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September  2017, 22(7): 2543-2567. doi: 10.3934/dcdsb.2017087

Bifurcation and final patterns of a modified Swift-Hohenberg equation

1. 

Division of General Education, Kwangwoon University, Seoul, 139-701, Korea

2. 

Division of Computational Sciences, National Institute for Mathematical Sciences, Daejeon, 305-811, Korea

3. 

Department of Mathematics and Research Institute of Basic Sciences, Kyung Hee University, Seoul, 02447, Korea

4. 

Department of Undergraduate Studies, Daegu Gyeongbuk Institute of Science and Technology, Daegu, 711-873, Korea

Received  July 2016 Revised  October 2016 Published  March 2017

Fund Project: Y. Choi was supported by the Research Grant of Kwangwoon University in 2015. T. Ha was partially supported by the National Institute for Mathematical Sciences (NIMS) grant funded by the Korean government (No. A21300000). J. Han was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(2011-0008557).

In this paper, we study the dynamical bifurcation and final patterns of a modified Swift-Hohenberg equation(MSHE). We prove that the MSHE bifurcates from the trivial solution to an $S^1$-attractor as the control parameter $\alpha $ passes through a critical number $\hat{\alpha }$. Using the center manifold analysis, we study the bifurcated attractor in detail by showing that it consists of finite number of singular points and their connecting orbits. We investigate the stability of those points. We also provide some numerical results supporting our analysis.

Citation: Yuncherl Choi, Taeyoung Ha, Jongmin Han, Doo Seok Lee. Bifurcation and final patterns of a modified Swift-Hohenberg equation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2543-2567. doi: 10.3934/dcdsb.2017087
References:
[1]

I. Aranson and L. Kramer, The world of the complex Ginzburg-Landau equation, Rev. Mod. Phys., 74 (2002), 99-143.  doi: 10.1103/RevModPhys.74.99.  Google Scholar

[2]

M. Bestehorn and H. Haken, Transient patterns of the convection instability: A model-calculation, Z. Phys. B, 57 (1984), 329-333.  doi: 10.1007/BF01470424.  Google Scholar

[3]

Y. Choi, Dynamical bifurcation of one dimensional modified Swift-Hohenberg equation, Bull. Korean Math. Soc., 52 (2015), 1241-1252.  doi: 10.4134/BKMS.2015.52.4.1241.  Google Scholar

[4]

Y. Choi and J. Han, Dynamical bifurcation of the damped Kuramoto-Sivashinsky equation, J. Math. Anal. Appl., 421 (2015), 383-398.  doi: 10.1016/j.jmaa.2014.07.009.  Google Scholar

[5]

Y. ChoiJ. Han and C. H. Hsia, Bifurcation analysis of the damped Kuramoto-Sivashinsky equation with respect to the period, Discr. Cont. Dyn. Syst. B., 20 (2015), 1933-1957.  doi: 10.3934/dcdsb.2015.20.1933.  Google Scholar

[6]

Y. Choi, J. Han and J. Park, Dynamical bifurcation of the generalized Swift-Hohenberg equation Intern. J. Bifur. Chaos, 25 (2015), 1550095, 16 pp. doi: 10.1142/S0218127415500959.  Google Scholar

[7]

M. Cross and P. Hohenbrg, Pattern formation outside equilibrium, Rev. Mod. Phys., 65 (1993), 851-1112.  doi: 10.1103/RevModPhys.65.851.  Google Scholar

[8]

A. DolemanB. SandstedeA. Scheel and G. Schneider, Propagation of hexagonal patterns near onset, Euro. J. Appl. Math., 14 (2003), 85-110.  doi: 10.1017/S095679250200503X.  Google Scholar

[9]

N. Duan and W. Gao, Optimal control of a modified Swift-Hohenberg equation, Electr. J. Diff. Eqns., 2012 (2012), 1-12.   Google Scholar

[10]

K. R. EdlerJ. D. Gunton and N. Goldenfled, Transition to spatiotemporal chaos in the damped Kuramoto-Sivashinky equation, Phys. Rev. E, 56 (1997), 1631-1634.   Google Scholar

[11]

H. Gao and Q. Xiao, Bifurcation analysis of the 1D and 2D generalized Swift-Hohenberg equation, Intern. J. Bifur. Chaos, 20 (2010), 619-643.  doi: 10.1142/S0218127410025922.  Google Scholar

[12]

J. P. Gollub and J. S. Langer, Pattern formation in nonequilibrium physics, More Things in Heaven and Earth, (1999), 665-676.  doi: 10.1007/978-1-4612-1512-7_43.  Google Scholar

[13]

J. Han and C.-H. Hsia, Dynamical bifurcation of the two dimensional Swift-Hohenberg equation with odd periodic condition, Discr. Cont. Dyn. Syst. B., 17 (2012), 2431-2449.  doi: 10.3934/dcdsb.2012.17.2431.  Google Scholar

[14]

