Stabilization of unstable steady states and periodic orbits in an electrochemical system using delayed-feedback control

Mitigation of limit cycle oscillations in a turbulent thermoacoustic system via delayed acoustic self-feedback

We report the occurrence of amplitude death (AD) of limit cycle oscillations in a bluff body stabilized turbulent combustor through delayed acoustic self-feedback. Such feedback control is achieved by coupling the acoustic field of the combustor to itself through a single coupling tube attached near the anti-node position of the acoustic standing wave. We observe that the amplitude and dominant frequency of the limit cycle oscillations gradually decrease as the length of the coupling tube is increased. Complete suppression (AD) of these oscillations is observed when the length of the coupling tube is nearly 3 / 8 times the wavelength of the fundamental acoustic mode of the combustor. Meanwhile, as we approach this state of amplitude death, the dynamical behavior of acoustic pressure changes from the state of limit cycle oscillations to low-amplitude chaotic oscillations via intermittency. We also study the change in the nature of the coupling between the unsteady flame dynamics and the acoustic field as the length of the coupling tube is increased. We find that the temporal synchrony between these oscillations changes from the state of synchronized periodicity to desynchronized aperiodicity through intermittent synchronization. Furthermore, we reveal that the application of delayed acoustic self-feedback with optimum feedback parameters completely disrupts the positive feedback loop between hydrodynamic, acoustic, and heat release rate fluctuations present in the combustor during thermoacoustic instability, thus mitigating instability. We anticipate this method to be a viable and cost-effective option to mitigate thermoacoustic oscillations in turbulent combustion systems used in practical propulsion and power systems.

Delayed feedback induced multirhythmicity in the oscillatory electrodissolution of copper

Occurrence of bi- and trirhythmicities (coexistence of two or three stable limit cycles, respectively, with distinctly different periods) has been studied experimentally by applying delayed feedback control to the copper-phosphoric acid electrochemical system oscillating close to a Hopf bifurcation point under potentiostatic condition. The oscillating electrode potential is delayed by τ and the difference between the present and delayed values is fed back to the circuit potential with a feedback gain K. The experiments were performed by determining the period of current oscillations T as a function of (both increasing and decreasing) τ at several fixed values of K. With small delay times, the period exhibits a sinusoidal type dependence on τ. However, with relatively large delays (typically τ ≫ T) for each feedback gain K, there exists a critical delay τcrit above which birhythmicity emerges. The experiments show that for weak feedback, Kτcrit is approximately constant. At very large delays, the dynamics becomes even more complex, and trirhythmicity could be observed. Results of numerical simulations based on a general kinetic model for metal electrodissolution were consistent with the experimental observations. The experimental and numerical results are also interpreted by using a phase model; the model parameters can be obtained from experimental data measured at small delay times. Analytical solutions to the phase model quantitatively predict the parameter regions for the appearance of birhythmicity in the experiments, and explain the almost constant value of Kτcrit for weak feedback.

Adaptive time-delayed stabilization of steady states and periodic orbits

We derive adaptive time-delayed feedback controllers that stabilize fixed points and periodic orbits. First, we develop an adaptive controller for stabilization of a steady state by applying the speed-gradient method to an appropriate goal function and prove global asymptotic stability of the resulting system. For an example we show that the advantage of the adaptive controller over the nonadaptive one is in a smaller controller gain. Second, we propose adaptive time-delayed algorithms for stabilization of periodic orbits. Their efficiency is confirmed by local stability analysis. Numerical examples demonstrate the applicability of the proposed controllers.

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.

Time-delayed feedback control design beyond the odd-number limitation

We present an algorithm for a time-delayed feedback control design to stabilize periodic orbits with an odd number of positive Floquet exponents in autonomous systems. Due to the so-called odd-number theorem such orbits have been considered as uncontrollable by time-delayed feedback methods. However, this theorem has been refuted by a counterexample and recently a corrected version of the theorem has been proved. In our algorithm, the control matrix is designed using a relationship between Floquet multipliers of the systems controlled by time-delayed and proportional feedback. The efficacy of the algorithm is demonstrated with the Lorenz and Chua systems.

Structures of chaos in open reaction systems

By numerically simulating the Bray-Liebhafsky (BL) reaction (the hydrogen peroxide decomposition in the presence of hydrogen and iodate ions) in a continuously fed well stirred tank reactor (CSTR), we find "structured" types of chaos emerging in regular order with respect to flow rate as the control parameter. These chaotic "structures" appear between each two successive periodic states, and have forms and evolution resembling to the neighboring periodic dynamics. More precisely, in the transition from period-doubling route to chaos to the arising periodic mixture of different mixed-mode oscillations, we are able to recognize and qualitatively and quantitatively distinguish the sequence of "period-doubling" chaos and chaos consisted of mixed-mode oscillations (the "mixed-mode structured" chaos), both appearing in regular order between succeeding periodic states. Additionally, between these types of chaos, the chaos without such recognizable "structures" ("unstructured" chaos) is also distinguished. Furthermore, all transitions between two successive periodic states are realized through bifurcation of chaotic states. This scenario is a universal feature throughout the whole mixed-mode region, as well as throughout other mixed-mode regions obtained under different initial conditions.

Complete chaotic synchronization and exclusion of mutual Pyragas control in two delay-coupled Rössler-type oscillators

Two identical chaotic oscillators that are mutually coupled via time delayed signals show very complex patterns of completely synchronized dynamics including stationary states and periodic as well as chaotic oscillations. We have experimentally observed these synchronized states in delay-coupled electronic circuits and have analyzed their stability by numerical simulations and analytical calculations. We found that the conditions for longitudinal and transversal stability largely exclude each other and prevent, e.g., the synchronization of Pyragas-controlled orbits. Most striking is the observation of complete chaotic synchronization for large delay times, which should not be allowed in the given coupling scheme on the background of the actual paradigm.

Interaction of noise with excitable dynamics

In this paper, the interaction of noise with excitable dynamics of a three-electrode electrochemical cell is examined. Different scenarios involving both external and internal noise sources are considered. In the case of external noise, aperiodic stochastic resonance and regulation of the noise-induced spiking behaviour are investigated. In the case of internal noise, the interaction of intrinsic electrochemical noise with autonomous nonlinear dynamics is studied. The amplitude of this internal noise, determined by the concentration of chloride ions, is monotonically increased and the provoked dynamics are analysed. Our results indicate that internal noise, similar to its external counterpart, is able to induce regularity in the system response.

Control of turbulence in oscillatory reaction-diffusion systems through a combination of global and local feedback

Global time-delay autosynchronization is known to control spatiotemporal turbulence in oscillatory reaction-diffusion systems. Here, we investigate the complex Ginzburg-Landau equation in the regime of spatiotemporal turbulence and study numerically how local or a combination of global and local time-delay autosynchronization can be used to suppress turbulence by inducing uniform oscillations. Numerical simulations show that while a purely local control is unsuitable to produce uniform oscillations, a mixed local and global control can be efficient and also able to create other patterns such as standing waves, amplitude death, or traveling waves.

Refuting the odd-number limitation of time-delayed feedback control

We refute an often invoked theorem which claims that a periodic orbit with an odd number of real Floquet multipliers greater than unity can never be stabilized by time-delayed feedback control in the form proposed by Pyragas. Using a generic normal form, we demonstrate that the unstable periodic orbit generated by a subcritical Hopf bifurcation, which has a single real unstable Floquet multiplier, can in fact be stabilized. We derive explicit analytical conditions for the control matrix in terms of the amplitude and the phase of the feedback control gain, and present a numerical example. Our results are of relevance for a wide range of systems in physics, chemistry, technology, and life sciences, where subcritical Hopf bifurcations occur.

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.

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.

Variational calculation of the limit cycle and its frequency in a two-neuron model with delay

We consider a model system of two coupled Hopfield neurons, which is described by delay differential equations taking into account the finite signal propagation and processing times. When the delay exceeds a critical value, a limit cycle emerges via a supercritical Hopf bifurcation. First, we calculate its frequency and trajectory perturbatively by applying the Poincaré-Lindstedt method. Then, the perturbation series are resummed by means of the Shohat expansion in good agreement with numerical values. However, with increasing delay, the accuracy of the results from the Shohat expansion worsens. We thus apply variational perturbation theory (VPT) to the perturbation expansions to obtain more accurate results, which moreover hold even in the limit of large delays.

Dynamical behavior of the motions associated with the nonlinear periodic regime in a laboratory plasma subject to delayed feedback

Time-delayed feedback is applied to the motions associated with the nonlinear periodic regime generated due to current-driven ion acoustic instability; this is a typical instability in a laboratory plasma, and the dynamical behavior is experimentally investigated using delayed feedback. A time-delayed autosynchronization method is applied. When delayed feedback is applied to the nonlinear periodic orbit, the periodic state changes to various motions depending on the control parameters, namely, the arbitrary time delay and the proportionality constant. Lyapunov exponents are calculated in order to examine the dynamical behavior.

Controlling drift-wave turbulence using time-delay and space-shift autosynchronization feedback

Drift-wave turbulence control in a one-dimensional nonlinear drift-wave equation driven by a sinusoidal wave is considered. We apply time-delay and space-shift feedback signals, to suppress turbulence. By using global and local pinning strategies, we show numerically that the turbulent state can be controlled to periodic states effectively if appropriate time-delay length and space-shift distance are chosen. The physical mechanism of the control scheme is understood based on the energy-minimum principle.

Regulating noise-induced spiking using feedback

We report successful manipulation of the noise provoked spiking behavior using delayed feedback control. Experiments were performed in a three electrode electrochemical cell under potentiostatic conditions. The uncontrolled system exhibited noise invoked oscillations whose regularity was quantified using normalized variance (NV). Superimposing delayed feedback, for appropriate values of delay (t), an enhancement in the regularity of the spike sequence was attained. Numerical simulations corroborated experimental observations.

Persistence of chaos in a time-delayed-feedback controlled Duffing system

This paper concerns global phase structures of a time-delayed-feedback controlled two-well Duffing system. The remains of a global stretch and fold structure along an unstable manifold, which develops from an unstable fixed point in function space, reveals that the global chaotic dynamics is inherited from the original system by the controlled system. The remains of the original chaotic dynamics causes a highly complicated domain of attraction for target orbits and a long chaotic transient before convergence.

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.

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 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.

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.

Effects of time-delayed feedback on chaotic oscillators

We study the effects of time-delayed feedback on chaotic systems where the delay time is both fixed (static case) and varying (dynamic case) in time. For the static case, typical phase coherent and incoherent chaotic oscillators are investigated. Detailed phase diagrams are investigated in the parameter space of feedback gain ( K ) and delay time ( tau ). Linear stability analysis, by assuming the time-delayed perturbation, varies as e(lambdat) where lambda is the eigenvalue, gives the boundaries of the stability islands and critical feedback gains ( K(c) ) for both Rössler oscillators and Lorenz oscillators. We also found that the stability island are found when the delay time is about tau= (n+ 1 / 2 ) T , where n is an integer and T is the average period of the chaotic oscillator. It is shown that these analytical predictions agree well with the numerical results. For the dynamic case, we investigate Rössler oscillator with periodically modulated delay time. Stability regimes are found for parameter space of feedback gain and modulation frequency in which it was impossible to be stabilized for a fixed delay time. We also trace the detailed routes to the stability near the island boundaries for both cases by investigating bifurcation diagrams.

Domain of attraction for stabilized orbits in time delayed feedback controlled Duffing systems

Time delayed feedback control is well known as a practical method for stabilizing unstable periodic orbits embedded in chaotic attractors. However, this control method still has an open problem of estimating domain of attraction for target unstable periodic orbits. In this paper, we numerically discuss the domain of attraction in Duffing systems under the control method. The disturbance to initial conditions reveals that the domain of attraction possibly exhibits self-similar structures in its boundaries.

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.

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

We present a scheme to stabilize high-frequency domain oscillations in semiconductor superlattices by a time-delayed feedback loop. Applying concepts from chaos control theory we propose to control the spatiotemporal dynamics of fronts of accumulation and depletion layers which are generated at the emitter and may collide and annihilate during their transit, and thereby suppress chaos. The proposed method only requires the feedback of internal global electrical variables, viz., current and voltage, which makes the practical implementation very easy.

Time-delay autosynchronization of the spatiotemporal dynamics in resonant tunneling diodes

The double barrier resonant tunneling diode exhibits complex spatiotemporal patterns including low-dimensional chaos when operated in an active external circuit. We demonstrate how autosynchronization by time-delayed feedback control can be used to select and stabilize specific current density patterns in a noninvasive way. We compare the efficiency of different control schemes involving feedback in either local spatial or global degrees of freedom. The numerically obtained Floquet exponents are explained by analytical results from linear stability 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.

Stabilizing coupled map lattice systems with adaptive adjustment

The adaptive adjustment mechanism is applied to stabilization of a general coupled-map lattice system defined by x(i,t+1)=f(x(i,t))+C(i)(x(i,t),x(i-1,t))+D(x(i,t),x(i-1,t)), where f: R-->R is a nonlinear map, and C(i),D(i): R2-->R are coupling functions that satisfy C(i)(x,x)=0 and D(i)(x,x)=0, for all x in R, i=1,2, em leader,n. Sufficient conditions and ranges of adjustment parameters that guarantee the local stability of a synchronized fixed point are provided. Numerical simulations demonstrate the effectiveness and efficiency for this mechanism to stabilize the system to an originally unstable synchronized fixed point or a periodic orbit.

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.

Populations of coupled electrochemical oscillators

Experiments were carried out on arrays of chaotic electrochemical oscillators to which global coupling, periodic forcing, and feedback were applied. The global coupling converts a very weakly coupled set of chaotic oscillators to a synchronized state with sufficiently large values of coupling strength; at intermediate values both intermittent and stable chaotic cluster states occur. Cluster formation and synchronization were also obtained by applying feedback and forcing to a moderately coupled base state. The three cases differ, however, in other details. The feedback and forcing also produce periodic cluster states and more than two clusters. Configurations of two (chaotic) clusters and two, three, or four (periodic) clusters were observed. (c) 2002 American Institute of Physics.

Controlling high-order chaotic discrete systems by lagged adaptive adjustment

A lagged adaptive adjustment mechanism has been developed to stabilize a high-order discrete system. Theoretical analysis and computer simulations have been provided to show the effectiveness and efficiency of this mechanism in practice.

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.