  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, when the control parameter crosses the principal
eigenvalue of the linearized equation. The local behavior of solutions and
their bifurcation to an invariant set near higher eigenvalues are analyzed as
well.
