6]

H. Kang and Y. Pesin, Dynamics of a discrete brusselator model: Escape to infinity and julia set, Milan J. Math., 73 (2005), 1-17.  doi: 10.1007/s00032-005-0036-y.  Google Scholar

[7]

H. Shoji, K. Yamada, D. Ueyama and T. Ohta, Turing patterns in three dimensions, Phys. Rev. E, 75 (2007), 046212, 13 pp. doi: 10.1103/PhysRevE.75.046212.  Google Scholar

[8]

T. Ma and S. Wang, Bifurcation Theory and Applications, World Scientific, 2005. doi: 10.1142/9789812701152.  Google Scholar

[9]

T. Ma and S. Wang, Phase transitions for the Brusselator model, J. Math. Phys., 52 (2011), 033501, 23 pp. doi: 10.1063/1.3559120.  Google Scholar

[10]

T. Ma and S. Wang, Phase Transition Dynamics 2nd ed., Springer, 2019. doi: 10.1007/978-3-030-29260-7.  Google Scholar

[11]

MathWorks, Matlab: Mathematics(R2020a), Retrieved from https://www.mathworks.com/help/pdf_doc/matlab_math.pdf Google Scholar

[12]

A. S. Mikhailov and K. Showalter, Control of waves, patterns and turbulence in chemical systems, Physics Reports, 425 (2006), 79-194.  doi: 10.1016/j.physrep.2005.11.003.  Google Scholar

[13]

L. A. Peletier and W. C. Troy, Spatial Patterns: Higher Order Models in Physics and Mechanics, Birkhauser, 2001. doi: 10.1007/978-1-4612-0135-9.  Google Scholar

[14]

R. Peng and M. X. Wang, Pattern formation in the Brusselator system, J. Math. Anal. Appl., 309 (2005), 151-166.  doi: 10.1016/j.jmaa.2004.12.026.  Google Scholar

[15]

R. Peng and M. X. Wang, On steady-state solutions of the Brusselator-type system, Nonlin. Anal., 71 (2009), 1389-1394.  doi: 10.1016/j.na.2008.12.003.  Google Scholar

[16]

I. Prigogine and R. Lefever, Symmetry breaking instabilities in dissipative systems, J. Chem. Phys., 48 (1968), 1695-1700.   Google Scholar

[17]

A. TothV. Gaspar and K. Showalter, Signal transmission in chemical systems: Propagation of chemical waves through capillary tubes, J. Phys. Chem., 98 (1994), 522-531.  doi: 10.1021/j100053a029.  Google Scholar

[18]

A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.  Google Scholar

[19]

Y. You, Global Dynamics of the Brusselator equations, Dynamics of PDE, 4 (2007), 167-196.  doi: 10.4310/DPDE.2007.v4.n2.a4.  Google Scholar

show all references

References:
[1]

R. Anguelov and S. M. Stoltz, Stationary and oscillatory patterns in a coupled Brusselator model, Math. Computers Simul., 133 (2017), 39-46.  doi: 10.1016/j.matcom.2015.06.002.  Google Scholar

[2]

K. J. Brown and F. A. Davidson, Global bifurcation in the Brusselator system, Nonlin. Anal., 12 (1995), 1713-1725.  doi: 10.1016/0362-546X(94)00218-7.  Google Scholar

[3] I. R. Epstein and J. A. Pojman, An Introduction to Nonlinear Chemical Dynamics,, Oxford Univ. Press, 1998.   Google Scholar
[4]

M. Ghergu, Non-constant steady-state solutions for Brusselator type systems, Nonlinearity, 21 (2008), 2331-2345.  doi: 10.1088/0951-7715/21/10/007.  Google Scholar

[5]

B. Guo and Y. Han, Attractor and spatial chaos for the Brusselator in $ \mathbb{R}^N$, Nonlin. Anal., 70 (2009), 3917-3931.  doi: 10.1016/j.na.2008.08.002.  Google Scholar

[6]

H. Kang and Y. Pesin, Dynamics of a discrete brusselator model: Escape to infinity and julia set, Milan J. Math., 73 (2005), 1-17.  doi: 10.1007/s00032-005-0036-y.  Google Scholar

[7]

H. Shoji, K. Yamada, D. Ueyama and T. Ohta, Turing patterns in three dimensions, Phys. Rev. E, 75 (2007), 046212, 13 pp. doi: 10.1103/PhysRevE.75.046212.  Google Scholar

[8]

T. Ma and S. Wang, Bifurcation Theory and Applications, World Scientific, 2005. doi: 10.1142/9789812701152.  Google Scholar

[9]

T. Ma and S. Wang, Phase transitions for the Brusselator model, J. Math. Phys., 52 (2011), 033501, 23 pp. doi: 10.1063/1.3559120.  Google Scholar

[10]

T. Ma and S. Wang, Phase Transition Dynamics 2nd ed., Springer, 2019. doi: 10.1007/978-3-030-29260-7.  Google Scholar

[11]

MathWorks, Matlab: Mathematics(R2020a), Retrieved from https://www.mathworks.com/help/pdf_doc/matlab_math.pdf Google Scholar

[12]

A. S. Mikhailov and K. Showalter, Control of waves, patterns and turbulence in chemical systems, Physics Reports, 425 (2006), 79-194.  doi: 10.1016/j.physrep.2005.11.003.  Google Scholar

[13]

L. A. Peletier and W. C. Troy, Spatial Patterns: Higher Order Models in Physics and Mechanics, Birkhauser, 2001. doi: 10.1007/978-1-4612-0135-9.  Google Scholar

[14]

