scholarly journals Bifurcation of a limit cycle in the ac-driven complex Ginzburg-Landau equation

2010 ◽  
Vol 14 (1) ◽  
pp. 129-141 ◽  
Author(s):  
Qiongwei Huang ◽  
◽  
Jiashi Tang
Keyword(s):  
Limit Cycle ◽  
Landau Equation ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation

scholarly journals Localized Pulsating Solutions of the Generalized Complex Cubic-Quintic Ginzburg-Landau Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Ivan M. Uzunov ◽  
Zhivko D. Georgiev
Keyword(s):  
Limit Cycle ◽  
Solitary Wave Solution ◽  
Landau Equation ◽  
Melnikov Method ◽  
Van Der Pol Oscillator ◽  
Unperturbed System ◽  
Van Der Pol ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation ◽  
Generalized Complex

We study the dynamics of the localized pulsating solutions of generalized complex cubic-quintic Ginzburg-Landau equation (CCQGLE) in the presence of intrapulse Raman scattering (IRS). We present an approach for identification of periodic attractors of the generalized CCQGLE. Using ansatz of the travelling wave and fixing some relations between the material parameters, we derive the strongly nonlinear Lienard-Van der Pol equation for the amplitude of the nonlinear wave. Next, we apply the Melnikov method to this equation to analyze the possibility of existence of limit cycles. For a set of fixed parameters we show the existence of limit cycle that arises around a closed phase trajectory of the unperturbed system and prove its stability. We apply the Melnikov method also to the equation of Duffing-Van der Pol oscillator used for the investigation of the influence of the IRS on the bandwidth limited amplification. We prove the existence and stability of a limit cycle that arises in a neighborhood of a homoclinic trajectory of the corresponding unperturbed system. The condition of existence of the limit cycle derived here coincides with the relation between the critical value of velocity and the amplitude of the solitary wave solution (Uzunov, 2011).

Spiral Solutions of the Two-Dimensional Complex Ginzburg–Landau Equation

2001 ◽  
Vol 35 (2) ◽  
pp. 159-161
Author(s):  
Liu Shi-Da ◽  
Liu Shi-Kuo ◽  
Fu Zun-Tao ◽  
Zhao Qiang
Keyword(s):  
Landau Equation ◽  
Two Dimensional ◽  
Ginzburg Landau ◽  
Ginzburg Landau Equation
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