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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 and 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.

[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.

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

[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.

[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.

[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.

[8]

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

[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.

[10]

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

[11]

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

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[19]

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

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.

[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.

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

[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.

[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.

[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.

[8]

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

[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.

[10]

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

[11]

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

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[19]

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

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
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