Abstract
In this paper, we study the dynamical bifurcation and final patterns of a modified Swift-Hohenberg equation(MSHE). We prove that the MSHE bifurcates from the trivial solution to an \begin{document}$S^1$\end{document} -attractor as the control parameter \begin{document}$\alpha $\end{document} passes through a critical number \begin{document}$\hat{\alpha }$\end{document} . Using the center manifold analysis, we study the bifurcated attractor in detail by showing that it consists of finite number of singular points and their connecting orbits. We investigate the stability of those points. We also provide some numerical results supporting our analysis.
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