Self-stabilization of high-frequency oscillations in semiconductor superlattices by time-delay autosynchronization

Feedback control of flow alignment in sheared liquid crystals

Based on a continuum theory, we investigate the manipulation of the nonequilibrium behavior of a sheared liquid crystal via closed-loop feedback control. Our goal is to stabilize a specific dynamical state, that is, the stationary "flow alignment," under conditions where the uncontrolled system displays oscillatory director dynamics with in-plane symmetry. To this end we employ time-delayed feedback control (TDFC), where the equation of motion for the ith component q(i)(t) of the order parameter tensor is supplemented by a control term involving the difference q(i)(t)-q(i)(t-τ). In this diagonal scheme, τ is the delay time. We demonstrate that the TDFC method successfully stabilizes flow alignment for suitable values of the control strength K and τ; these values are determined by solving an exact eigenvalue equation. Moreover, our results show that only small values of K are needed when the system is sheared from an isotropic equilibrium state, contrary to the case where the equilibrium state is nematic.

Dynamics of localized structures in reaction-diffusion systems induced by delayed feedback

We are interested in stability properties of a single localized structure in a three-component reaction-diffusion system subjected to the time-delayed feedback. We shall show that variation in the product of the delay time and the feedback strength leads to complex dynamical behavior of the system, including formation of target patterns, spontaneous motion, and spontaneous breathing as well as various complex structures, arising from combination of different oscillatory instabilities. In the case of spontaneous motion, we provide a bifurcation analysis of the delayed system and derive an order parameter equation for the position of the localized structure, explicitly describing its temporal evolution in the vicinity of the bifurcation point. This equation is a subject to a nonlinear delay differential equation, which can be transformed to the normal form of the pitchfork drift bifurcation.

Instabilities of localized structures in dissipative systems with delayed feedback

We report on a novel behavior of solitary localized structures in a real Swift-Hohenberg equation subjected to a delayed feedback. We shall show that variation in the product of the delay time and the feedback strength leads to nontrivial instabilities resulting in the formation of oscillons, soliton rings, labyrinth patterns, or moving structures. We provide a bifurcation analysis of the delayed system and derive a system of order parameter equations explicitly describing the temporal behavior of the localized structure in the vicinity of the bifurcation point. We demonstrate that a normal form of the bifurcation, responsible for the emergence of moving solitary structures, can be obtained and show that spontaneous motion to the lowest order occurs without change of the shape.

CONTROLLING STOCHASTIC OSCILLATIONS CLOSE TO A HOPF BIFURCATION BY TIME-DELAYED FEEDBACK

We study the effect of a time-delayed feedback upon a Van der Pol oscillator under the influence of white noise in the regime below the Hopf bifurcation where the deterministic system has a stable fixed point. We show that both the coherence and the frequency of the noise-induced oscillations can be controlled by varying the delay time and the strength of the control force. Approximate analytical expressions for the power spectral density and the coherence properties of the stochastic delay differential equation are developed, and are in good agreement with our numerical simulations. Our analytical results elucidate how the correlation time of the controlled stochastic oscillations can be maximized as a function of delay and feedback strength.

Resonant control of stochastic spatiotemporal dynamics in a tunnel diode by multiple time-delayed feedback

We study the control of noise-induced spatiotemporal current density patterns in a semiconductor nanostructure (double-barrier resonant tunneling diode) by multiple time-delayed feedback. We find much more pronounced resonant features of noise-induced oscillations compared to single time feedback, rendering the system more sensitive to variations in the delay time tau . The coherence of noise-induced oscillations measured by the correlation time exhibits sharp resonances as a function of tau , and can be strongly increased by optimal choices of tau . Similarly, the peaks in the power spectral density are sharpened. We provide analytical insight into the control mechanism by relating the correlation times and mean frequencies of noise-induced breathing oscillations to the stability properties of the deterministic stationary current density filaments under the influence of the control loop. Moreover, we demonstrate that the use of multiple time delays enlarges the regime in which the deterministic dynamical properties of the system are not changed by delay-induced bifurcations.

Control of coherence resonance in semiconductor superlattices

We study the effect of time-delayed feedback control and Gaussian white noise on the spatiotemporal charge dynamics in a semiconductor superlattice. The system is prepared in a regime where the deterministic dynamics is close to a global bifurcation, namely, a saddle-node bifurcation on a limit cycle. In the absence of control, noise can induce electron charge front motion through the entire device, and coherence resonance is observed. We show that with appropriate selection of the time-delayed feedback parameters the effect of coherence resonance can be either enhanced or destroyed, and the coherence of stochastic domain motion at low noise intensity is dramatically increased. Additionally, the purely delay-induced dynamics in the system is investigated, and a homoclinic bifurcation of a limit cycle is found.

Failure of feedback as a putative common mechanism of spreading depolarizations in migraine and stroke

The stability of cortical function depends critically on proper regulation. Under conditions of migraine and stroke a breakdown of transmembrane chemical gradients can spread through cortical tissue. A concomitant component of this emergent spatio-temporal pattern is a depolarization of cells detected as slow voltage variations. The propagation velocity of approximately 3 mm/min indicates a contribution of diffusion. We propose a mechanism for spreading depolarizations (SD) that rests upon a nonlocal or noninstantaneous feedback in a reaction-diffusion system. Depending upon the characteristic space and time scales of the feedback, the propagation of cortical SD can be suppressed by shifting the bifurcation line, which separates the parameter regime of pulse propagation from the regime where a local disturbance dies out. The optimization of this feedback is elaborated for different control schemes and ranges of control parameters.

Control of unstable steady states by extended time-delayed feedback

Time-delayed feedback methods can be used to control unstable periodic orbits as well as unstable steady states. We present an application of extended time delay autosynchronization introduced by Socolar [Phys. Rev. E 50, 3245 (1994)] to an unstable focus. This system represents a generic model of an unstable steady state which can be found, for instance, in Hopf bifurcation. In addition to the original controller design, we investigate effects of control loop latency and a bandpass filter on the domain of control. Furthermore, we consider coupling of the control force to the system via a rotational coupling matrix parametrized by a variable phase. We present an analysis of the domain of control and support our results by numerical calculations.

Beyond the odd number limitation: a bifurcation analysis of time-delayed feedback control

We investigate the normal form of a subcritical Hopf bifurcation subjected to time-delayed feedback control. Bifurcation diagrams which cover time-dependent states as well are obtained by analytical means. The computations show that unstable limit cycles with an odd number of positive Floquet exponents can be stabilized by time-delayed feedback control, contrary to incorrect claims in the literature. The model system constitutes one of the few examples where a nonlinear time delay system can be treated entirely by analytical means.

Stabilization of unstable rigid rotation of spiral waves in excitable media

Depending on the parameters of two-dimensional excitable or oscillatory media rigidly rotating or meandering spiral waves are observed. The transition from rigid rotation to meandering motion occurs via a supercritical Hopf bifurcation. To stabilize rigid rotation in a parameter range beyond the Hopf bifurcation, we propose and successfully apply a proportional control algorithm as well as time delay autosynchronization. Both control methods are noninvasive. This allows for determination of the parameters of unstable rigid rotation of spiral waves either for a model or an experimental system. Using the Oregonator model for the light-sensitive Belousov-Zhabotinsky reaction as a representative example we show that quite naturally some latency time appears in the control loop, and propose an efficient method to overcome its destabilizing influence.

Tracking unstable steady states and periodic orbits of oscillatory and chaotic electrochemical systems using delayed feedback control

Experimental results are presented on successful application of delayed-feedback control algorithms for tracking unstable steady states and periodic orbits of electrochemical dissolution systems. Time-delay autosynchronization and delay optimization with a descent gradient method were applied for stationary states and periodic orbits, respectively. These tracking algorithms are utilized in constructing experimental bifurcation diagrams of the studied electrochemical systems in which Hopf, saddle-node, saddle-loop, and period-doubling bifurcations take place.

Control of unstable steady states by time-delayed feedback methods

We show that time-delayed feedback methods, which have successfully been used to control unstable periodic orbits, provide a tool to stabilize unstable steady states. We present an analytical investigation of the feedback scheme using the Lambert function and discuss effects of both a low-pass filter included in the control loop and nonzero latency times associated with the generation and injection of the feedback signal.

Long-lived transient behavior in an n+-n-n+ semiconductor device with optical stochasticity

We numerically study the response time of an ultrafast microwave switch, i.e., a GaAs-based Gunn device, under stochastic stimuli. The conducting states in this semiconductor device can be controlled in the presence of laser illumination near the doping notch. With the consideration of the additional randomness in laser intensity, the switching time from the initial unstable state to the final stable state will increase. Therefore, the so-called noise delayed decay of unstable states in an n+-n-n+ semiconductor device is demonstrated.

Delayed feedback control of chaos: bifurcation analysis

We study the effect of time delayed feedback control in the form proposed by Pyragas on deterministic chaos in the Rössler system. We reveal the general bifurcation diagram in the parameter plane of time delay tau and feedback strength K which allows one to explain the phenomena that have been discovered in some previous works. We show that the bifurcation diagram has essentially a multileaf structure that constitutes multistability: the larger the tau, the larger the number of attractors that can coexist in the phase space. Feedback induces a large variety of regimes nonexistent in the original system, among them tori and chaotic attractors born from them. Finally, we estimate how the parameters of delayed feedback influence the periods of limit cycles in the system.