Bifurcation in the Swift–Hohenberg Equation

This paper is concerned with the asymptotic behavior of the solutions u(x, t) of the Swift–Hohenberg equation with quintic polynomial on the cylindrical domain Q=(0,L)×R+. With the control parameter α in the Swift–Hohenberg equation and the length L of the domain regarded as bifurcation parameters, branches of nontrivial solutions bifurcating from the trivial solution at certain points are shown. Local behavior of these branches is also investigated. With the help of a center manifold analysis, two types of structures in the bifurcation diagrams are presented when the bifurcation points are close, and their stabilities are analyzed.