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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.  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.  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.   Google Scholar

[4]

J. P. Gollub and J. S. Langer, Pattern formation in nonequilibrium physics,, Rev. Mod. Phys., 71 (1999), 396.  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 (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 (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.  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 (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.   Google Scholar

[12]

J. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability,, Phys. Rev. A, 15 (1977), 319.  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.   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.  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.  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.   Google Scholar

[4]

J. P. Gollub and J. S. Langer, Pattern formation in nonequilibrium physics,, Rev. Mod. Phys., 71 (1999), 396.  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 (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 (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.  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 (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.   Google Scholar

[12]

J. Swift and P. C. Hohenberg, Hydrodynamic fluctuations at the convective instability,, Phys. Rev. A, 15 (1977), 319.  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.   Google Scholar

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