J. Han and M. Yari, Dynamic bifurcation of the periodic Swift-Hohenberg equation, Bull. Korean Math. Soc., 49 (2012), 923-937.  doi: 10.4134/BKMS.2012.49.5.923.  Google Scholar

[15]

M. HilaliS. MétensP. Borckmans and G. Dewel, Pattern selection in the generalized Swift-Hohenberg model, Phys. Rev. E, 51 (1995), 2046-2052.  doi: 10.1103/PhysRevE.51.2046.  Google Scholar

[16]

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

[17]

T. Ma and S. Wang, Phase Transition Dynamics in Nonlinear Sciences Springer, 2014. doi: 10.1007/978-1-4614-8963-4.  Google Scholar

[18]

L. A. Peletier and V. Rottschäfer, Pattern selection of solutions of the Swift-Hohenberg equation, Physica D, 194 (2004), 95-126.  doi: 10.1016/j.physd.2004.01.043.  Google Scholar

[19]

L. Peletier and J. Williams, Some canonical bifurcations in the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Sys., 6 (2007), 208-235.  doi: 10.1137/050647232.  Google Scholar

[20]

M. Polat, Global attractor for a modified Swift-Hohenberg equation, Computers Math. Appl., 57 (2009), 62-66.  doi: 10.1016/j.camwa.2008.09.028.  Google Scholar

[21]

L. SongY. Zhang and T. Ma, Global attractor for a modified Swift-Hohenberg equation in $H^k$ spaces, Nonlin. Anal., 72 (2010), 183-191.  doi: 10.1016/j.na.2009.06.103.  Google Scholar

[22]

J. Swift and P. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15 (1977), 319-328.  doi: 10.1103/PhysRevA.15.319.  Google Scholar

[23]

Q. Xiao and H. Gao, Bifurcation analysis of the Swift-Hohenberg equation with quintic nonlinearity, Intern. J. Bifur. Chaos, 19 (2009), 2927-2937.  doi: 10.1142/S0218127409024542.  Google Scholar

[24]

Q. Xiao and H. Gao, Bifurcation analysis of a modified Swift-Hohenberg equation, Nonlin. Anal. Real World Appl., 11 (2010), 4451-4464.  doi: 10.1016/j.nonrwa.2010.05.028.  Google Scholar

[25]

M. Yari, Attractor bifurcation and final patterns of the $N$-dimensional and generalized Swift-Hohenberg equations, Discr. Cont. Dyn. Sys. B, 7 (2007), 441-456.  doi: 10.3934/dcdsb.2007.7.441.  Google Scholar

[26]

X. ZhaoB. LiuP. ZhangW. Zhang and F. Liu, Fourier spectral method for the modified Swift-Hohenberg equation, Adv. Difference Eqns., 2013 (2013), 1-19.  doi: 10.1186/1687-1847-2013-156.  Google Scholar

show all references

References:
[1]

I. Aranson and L. Kramer, The world of the complex Ginzburg-Landau equation, Rev. Mod. Phys., 74 (2002), 99-143.  doi: 10.1103/RevModPhys.74.99.  Google Scholar

[2]

M. Bestehorn and H. Haken, Transient patterns of the convection instability: A model-calculation, Z. Phys. B, 57 (1984), 329-333.  doi: 10.1007/BF01470424.  Google Scholar

[3]

Y. Choi, Dynamical bifurcation of one dimensional modified Swift-Hohenberg equation, Bull. Korean Math. Soc., 52 (2015), 1241-1252.  doi: 10.4134/BKMS.2015.52.4.1241.  Google Scholar

[4]

Y. Choi and J. Han, Dynamical bifurcation of the damped Kuramoto-Sivashinsky equation, J. Math. Anal. Appl., 421 (2015), 383-398.  doi: 10.1016/j.jmaa.2014.07.009.  Google Scholar

[5]

Y. ChoiJ. Han and C. H. Hsia, Bifurcation analysis of the damped Kuramoto-Sivashinsky equation with respect to the period, Discr. Cont. Dyn. Syst. B., 20 (2015), 1933-1957.  doi: 10.3934/dcdsb.2015.20.1933.  Google Scholar

[6]

Y. Choi, J. Han and J. Park, Dynamical bifurcation of the generalized Swift-Hohenberg equation Intern. J. Bifur. Chaos, 25 (2015), 1550095, 16 pp. doi: 10.1142/S0218127415500959.  Google Scholar

[7]

M. Cross and P. Hohenbrg, Pattern formation outside equilibrium, Rev. Mod. Phys., 65 (1993), 851-1112.  doi: 10.1103/RevModPhys.65.851.  Google Scholar

[8]

A. DolemanB. SandstedeA. Scheel and G. Schneider, Propagation of hexagonal patterns near onset, Euro. J. Appl. Math., 14 (2003), 85-110.  doi: 10.1017/S095679250200503X.  Google Scholar

[9]

N. Duan and W. Gao, Optimal control of a modified Swift-Hohenberg equation, Electr. J. Diff. Eqns., 2012 (2012), 1-12.   Google Scholar

[10]

