Bulletin of the
Korean Mathematical Society BKMS

In this paper we study the dynamic bifurcation of the Swift-Hohenberg equation on a periodic cell $\Omega =[-L,L]$. It is shown that the equations bifurcates from the trivial solution to an attractor $\mathcal A_\lambda$ when the control parameter $\lambda$ crosses the critical value. In the odd periodic case, $\mathcal A_\lambda$ is homeomorphic to $S^1$ and consists of eight singular points and their connecting orbits. In the periodic case, $\mathcal A_\lambda$ is homeomorphic to $S^1$, and contains a torus and two circles which consist of singular points.

Keywords: Swift-Hohenberg equation, attractor bifurcation

MSC numbers: Primary 37G35, 35G25