Stabilizing unstable periodic orbits in a fast diode resonator using continuous time-delay autosynchronization

Time-Delayed Quantum Feedback Control

A theory of time-delayed coherent quantum feedback is developed. More specifically, we consider a quantum system coupled to a bosonic reservoir creating a unidirectional feedback loop. It is shown that the dynamics can be mapped onto a fictitious series of cascaded quantum systems, where the system is driven by past versions of itself. The derivation of this model relies on a tensor network representation of the system-reservoir time propagator. For concreteness, this general theory is applied to a driven two-level atom scattering into a coherent feedback loop. We demonstrate how delay effects can qualitatively change the dynamics of the atom and how quantum control can be implemented in the presence of time delays.

Effect of delay mismatch in Pyragas feedback control

Pyragas time-delayed feedback is a control scheme designed to stabilize unstable periodic orbits, which occur naturally in many nonlinear dynamical systems. It has been successfully implemented in a number of applications, including lasers and chemical systems. The control scheme targets a specific unstable periodic orbit by adding a feedback term with a delay chosen as the period of the unstable periodic orbit. However, in an experimental or industrial environment, obtaining the exact period or setting the delay equal to the exact period of the target periodic orbit may be difficult. This could be due to a number of factors, such as incomplete information on the system or the delay being set by inaccurate equipment. In this paper, we evaluate the effect of Pyragas control on the prototypical generic subcritical Hopf normal form when the delay is close to but not equal to the period of the target periodic orbit. Specifically, we consider two cases: first, a constant, and second, a linear approximation of the period. We compare these two cases to the case where the delay is set exactly to the target period, which serves as the benchmark case. For this comparison, we construct bifurcation diagrams and determine any regions where a stable periodic orbit close to the target is stabilized by the control scheme. In this way, we find that at least a linear approximation of the period is required for successful stabilization by Pyragas control.

Delayed feedback control of unstable steady states with high-frequency modulation of the delay

We analyze the stabilization of unstable steady states by delayed feedback control with a periodic time-varying delay in the regime of a high-frequency modulation of the delay. The average effect of the delayed feedback term in the control force is equivalent to a distributed delay in the interval of the modulation, and the obtained distribution depends on the type of the modulation. In our analysis we use a simple generic normal form of an unstable focus, and investigate the effects of phase-dependent coupling and the influence of the control loop latency on the controllability. In addition, we have explored the influence of the modulation of the delays in multiple delay feedback schemes consisting of two independent delay lines of Pyragas type. A main advantage of the variable delay is the considerably larger domain of stabilization in parameter space.

Time-delay autosynchronization control of defect turbulence in the cubic-quintic complex Ginzburg-Landau equation

We investigate the effectiveness of a Global time-delay autosynchronization control scheme aimed at stabilizing traveling wave solutions of the cubic-quintic Ginzburg-Landau equation in the Benjamin-Feir-Newell unstable regime. Numerical simulations show that a global control can be efficient and also can create other patterns such as spatiotemporal intermittency regimes, standing waves, or uniform oscillations.

Extending anticipation horizon of chaos synchronization schemes with time-delay coupling

We analyse anticipating synchronization in chaotic systems with time-delay coupling. Two algorithms for extending the prediction horizon are considered. One of them is based on the design of a suitable coupling matrix compensating the phase lag in the time-delay feedback term of the slave system. The second algorithm extends the first by incorporating, in the coupling law, information from many previous states of the master and slave systems. We demonstrate the efficiency of both algorithms with the simple dynamical model of coupled unstable spirals, as well as with the coupled Rössler systems. The maximum prediction time attained for the Rössler system is equal to the characteristic period of chaotic oscillations.

Experimental continuation of periodic orbits through a fold

We present a continuation method that enables one to track or continue branches of periodic orbits directly in an experiment when a parameter is changed. A control-based setup in combination with Newton iterations ensures that the periodic orbit can be continued even when it is unstable. This is demonstrated with the continuation of initially stable rotations of a vertically forced pendulum experiment through a fold bifurcation to find the unstable part of the branch.

