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

Jongmin Han; Chun-Hsiung Hsia

doi:10.3934/dcdsb.2012.17.2431

Article Contents

2012, Volume 17, Issue 7: 2431-2449. Doi: 10.3934/dcdsb.2012.17.2431

This issue Previous Article Impacts of migration and immigration on disease transmission dynamics in heterogeneous populations Next Article Stability conditions for a class of delay differential equations in single species population dynamics

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
2.
Department of Mathematics, National Taiwan University, Taipei, 10617

Received: May 31, 2011

Revised: February 29, 2012

Published: September 2012

Abstract Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 35B32, 35B41.

Citation:

Related Papers

Cited by

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.
[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.
[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.
[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.
[5]	J. Han and M. Yari, Dynamic bifurcation of the one-dimensional periodic Swift-Hohenberg equation, Bull. Korean Math. Soc., to appear.
[6]	D. Henry, "Geometric Theory of Semilinear Parabolic Equations," Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981.
[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.
[8]	T. Ma and S. Wang, "Phase Transition Dynamics in Nonlinear Sciences," Springer, in press.
[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.
[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.
[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.
[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.
[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.