K. R. EdlerJ. D. Gunton and N. Goldenfled, Transition to spatiotemporal chaos in the damped Kuramoto-Sivashinky equation, Phys. Rev. E, 56 (1997), 1631-1634.   Google Scholar

[11]

H. Gao and Q. Xiao, Bifurcation analysis of the 1D and 2D generalized Swift-Hohenberg equation, Intern. J. Bifur. Chaos, 20 (2010), 619-643.  doi: 10.1142/S0218127410025922.  Google Scholar

[12]

J. P. Gollub and J. S. Langer, Pattern formation in nonequilibrium physics, More Things in Heaven and Earth, (1999), 665-676.  doi: 10.1007/978-1-4612-1512-7_43.  Google Scholar

[13]

J. Han and C.-H. Hsia, Dynamical bifurcation of the two dimensional Swift-Hohenberg equation with odd periodic condition, Discr. Cont. Dyn. Syst. B., 17 (2012), 2431-2449.  doi: 10.3934/dcdsb.2012.17.2431.  Google Scholar

[14]

J. Han and M. Yari, Dynamic bifurcation of the periodic Swift-Hohenberg equation, Bull. Korean Math. Soc., 49 (2012), 923-937.  doi: 10.4134/BKMS.2012.49.5.923.  Google Scholar

[15]

M. HilaliS. MétensP. Borckmans and G. Dewel, Pattern selection in the generalized Swift-Hohenberg model, Phys. Rev. E, 51 (1995), 2046-2052.  doi: 10.1103/PhysRevE.51.2046.  Google Scholar

[16]

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

[17]

T. Ma and S. Wang, Phase Transition Dynamics in Nonlinear Sciences Springer, 2014. doi: 10.1007/978-1-4614-8963-4.  Google Scholar

[18]

L. A. Peletier and V. Rottschäfer, Pattern selection of solutions of the Swift-Hohenberg equation, Physica D, 194 (2004), 95-126.  doi: 10.1016/j.physd.2004.01.043.  Google Scholar

[19]

L. Peletier and J. Williams, Some canonical bifurcations in the Swift-Hohenberg equation, SIAM J. Appl. Dyn. Sys., 6 (2007), 208-235.  doi: 10.1137/050647232.  Google Scholar

[20]

M. Polat, Global attractor for a modified Swift-Hohenberg equation, Computers Math. Appl., 57 (2009), 62-66.  doi: 10.1016/j.camwa.2008.09.028.  Google Scholar

[21]

L. SongY. Zhang and T. Ma, Global attractor for a modified Swift-Hohenberg equation in $H^k$ spaces, Nonlin. Anal., 72 (2010), 183-191.  doi: 10.1016/j.na.2009.06.103.  Google Scholar

[22]

J. Swift and P. Hohenberg, Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15 (1977), 319-328.  doi: 10.1103/PhysRevA.15.319.  Google Scholar

[23]

Q. Xiao and H. Gao, Bifurcation analysis of the Swift-Hohenberg equation with quintic nonlinearity, Intern. J. Bifur. Chaos, 19 (2009), 2927-2937.  doi: 10.1142/S0218127409024542.  Google Scholar

[24]

Q. Xiao and H. Gao, Bifurcation analysis of a modified Swift-Hohenberg equation, Nonlin. Anal. Real World Appl., 11 (2010), 4451-4464.  doi: 10.1016/j.nonrwa.2010.05.028.  Google Scholar

[25]

M. Yari, Attractor bifurcation and final patterns of the $N$-dimensional and generalized Swift-Hohenberg equations, Discr. Cont. Dyn. Sys. B, 7 (2007), 441-456.  doi: 10.3934/dcdsb.2007.7.441.  Google Scholar

[26]

X. ZhaoB. LiuP. ZhangW. Zhang and F. Liu, Fourier spectral method for the modified Swift-Hohenberg equation, Adv. Difference Eqns., 2013 (2013), 1-19.  doi: 10.1186/1687-1847-2013-156.  Google Scholar

Figure 1.  Truncated system for $N = 0$ with $\beta=0.0001$
Figure 2.  Structure of the bifurcated attractor for $N = 0$
Figure 3.  Truncated system for $N = 1$ with $\beta=0.0001$ and $\mu = 1$
Figure 4.  Structure of the bifurcated attractor for $N = 1$
Figure 5.  Structure of the bifurcated attractor for $N \geq 2$
Figure 6.  Initial conditions
Figure 7.  Tests for (a) $N=0$ and (b) $N=1$ given the initial value $u_0(x)$ and $\mu = 1$
Figure 8.  Tests for $N=2$ given by (a) the initial value $u_0(x)$ and $\mu = 1/2$, (b) the initial value $u_1(x)$ and $\mu = 1/2$ and (c) the initial value $u_1(x)$ and $\mu = 2$
Figure 9.  Tests for $N=8$ given by (a) the initial value $u_1(x)$ and $\mu = 1$ and (b) the initial value $u_2(x)$ and $\mu = 1$
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