Stabilizing unstable periodic orbits in the Lorenz equations using time-delayed feedback control

For many years it was believed that an unstable periodic orbit with an odd number of real Floquet multipliers greater than unity cannot be stabilized by the time-delayed feedback control mechanism of Pyragas. A recent paper by Fiedler et al. Phys. Rev. Lett. 98, 114101 (2007) uses the normal form of a subcritical Hopf bifurcation to give a counterexample to this theorem. Using the Lorenz equations as an example, we demonstrate that the stabilization mechanism identified by Fiedler et al. for the Hopf normal form can also apply to unstable periodic orbits created by subcritical Hopf bifurcations in higher-dimensional dynamical systems. Our analysis focuses on a particular codimension-two bifurcation that captures the stabilization mechanism in the Hopf normal form example, and we show that the same codimension-two bifurcation is present in the Lorenz equations with appropriately chosen Pyragas-type time-delayed feedback. This example suggests a possible strategy for choosing the feedback gain matrix in Pyragas control of unstable periodic orbits that arise from a subcritical Hopf bifurcation of a stable equilibrium. In particular, our choice of feedback gain matrix is informed by the Fiedler et al. example, and it works over a broad range of parameters, despite the fact that a center-manifold reduction of the higher-dimensional problem does not lead to their model problem.

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.

Delayed feedback control of chaos

Time-delayed feedback control is well known as a practical method for stabilizing unstable periodic orbits embedded in chaotic attractors. The method is based on applying feedback perturbation proportional to the deviation of the current state of the system from its state one period in the past, so that the control signal vanishes when the stabilization of the target orbit is attained. A brief review on experimental implementations, applications for theoretical models and most important modifications of the method is presented. Recent advancements in the theory, as well as an idea of using an unstable degree of freedom in a feedback loop to avoid a well-known topological limitation of the method, are described in detail.

Delayed feedback control of the Lorenz system: an analytical treatment at a subcritical Hopf bifurcation

We develop an analytical approach for the delayed feedback control of the Lorenz system close to a subcritical Hopf bifurcation. The periodic orbits arising at this bifurcation have no torsion and cannot be stabilized by a conventional delayed feedback control technique. We utilize a modification based on an unstable delayed feedback controller. The analytical approach employs the center manifold theory and the near identity transformation. We derive the characteristic equation for the Floquet exponents of the controlled orbit in an analytical form and obtain simple expressions for the threshold of stability as well as for an optimal value of the control gain. The analytical results are supported by numerical analysis of the original system of nonlinear differential-difference equations.

Controlling chaos in spatially extended beam-plasma system by the continuous delayed feedback

In this paper we discuss the control of complex spatio-temporal dynamics in a spatially extended nonlinear system (fluid model of Pierce diode) based on the concepts of controlling chaos in the systems with few degrees of freedom. A presented method is connected with stabilization of unstable homogeneous equilibrium state and the unstable spatio-temporal periodical states analogous to unstable periodic orbits of chaotic dynamics of the systems with few degrees of freedom. We show that this method is effective and allows to achieve desired regular dynamics chosen from a number of possible in the considered system.

Delayed feedback control of forced self-sustained oscillations

We consider a weakly nonlinear van der Pol oscillator subjected to a periodic force and delayed feedback control. Without control, the oscillator can be synchronized by the periodic force only in a certain domain of parameters. However, outside of this domain the system possesses unstable periodic orbits that can be stabilized by delayed feedback perturbation. The feedback perturbation vanishes if the stabilization is successful and thus the domain of synchronization can be extended with only small control force. We take advantage of the fact that the system is close to a Hopf bifurcation and derive a simplified averaged equation which we are able to treat analytically even in the presence of the delayed feedback. As a result we obtain simple analytical expressions defining the domain of synchronization of the controlled system as well as an optimal value of the control gain. The analytical theory is supported by numerical simulations of the original delay-differential equations.

