American Institute of Mathematical Sciences

Advanced
March  2007, 7(2): 441-456. doi: 10.3934/dcdsb.2007.7.441

Attractor bifurcation and final patterns of the n-dimensional and generalized Swift-Hohenberg equations

Masoud Yari 1,

1. 

Department of Mathematics, Indiana University, Rawles Hall, Bloomington IN 47405, United States

Received  May 2006 Revised  October 2006 Published  December 2006

In this paper I will investigate the bifurcation and asymptotic behavior of solutions of the Swift-Hohenberg equation and the generalized Swift-Hohenberg equation with the Dirichlet boundary condition on a one-dimensional domain $(0,L)$. I will also study the bifurcation and stability of patterns in the $n$-dimensional Swift-Hohenberg equation with the odd-periodic and periodic boundary conditions. It is shown that each equation bifurcates from the trivial solution to an attractor $\mathcal A_\lambda$ when the control parameter $\lambda$ crosses $\lambda _{c} $, the principal eigenvalue of $(I+\Delta)^2$. The local behavior of solutions and their bifurcation to an invariant set near higher eigenvalues are analyzed as well.
Keywords: Attractor bifurcation, Pattern formation, Swift-Hohenberg equation.
Mathematics Subject Classification: Primary: 35G25,37G35; Secondary: 35B4.
Citation: Masoud Yari. Attractor bifurcation and final patterns of the n-dimensional and generalized Swift-Hohenberg equations. Discrete and Continuous Dynamical Systems - B, 2007, 7 (2) : 441-456. doi: 10.3934/dcdsb.2007.7.441
[1]

Jongmin Han, Masoud Yari. Dynamic bifurcation of the complex Swift-Hohenberg equation. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 875-891. doi: 10.3934/dcdsb.2009.11.875
[2]

Yuncherl Choi, Taeyoung Ha, Jongmin Han, Doo Seok Lee. Bifurcation and final patterns of a modified Swift-Hohenberg equation. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2543-2567. doi: 10.3934/dcdsb.2017087
[3]

Toshiyuki Ogawa, Takashi Okuda. Bifurcation analysis to Swift-Hohenberg equation with Steklov type boundary conditions. Discrete and Continuous Dynamical Systems, 2009, 25 (1) : 273-297. doi: 10.3934/dcds.2009.25.273
[4]

J. Burke, Edgar Knobloch. Multipulse states in the Swift-Hohenberg equation. Conference Publications, 2009, 2009 (Special) : 109-117. doi: 10.3934/proc.2009.2009.109
[5]

Peng Gao. Averaging principles for the Swift-Hohenberg equation. Communications on Pure and Applied Analysis, 2020, 19 (1) : 293-310. doi: 10.3934/cpaa.2020016
[6]

Jongmin Han, Chun-Hsiung Hsia. Dynamical bifurcation of the two dimensional Swift-Hohenberg equation with odd periodic condition. Discrete and Continuous Dynamical Systems - B, 2012, 17 (7) : 2431-2449. doi: 10.3934/dcdsb.2012.17.2431
[7]

Ling-Jun Wang. The dynamics of small amplitude solutions of the Swift-Hohenberg equation on a large interval. Communications on Pure and Applied Analysis, 2012, 11 (3) : 1129-1156. doi: 10.3934/cpaa.2012.11.1129
[8]

Yanfeng Guo, Jinqiao Duan, Donglong Li. Approximation of random invariant manifolds for a stochastic Swift-Hohenberg equation. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1701-1715. doi: 10.3934/dcdss.2016071
[9]

Yixia Shi, Maoan Han. Existence of generalized homoclinic solutions for a modified Swift-Hohenberg equation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3189-3204. doi: 10.3934/dcdss.2020114
[10]

Shengfu Deng. Periodic solutions and homoclinic solutions for a Swift-Hohenberg equation with dispersion. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 1647-1662. doi: 10.3934/dcdss.2016068
[11]

John Burke, Edgar Knobloch. Normal form for spatial dynamics in the Swift-Hohenberg equation. Conference Publications, 2007, 2007 (Special) : 170-180. doi: 10.3934/proc.2007.2007.170
[12]

Kevin Li. Dynamic transitions of the Swift-Hohenberg equation with third-order dispersion. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6069-6090. doi: 10.3934/dcdsb.2021003
[13]

Andrea Giorgini. On the Swift-Hohenberg equation with slow and fast dynamics: well-posedness and long-time behavior. Communications on Pure and Applied Analysis, 2016, 15 (1) : 219-241. doi: 10.3934/cpaa.2016.15.219
[14]

Fengqi Yi, Eamonn A. Gaffney, Sungrim Seirin-Lee. The bifurcation analysis of turing pattern formation induced by delay and diffusion in the Schnakenberg system. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 647-668. doi: 10.3934/dcdsb.2017031
[15]

Qingyan Shi, Junping Shi, Yongli Song. Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity. Discrete and Continuous Dynamical Systems - B, 2019, 24 (2) : 467-486. doi: 10.3934/dcdsb.2018182
[16]

Christian Kuehn, Pasha Tkachov. Pattern formation in the doubly-nonlocal Fisher-KPP equation. Discrete and Continuous Dynamical Systems, 2019, 39 (4) : 2077-2100. doi: 10.3934/dcds.2019087
[17]

Julien Barré, Pierre Degond, Diane Peurichard, Ewelina Zatorska. Modelling pattern formation through differential repulsion. Networks and Heterogeneous Media, 2020, 15 (3) : 307-352. doi: 10.3934/nhm.2020021
[18]

Julien Cividini. Pattern formation in 2D traffic flows. Discrete and Continuous Dynamical Systems - S, 2014, 7 (3) : 395-409. doi: 10.3934/dcdss.2014.7.395
[19]

Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. Pattern formation in a cross-diffusion system. Discrete and Continuous Dynamical Systems, 2015, 35 (4) : 1589-1607. doi: 10.3934/dcds.2015.35.1589
[20]

Peter Rashkov. Remarks on pattern formation in a model for hair follicle spacing. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1555-1572. doi: 10.3934/dcdsb.2015.20.1555

2021 Impact Factor: 1.497

Tools

Metrics

  • PDF downloads (75)
  • HTML views (0)
  • Cited by (12)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy