Delay stabilization of rotating waves near fold bifurcation and application to all-optical control of a semiconductor laser

Abstract

We consider the delayed feedback control method for stabilization of unstable rotating waves near a fold bifurcation. Theoretical analysis of a generic model and numerical bifurcation analysis of the rate-equations model demonstrate that such orbits can always be stabilized by a proper choice of control parameters. Our paper confirms the recently discovered invalidity of the so-called "odd-number limitation" of delayed feedback control. Previous results have been restricted to the vicinity of a subcritical Hopf bifurcation. We now refute such a limitation for rotating waves near a fold bifurcation. We include an application to all-optical realization of the control in three-section semiconductor lasers.

Control of Complex Nonlinear Systems with Delay

“…One such system, a threesection semiconductor laser, was discussed recently in Ref. [FIE08]. In this work, numerical bifurcation analysis confirmed that an all-optical delayed feedback control can successfully stabilize rotating waves close to a fold bifurcation in this system.…”

“…In this Section, I will stabilization of periodic orbits which occur due to a fold bifurcation following Ref. [FIE08]. This bifurcation is sometimes called saddle-node bifurcation of limit cycles because the periodic orbit generated at the bifurcation point is repulsive for points outside the limit cycle and attracting for points from the inside.…”

Time-Delayed Feedback Control: From Simple Models to Lasers and Neural Systems

“…In this section we review basic properties of time-delayed feedback control, using simple normal form models which are representative of a large class of nonlinear dynamic systems [48,49,50,52,54].…”

“…Analytical insight into those schemes has been obtained by several theoretical studies, e.g. [42,43,44,45,46,47,48,49,50,51,52,53,54] as well as by numerical bifurcation analysis, e.g. [55,56].…”

Dynamics and Stability of Permanent-Magnet Synchronous Motor

“…By the bifurcation theory of the equilibrium points, the equilibrium points may lose stability when the parameters pass through the key values and the bifurcation behavior occurs [20][21][22][23]. In order to obtain the conditions of Hopf bifurcation, setting = ( ̸ = 0) and substituting it into characteristic equation (6), we obtain…”