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Article Contents

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

• * Corresponding author: Jongmin Han
• 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.

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

 Citation:

• 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

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

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

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
•  [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. Toth, V. 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.

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