R. Peng and M. X. Wang, Pattern formation in the Brusselator system, J. Math. Anal. Appl., 309 (2005), 151-166.  doi: 10.1016/j.jmaa.2004.12.026.  Google Scholar

[15]

R. Peng and M. X. Wang, On steady-state solutions of the Brusselator-type system, Nonlin. Anal., 71 (2009), 1389-1394.  doi: 10.1016/j.na.2008.12.003.  Google Scholar

[16]

I. Prigogine and R. Lefever, Symmetry breaking instabilities in dissipative systems, J. Chem. Phys., 48 (1968), 1695-1700.   Google Scholar

[17]

A. TothV. Gaspar and K. Showalter, Signal transmission in chemical systems: Propagation of chemical waves through capillary tubes, J. Phys. Chem., 98 (1994), 522-531.  doi: 10.1021/j100053a029.  Google Scholar

[18]

A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.  Google Scholar

[19]

Y. You, Global Dynamics of the Brusselator equations, Dynamics of PDE, 4 (2007), 167-196.  doi: 10.4310/DPDE.2007.v4.n2.a4.  Google Scholar

Figure 1.  Examples of Structure of $ {\mathcal A}_N $ in Table 1, 2 and 3
Figure 2.  Examples of Structure of $ {\mathcal A}_N $ in Table 4
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) $
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) $
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) $
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) $
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) $
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) $
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
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
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
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
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
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
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
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
[1]

I-Liang Chern, Chun-Hsiung Hsia. Dynamic phase transition for binary systems in cylindrical geometry. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 173-188. doi: 10.3934/dcdsb.2011.16.173

[2]

Camillo De Lellis, Emanuele Spadaro. Center manifold: A case study. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1249-1272. doi: 10.3934/dcds.2011.31.1249

[3]

Junyi Tu, Yuncheng You. Random attractor of stochastic Brusselator system with multiplicative noise. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2757-2779. doi: 10.3934/dcds.2016.36.2757

[4]

Jun Yang. Coexistence phenomenon of concentration and transition of an inhomogeneous phase transition model on surfaces. Discrete & Continuous Dynamical Systems, 2011, 30 (3) : 965-994. doi: 10.3934/dcds.2011.30.965

[5]

Pavel Krejčí, Jürgen Sprekels. Long time behaviour of a singular phase transition model. Discrete & Continuous Dynamical Systems, 2006, 15 (4) : 1119-1135. doi: 10.3934/dcds.2006.15.1119

[6]

Mauro Garavello. Boundary value problem for a phase transition model. Networks & Heterogeneous Media, 2016, 11 (1) : 89-105. doi: 10.3934/nhm.2016.11.89

[7]

Mauro Garavello, Francesca Marcellini. The Riemann Problem at a Junction for a Phase Transition Traffic Model. Discrete & Continuous Dynamical Systems, 2017, 37 (10) : 5191-5209. doi: 10.3934/dcds.2017225

[8]

Pierluigi Colli, Antonio Segatti. Uniform attractors for a phase transition model coupling momentum balance and phase dynamics. Discrete & Continuous Dynamical Systems, 2008, 22 (4) : 909-932. doi: 10.3934/dcds.2008.22.909

[9]

Claudia Valls. The Boussinesq system:dynamics on the center manifold. Communications on Pure & Applied Analysis, 2005, 4 (4) : 839-860. doi: 10.3934/cpaa.2005.4.839

[10]

Hongyu Cheng, Rafael de la Llave. Time dependent center manifold in PDEs. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6709-6745. doi: 10.3934/dcds.2020213

[11]

Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the kuramoto model on graphs Ⅱ. asymptotic stability of the incoherent state, center manifold reduction, and bifurcations. Discrete & Continuous Dynamical Systems, 2019, 39 (7) : 3897-3921. doi: 10.3934/dcds.2019157

[12]

Lan Jia, Liang Li. Stability and dynamic transition of vegetation model for flat arid terrains. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021189

[13]

Francesca Marcellini. Existence of solutions to a boundary value problem for a phase transition traffic model. Networks & Heterogeneous Media, 2017, 12 (2) : 259-275. doi: 10.3934/nhm.2017011

[14]

Kousuke Kuto, Tohru Tsujikawa. Stationary patterns for an adsorbate-induced phase transition model I: Existence. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1105-1117. doi: 10.3934/dcdsb.2010.14.1105

[15]

Sergey V. Bolotin, Piero Negrini. Global regularization for the $n$-center problem on a manifold. Discrete & Continuous Dynamical Systems, 2002, 8 (4) : 873-892. doi: 10.3934/dcds.2002.8.873

[16]

Stefano Bianchini, Alberto Bressan. A center manifold technique for tracing viscous waves. Communications on Pure & Applied Analysis, 2002, 1 (2) : 161-190. doi: 10.3934/cpaa.2002.1.161

[17]

Jixun Chu, Pierre Magal. Hopf bifurcation for a size-structured model with resting phase. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 4891-4921. doi: 10.3934/dcds.2013.33.4891

[18]

Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, 2021, 20 (1) : 427-448. doi: 10.3934/cpaa.2020275

[19]

Matteo Novaga, Enrico Valdinoci. The geometry of mesoscopic phase transition interfaces. Discrete & Continuous Dynamical Systems, 2007, 19 (4) : 777-798. doi: 10.3934/dcds.2007.19.777

[20]

Alain Miranville. Some mathematical models in phase transition. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : 271-306. doi: 10.3934/dcdss.2014.7.271

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (101)
  • HTML views (215)
  • Cited by (0)

[Back to Top]