Bifurcation analysis of the  damped Kuramoto-Sivashinsky equation with respect to the period

Yuncherl  Choi; Jongmin Han; Chun-Hsiung Hsia

doi:10.3934/dcdsb.2015.20.1933

Article Contents

2015, Volume 20, Issue 7: 1933-1957. Doi: 10.3934/dcdsb.2015.20.1933

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Bifurcation analysis of the damped Kuramoto-Sivashinsky equation with respect to the period

1.
Division of General Education, Kwangwoon University, Seoul, 139-701
2.
Department of Mathematics and Research Institute for Basic Sciences, Kyung Hee University, 1 Hoegi-Dong, Dongdaemun-Gu, Seoul, 130-701
3.
Department of Mathematics and Taidar Institute of Mthematical Science, National Taiwan University, Taipei, 10617

Received: March 31, 2014

Revised: February 28, 2015

Published: August 2015

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Abstract

In this paper, we study bifurcation of the damped Kuramoto-Sivashinsky equation on an odd periodic interval of period $2\lambda$. We fix the control parameter $\alpha \in (0,1)$ and study how the equation bifurcates to attractors as $\lambda$ varies. Using the center manifold analysis, we prove that the bifurcated attractors are homeomorphic to $S^1$ and consist of four or eight singular points and their connecting orbits. We verify the structure of the bifurcated attractors by investigating the stability of each singular point.

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Mathematics Subject Classification: Primary: 37G35, 35B32.

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