Dynamical Bifurcation of the Generalized Swift–Hohenberg Equation

Abstract

In this paper, we prove that the generalized Swift–Hohenberg equation bifurcates from the trivial states to an attractor as the control parameter α passes through critical points. The bifurcation is divided into two groups according to the dimension of the center manifolds. We show that the bifurcated attractor is homeomorphic to S1 or S3 and it contains invariant circles of static solutions. We provide a criterion on the quadratic instability parameter μ which determines the bifurcation to be supercritical or subcritical.