Feedback control of travelling wave solutions of the complex Ginzburg–Landau equation

Abstract

Through a linear stability analysis, we investigate the effectiveness of a noninvasive feedback control scheme in stabilizing travelling wave solutions, R eiKx+iωt, of the one-dimensional complex Ginzburg–Landau equation (CGLE) in the Benjamin–Feir unstable regime. The feedback control, a generalization of the time-delay method of Pyragas (1992 Phys. Lett. A 170 421), was proposed by Lu et al (1996 Phys. Rev. Lett. 76 3316) in the setting of nonlinear optics. It involves both spatial shifts, by the wavelength of the targeted travelling wave, and a time delay that coincides with the temporal period of the travelling wave. We derive a single necessary and sufficient stability criterion that determines whether a travelling wave is stable to all perturbation wavenumbers. This criterion has the benefit that it determines an optimal value for the time-delay feedback parameter. For various coefficients in the CGLE we use this algebraic stability criterion to numerically determine stable regions in the (K, ρ)-parameter plane, where ρ is the feedback parameter associated with the spatial translation. We find that the combination of the two feedbacks greatly enlarges the parameter regime where stabilization is possible, and that the stable regions take the form of stability tongues in the (K, ρ)-plane. We discuss possible resonance mechanisms that could account for the spacing with K of the stability tongues.