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September  2008, 7(5): 1237-1253. doi: 10.3934/cpaa.2008.7.1237

Bifurcation and stability of the generalized complex Ginzburg--Landau equation

Jungho Park 1,

1. 

Department of Mathematics, New York Institute of Technology, Old Westbury, NY 11568, United States

Received  June 2007 Revised  January 2008 Published  June 2008

This paper examines the bifurcation and stability of the solutions of the complex Ginzburg--Landau equation(CGLE). The structure of the bifurcated solutions shall be explored as well. We investigate two different modes of the CGLE. The first mode of the CGLE contains only an unstable cubic term and the second mode contains not only a cubic term but a quintic term. The solutions of the cubic CGLE bifurcate from the trivial solution to an attractor supercritically in some parameter range. However, for the cubic-quintic CGLE, a subcritical bifurcation is obtained. Due to the global attractor, we obtain a saddle node bifurcation point $\lambda_c$. By thoroughly investigating the structure and transition of the solutions of the CGLE, we confirm that the bifurcated solutions are homeomorphic to $S^1$ and contain steady state solutions.
Keywords: Generalized Ginzburg--Landau equation, saddle node, subcritical and supercritical bifurcation, stability..
Mathematics Subject Classification: Primary: 35Q, 35B, 37L; Secondary: 76.
Citation: Jungho Park. Bifurcation and stability of the generalized complex Ginzburg--Landau equation. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1237-1253. doi: 10.3934/cpaa.2008.7.1237
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