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

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.