Delayed feedback control of dynamical systems at a subcritical Hopf bifurcation

We consider the delayed feedback control of a torsion-free unstable periodic orbit originated in a dynamical system at a subcritical Hopf bifurcation. Close to the bifurcation point the problem is treated analytically using the method of averaging. We discuss the necessity of employing an unstable degree of freedom in the feedback loop as well as a nonlinear coupling between the controlled system and controller. To demonstrate our analytical approach the specific example of a nonlinear electronic circuit is taken as a model of a subcritical Hopf bifurcation.

Experimental control of coherence of a chaotic oscillator

We give experimental evidence that a delayed feedback control strategy is able to efficiently enhance the coherence of an experimental self-sustained chaotic oscillator obtained from a CO2 laser with electro-optical feedback. We demonstrate that coherence control is achieved for various choices of the delay time in the feedback control, including values that would lead to the stabilization of an unstable periodic orbit embedded within the chaotic attractor. The relationship between the two processes is discussed.

Controlling fast chaos in delay dynamical systems

We introduce a novel approach for controlling fast chaos in time-delay dynamical systems and use it to control a chaotic photonic device with a characteristic time scale of approximately 12 ns. Our approach is a prescription for how to implement existing chaos-control algorithms in a way that exploits the system's inherent time delay and allows control even in the presence of substantial control-loop latency (the finite time it takes signals to propagate through the components in the controller). This research paves the way for applications exploiting fast control of chaos, such as chaos-based communication schemes and stabilizing the behavior of ultrafast lasers.

Design and robustness of delayed feedback controllers for discrete systems

We study a matrix form of time-delay feedback control in the context of discrete time maps of high dimension. In almost all cases where standard proportional feedback control methods can achieve control, time-delay feedback controllers containing only static elements can be designed to achieve identical linear stability properties. Analysis of an example involving a ring of coupled maps that can be controlled at only two sites demonstrates that the time-delay controller equivalent to a standard optimal controller can be equally robust in the presence of noise, except at special points in parameter space where the uncontrolled system has a mode with Floquet multiplier exactly equal to 1. Numerical simulations confirm the results of the analysis.

Controlling turbulence in a surface chemical reaction by time-delay autosynchronization

A global time-delay feedback scheme is implemented experimentally to control chemical turbulence in the catalytic CO oxidation on a Pt(110) single crystal surface. The reaction is investigated under ultrahigh vacuum conditions by means of photoemission electron microscopy. We present results showing that turbulence can be efficiently suppressed by applying time-delay autosynchronization. Hysteresis effects are found in the transition regime from turbulence to homogeneous oscillations. At optimal delay time, we find a discontinuity in the oscillation period that can be understood in terms of an analytical investigation of a phase equation with time-delay autosynchronization. The experimental results are reproduced in numerical simulations of a realistic reaction model.

Analytical properties and optimization of time-delayed feedback control

Time-delayed feedback control is an efficient method for stabilizing unstable periodic orbits of chaotic systems. If the equations governing the system dynamics are known, the success of the method can be predicted by a linear stability analysis of the desired orbit. Unfortunately, the usual procedures for evaluating the Floquet exponents of such systems are rather intricate. We show that the main stability properties of the system controlled by time-delayed feedback can be simply derived from a leading Floquet exponent defining the system behavior under proportional feedback control. Optimal parameters of the delayed feedback controller can be evaluated without an explicit integration of delay-differential equations. The method is valid for low-dimensional systems whose unstable periodic orbits are originated from a period doubling bifurcation and is demonstrated for the Rössler system and the Duffing oscillator.

Bridges of periodic solutions and tori in semiconductor lasers subject to delay

For semiconductor lasers subject to a delayed optical feedback, branches of steady states sequentially appear as the feedback rate is increased. But branches of time-periodic solutions are connecting pairs of steady states and provide bridges between stable and unstable modes. All bridges experience a change of stability through a torus bifurcation point. Close to the bifurcation point, the torus remains localized near a specific fixed point in phase space. As the feedback rate increases, the torus envelope suddenly unfolds and its trajectory visits two or more unstable fixed points, anticipating the rich dynamics observed at larger feedback rates.

