DYNAMICAL BIFURCATION OF THE ONE DIMENSIONAL MODIFIED SWIFT-HOHENBERG EQUATION

Abstract

Abstract. In this paper, we study the dynamical bifurcation of the mod-ified Swift-Hohenberg equation on a periodic interval as the system con-trol parameter crosses through a critical number. This critical numberdepends on the period. We show that there happens the pitchfork bifur-cation under the spatially even periodic condition. We also prove thatin the general periodic condition the equation bifurcates to an attractorwhich is homeomorphic to a circle and consists of steady states solutions. 1. IntroductionThe formation of patterns in non-equilibrium systems is closely related tothe instability ([9]) and the bifurcation analysis plays an important role inunderstanding the instability. Indeed, the instability arises when stable statesare driven into unstable states during phases transition. As a control parameterrelated to the instability passes through critical values, the trivial state losesits stability and bifurcates to nontrivial states which form patterns. Thenthe dynamics of the system after the threshold of bifurcation is completelydetermined by its behavior on the center manifold. In particular, Ma andWang showed in [13] that the system bifurcates to a nontrivial attractor on thecenter manifold which determines the final patterns of the system.The Swift-Hohenberg equation is a widely accepted model in the study of theformation of patterns [1, 12]. It was derived in [18] as an approximate model forthe Rayleigh-B´enardconvection describing the pattern formation in layer fluidsbetween horizontal plates. It has attracted a lot of interest in various areas ofapplication regardingpattern formations such as Taylor-Couetteflow and lasers[5]. In particular, there has been much efforts on the bifurcation analysis as away of understanding pattern formations. See [4, 8, 10, 11, 14, 15, 19, 20] forrecent development in this direction.