American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Stability conditions for a class of delay differential equations in single species population dynamics
  • DCDS-B Home
  • This Issue
  • Next Article
    Impacts of migration and immigration on disease transmission dynamics in heterogeneous populations
October  2012, 17(7): 2431-2449. doi: 10.3934/dcdsb.2012.17.2431

Dynamical bifurcation of the two dimensional Swift-Hohenberg equation with odd periodic condition

Jongmin Han 1, and  Chun-Hsiung Hsia 2,

1. 

Department of Mathematics and Research Institute for Basic Sciences, Kyung Hee University, 1 Hoegi-Dong, Dongdaemun-Gu, Seoul, 130-701, South Korea
2. 

Department of Mathematics, National Taiwan University, Taipei, 10617

Received  June 2011 Revised  March 2012 Published  July 2012

In this article, we study the stability and dynamic bifurcation for the two dimensional Swift-Hohenberg equation with an odd periodic condition. It is shown that an attractor bifurcates from the trivial solution as the control parameter crosses the critical value. The bifurcated attractor consists of finite number of singular points and their connecting orbits. Using the center manifold theory, we verify the nondegeneracy and the stability of the singular points.
Keywords: Phase transition, Swift-Hohenberg equation, attractor bifurcation., dynamical bifurcation, center manifold.
Mathematics Subject Classification: Primary: 35B32, 35B4.
Citation: Jongmin Han, Chun-Hsiung Hsia. Dynamical bifurcation of the two dimensional Swift-Hohenberg equation with odd periodic condition. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2431-2449. doi: 10.3934/dcdsb.2012.17.2431
References:
[1]

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

[2]

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

[3]

S. Day, Y. Hiraoka, K. Mischaikow and T. Ogawa, Rigorous numerics for global dynamics: A study of the Swift-Hohenberg equationm, SIAM J. Appl. Dyn. Sys., 4 (2005), 1-31.  Google Scholar

[4]

J. P. Gollub and J. S. Langer, Pattern formation in nonequilibrium physics, Rev. Mod. Phys., 71 (1999), 396-403. doi: 10.1103/RevModPhys.71.S396.  Google Scholar

[5]

J. Han and M. Yari, Dynamic bifurcation of the one-dimensional periodic Swift-Hohenberg equation,, Bull. Korean Math. Soc., ().   Google Scholar

[6]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[7]

T. Ma and S. Wang, "Bifurcation Theory and Applications," World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, 53, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005.  Google Scholar

[8]

T. Ma and S. Wang, "Phase Transition Dynamics in Nonlinear Sciences,", Springer, ().   Google Scholar

[9]

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

[10]

L. A. Peletier and W. C. Troy, "Spatial Patterns: Higher Order Models in Physics and Mecahnics," Progress in Nonlinear Differential Equations and their Applications, 45, Birkhäuser Boston, Inc., Boston, MA, 2001.  Google Scholar

[11]

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

[12]

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

[13]

M. Yari, Attractor bifurcation and final patterns of the n-dimensional and generalized Swift-Hohenberg equations, Dis. Cont. Dyn. Sys. Ser. B, 7 (2007), 441-456.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

S. Day, Y. Hiraoka, K. Mischaikow and T. Ogawa, Rigorous numerics for global dynamics: A study of the Swift-Hohenberg equationm, SIAM J. Appl. Dyn. Sys., 4 (2005), 1-31.  Google Scholar

[4]

J. P. Gollub and J. S. Langer, Pattern formation in nonequilibrium physics, Rev. Mod. Phys., 71 (1999), 396-403. doi: 10.1103/RevModPhys.71.S396.  Google Scholar

[5]

J. Han and M. Yari, Dynamic bifurcation of the one-dimensional periodic Swift-Hohenberg equation,, Bull. Korean Math. Soc., ().   Google Scholar

[6]

D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.  Google Scholar

[7]

T. Ma and S. Wang, "Bifurcation Theory and Applications," World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, 53, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005.  Google Scholar

[8]

T. Ma and S. Wang, "Phase Transition Dynamics in Nonlinear Sciences,", Springer, ().   Google Scholar

[9]

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

[10]

L. A. Peletier and W. C. Troy, "Spatial Patterns: Higher Order Models in Physics and Mecahnics," Progress in Nonlinear Differential Equations and their Applications, 45, Birkhäuser Boston, Inc., Boston, MA, 2001.  Google Scholar

[11]

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

[12]

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

[13]

M. Yari, Attractor bifurcation and final patterns of the n-dimensional and generalized Swift-Hohenberg equations, Dis. Cont. Dyn. Sys. Ser. B, 7 (2007), 441-456.  Google Scholar

[1]

Jongmin Han, Masoud Yari. Dynamic bifurcation of the complex Swift-Hohenberg equation. Discrete & 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 & 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 & Continuous Dynamical Systems, 2009, 25 (1) : 273-297. doi: 10.3934/dcds.2009.25.273
[4]

Masoud Yari. Attractor bifurcation and final patterns of the n-dimensional and generalized Swift-Hohenberg equations. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 441-456. doi: 10.3934/dcdsb.2007.7.441
[5]

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
[6]

Peng Gao. Averaging principles for the Swift-Hohenberg equation. Communications on Pure & Applied Analysis, 2020, 19 (1) : 293-310. doi: 10.3934/cpaa.2020016
[7]

Ling-Jun Wang. The dynamics of small amplitude solutions of the Swift-Hohenberg equation on a large interval. Communications on Pure & 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 & 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 & Continuous Dynamical Systems - S, 2020, 13 (11) : 3189-3204. doi: 10.3934/dcdss.2020114
[10]

Kevin Li. Dynamic transitions of the Swift-Hohenberg equation with third-order dispersion. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021003
[11]

Shengfu Deng. Periodic solutions and homoclinic solutions for a Swift-Hohenberg equation with dispersion. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1647-1662. doi: 10.3934/dcdss.2016068
[12]

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
[13]

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

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
[15]

Chaoqun Huang, Nung Kwan Yip. Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part II. Networks & Heterogeneous Media, 2015, 10 (4) : 897-948. doi: 10.3934/nhm.2015.10.897
[16]

Chaoqun Huang, Nung Kwan Yip. Singular perturbation and bifurcation of diffuse transition layers in inhomogeneous media, part I. Networks & Heterogeneous Media, 2013, 8 (4) : 1009-1034. doi: 10.3934/nhm.2013.8.1009
[17]

Robert Skiba, Nils Waterstraat. The index bundle and multiparameter bifurcation for discrete dynamical systems. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5603-5629. doi: 10.3934/dcds.2017243
[18]

Thai Son Doan, Martin Rasmussen, Peter E. Kloeden. The mean-square dichotomy spectrum and a bifurcation to a mean-square attractor. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 875-887. doi: 10.3934/dcdsb.2015.20.875
[19]

Tian Ma, Shouhong Wang. Attractor bifurcation theory and its applications to Rayleigh-Bénard convection. Communications on Pure & Applied Analysis, 2003, 2 (4) : 591-599. doi: 10.3934/cpaa.2003.2.591
[20]

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

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (79)
  • HTML views (0)
  • Cited by (10)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences