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Stabilization of fixed points by time-delayed feedback

Time-delayed feedback control was developed to stabilize unstable periodic orbits (UPOs), embedded in a chaotic attractor with the time delay set equal to the period of the target orbit. Less is known about the stabilization of deterministic fixed points with this control method. We were able to show in a systematic analytic and numerical study that the linear modes of the fixed point of the delay-differential equation and the control regime have a characteristic structure. We considered a generic case of a dynamical system which has an unstable focus at the origin. The corresponding characteristic equation for the complex eigenvalues can be solved using the Lambert function and the stabilization regimes, in the control parameter plane, can be analytically determined, even when including a low pass filter and latency time (e. g. through signal processing). For delay times equal to integer multiples of the intrinsic period, control is never possible.

On the basis of the Lambert function we have given an explicit eigenmode expansion of the time-delayed dynamics which, in addition to this linear deterministic delay equation, opens up a series of applications to stochastic delay-equations and weak nonlinear analysis. In collaboration with Subdivision B2 we have determined analytically, and compared to the numerical solution, the asymptotic scaling behaviour of the eigenmodes for large delay times and the decomposition of the spectrum into a pseudo-continuous part and most two strongly unstable eigenvalues.

In collaboration with Subdivision A6 we have applied this control method to a real physical system, namely a multisection semiconductor laser, in order to stabilize the stationary state (cw laser emission) and to suppress the intensity pulsations. This is the first purely optical realization of the noninvasive Pyragas control for stabilization of fixed points. This time-delayed feedback can be realized with the help of a coupled Fabry-Perot-resonator on a very short timescale and nanometer lengthscale, which yields promising application to ultrafast optical signal processing very. We could derive the crucial role of a phase-dependent feedback from our theoretical model. By substituting the unity matrix of the feedback scheme with a rotation matrix with a phase angle, the stabilization regimes in the control parameter plane are deformed and  control is successful for parameter values at which stabilization is impossible in the absence of the phase.

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