Bulletin of the
Korean Mathematical Society
BKMS

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Bull. Korean Math. Soc. 2015; 52(4): 1241-1252

Printed July 31, 2015

https://doi.org/10.4134/BKMS.2015.52.4.1241

Copyright © The Korean Mathematical Society.

Dynamical bifurcation of the one dimensional modified Swift-Hohenberg equation

Yuncherl Choi

Kwangwoon University

Abstract

In this paper, we study the dynamical bifurcation of the modified Swift-Hohenberg equation on a periodic interval as the system control parameter crosses through a critical number. This critical number depends on the period. We show that there happens the pitchfork bifurcation under the spatially even periodic condition. We also prove that in the general periodic condition the equation bifurcates to an attractor which is homeomorphic to a circle and consists of steady states solutions.

Keywords: modified Swift-Hohenberg equation, dynamic bifurcation, center manifold function

MSC numbers: Primary 37G35, 35B32

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