Feedback control of traveling wave solutions to the complex Ginzburg Landau equation, and, A nonlinear analysis of the amplification properties of auditory hair cells

Through a linear stability analysis, the effectiveness of a noninvasive feedback control scheme aimed at stabilizing traveling wave solutions to the one-dimensional complex Ginzburg Landau equation (OGLE) is investigated in the Benjamin-Feir unstable regime. The feedback control, which was proposed in the setting of nonlinear optics, is a generalization of the time-delay method of Pyragas. It involves both spatial shifts by the wavelength of the targeted traveling wave, and a time-delay that coincides with the temporal period of the traveling wave. A single necessary and sufficient stability criterion is derived which determines whether a traveling 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 OGLE we use this algebraic stability criterion to numerically determine stable regions in the (K,rho) parameter plane, where rho is the feedback parameter associated with the spatial translation and K is the wavenumber of the traveling wave. It is found that a combination of the two feedbacks greatly enlarges the parameter regime where stabilization is possible, and that the stability regions take the form of stability tongues in the (K,rho)-plane. Possible resonance mechanisms that could account for the spacing in K of the stability tongues are discussed.;A mathematical model describing the coupling between two independent amplification mechanisms in auditory hair cells is proposed and analyzed. Hair cells are cells in the inner ear responsible for translating sound-induced mechanical stimuli into an electrical signal that can then be carried away by the auditory nerve. In nonmammals, two separate mechanisms have been postulated to contribute to the amplification and tuning properties of the hair cells. Models of each of these mechanisms have been shown to be poised near a Hopf bifurcation. Through a weakly nonlinear analysis, that assumes weak periodic forcing, weak damping, and weak coupling, the physiologically-based models of the two mechanisms are reduced to a system of two coupled amplitude equations. The predictions that follow from an analysis of the reduced equations, as well as performance benefits due to the coupling of the two mechanisms, are discussed and compared with experimental data.