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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 & Continuous Dynamical Systems - B, 2007, 7 (2) : 441-456. doi: 10.3934/dcdsb.2007.7.441
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