Nonlinear-dynamical arrhythmia control in humans

Nonlinear-dynamical control techniques, also known as chaos control, have been used with great success to control a wide range of physical systems. Such techniques have been used to control the behavior of in vitro excitable biological tissue, suggesting their potential for clinical utility. However, the feasibility of using such techniques to control physiological processes has not been demonstrated in humans. Here we show that nonlinear-dynamical control can modulate human cardiac electrophysiological dynamics by rapidly stabilizing an unstable target rhythm. Specifically, in 52/54 control attempts in five patients, we successfully terminated pacing-induced period-2 atrioventricular-nodal conduction alternans by stabilizing the underlying unstable steady-state conduction. This proof-of-concept demonstration shows that nonlinear-dynamical control techniques are clinically feasible and provides a foundation for developing such techniques for more complex forms of clinical arrhythmia.

Restricted feedback control of one-dimensional maps

Dynamical control of biological systems is often restricted by the practical constraint of unidirectional parameter perturbations. We show that such a restriction introduces surprising complexity to the stability of one-dimensional map systems and can actually improve controllability. We present experimental cardiac control results that support these analyses. Finally, we develop new control algorithms that exploit the structure of the restricted-control stability zones to automatically adapt the control feedback parameter and thereby achieve improved robustness to noise and drifting system parameters.

Control of chaos via an unstable delayed feedback controller

Delayed feedback control of chaos is well known as an effective method for stabilizing unstable periodic orbits embedded in chaotic attractors. However, it had been shown that the method works only for a certain class of periodic orbits characterized by a finite torsion. Modification based on an unstable delayed feedback controller is proposed in order to overcome this topological limitation. An efficiency of the modified scheme is demonstrated for an unstable fixed point of a simple dynamic model as well as for an unstable periodic orbit of the Lorenz system.

Analysis of the bifurcation diagram of a hybrid bistable system with feedback controls of chaos

Based on the dynamic equation of a hybrid bistable system with a delayed feedback, we have studied changes of the bifurcation diagram of its output oscillation under chaos suppression and delayed feedback control of chaos, respectively, and the physical origin of these changes. The result clearly shows that, in this case, the input intensity of the system is replaced by a smaller effective input intensity. So the bifurcation diagram is shifted to its right side, and a certain part of the chaotic oscillation becomes periodic oscillation.

Controlling chaos in a fast diode resonator using extended time-delay autosynchronization: Experimental observations and theoretical analysis

We stabilize unstable periodic orbits of a fast diode resonator driven at 10.1 MHz (corresponding to a drive period under 100 ns) using extended time-delay autosynchronization. Stabilization is achieved by feedback of an error signal that is proportional to the difference between the value of a state variable and an infinite series of values of the state variable delayed in time by integral multiples of the period of the orbit. The technique is easy to implement electronically and it has an all-optical counterpart that may be useful for stabilizing the dynamics of fast chaotic lasers. We show that increasing the weights given to temporally distant states enlarges the domain of control and reduces the sensitivity of the domain of control on the propagation delays in the feedback loop. We determine the average time to obtain control as a function of the feedback gain and identify the mechanisms that destabilize the system at the boundaries of the domain of control. A theoretical stability analysis of a model of the diode resonator in the presence of time-delay feedback is in good agreement with the experimental results for the size and shape of the domain of control. (c) 1997 American Institute of Physics.

Tracking controlled chaos: Theoretical foundations and applications

Tracking controlled states over a large range of accessible parameters is a process which allows for the experimental continuation of unstable states in both chaotic and non-chaotic parameter regions of interest. In algorithmic form, tracking allows experimentalists to examine many of the unstable states responsible for much of the observed nonlinear dynamic phenomena. Here we present a theoretical foundation for tracking controlled states from both dynamical systems as well as control theoretic viewpoints. The theory is constructive and shows explicitly how to track a curve of unstable states as a parameter is changed. Applications of the theory to various forms of control currently used in dynamical system experiments are discussed. Examples from both numerical and physical experiments are given to illustrate the wide range of tracking applications. (c) 1997 American Institute of Physics.