Bifurcation Analysis and Pattern Selection of Solutions for the Modified Swift–Hohenberg Equation

Abstract

In this paper, motivated by [Peletier & Rottschäfer, 2004; Peletier & Williams, 2007], we study the dynamical bifurcation of the modified Swift–Hohenberg equation endowed with an evenly periodic condition on the interval [Formula: see text]. As [Formula: see text] crosses over the critical points, the trivial solution bifurcates to an attractor and some new patterns of solutions emerge. We provide detailed descriptions of all possible final patterns of solutions on the overlapped intervals of [Formula: see text], which emerge after a gap collapses to a point. We also compute all critical values of [Formula: see text], [Formula: see text] and [Formula: see text] precisely, which are responsible for bifurcation and pattern formations. We finally provide numerical results that explain the main theorems.