The Pyragas method of feedback control has attracted much interest as a method of stabilising unstable periodic orbits in a number of situations. We show that a time-delayed feedback control similar to the Pyragas method can be used to stabilise periodic orbits with arbitrarily large period, specifically those resulting from a resonant bifurcation of a heteroclinic cycle. Our analysis reduces the infinite-dimensional delay-equation governing the system with feedback to a three-dimensional map, by making certain assumptions about the form of the solutions. The stability of a fixed point in this map corresponds to the stability of the periodic orbit in the flow, and can be computed analytically. We compare the analytic results to a numerical example and find very good agreement.
We propose a necessary condition for the successful stabilisation of a periodic orbit using the extended version of time-delayed feedback control. This condition depends on the number of real Floquet multipliers larger than unity and is therefore related to the well-known odd-number limitation in non-autonomous systems. We show that the period of the orbit which is induced by mismatching the delay-time of the control scheme and the period of the uncontrolled orbit plays an important role in the formulation of the odd-number limitation in the autonomous case.
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 …
We discuss applications of time-delayed feedback control to delay-coupled neural systems and lasers, in the framework of the FitzHugh-Nagumo neuron model and the Lang-Kobayashi laser model, respectively. In the context of neural systems, we will point out some complex scenarios of synchronized in-phase or antiphase oscillations, bursting patterns, or amplitude death, induced by delayed coupling in combination with delayed self-feedback in simple network motifs. For optical systems, we will show that multiple time-delayed feedback, realized by a Fabry-Perot resonator coupled to the laser, provi…
We prove an analytical limitation on the use of time-delayed feedback control for the stabilization of periodic orbits in autonomous systems. This limitation depends on the number of real Floquet multipliers larger than unity, and is therefore similar to the well-known odd number limitation of time-delayed feedback control. Recently, a two-dimensional example has been found, which explicitly demonstrates that the unmodified odd number limitation does not apply in the case of autonomous systems. We show that our limitation correctly predicts the stability boundaries in this case.
We analyze the vibrational resonance in the Duffing oscillator system in the presence of (i) a gamma distributed time-delayed feedback and (ii) integrative time-delayed (uniformly distributed time delays over a finite interval) feedback. Particularly, applying a theoretical procedure we obtain an expression for the response amplitude $Q$ at the low-frequency of the driving biharmonic force. For both double-well potential and single-well potential cases we are able to identify the regions in parameter space where either (i) two resonances, (ii) a single resonance or (iii) no resonance occur. Th…
We investigate the possibility to suppress noise-induced intensity pulsations (relaxation oscillations) in semiconductor lasers by means of a time-delayed feedback control scheme. This idea is first studied in a generic normal form model, where we derive an analytic expression for the mean amplitude of the oscillations and demonstrate that it can be strongly modulated by varying the delay time. We then investigate the control scheme analytically and numerically in a laser model of Lang-Kobayashi type and show that relaxation oscillations excited by noise can be very efficiently suppressed via …
The stabilization of steady states is studied in a modified Lang-Kobayashi model of a semiconductor laser. We show that multiple time-delayed feedback, realized by a Fabry-Perot resonator coupled to the laser, provides a valuable tool for the suppression of unwanted intensity pulsations, and leads to stable continuous-wave operation. The domains of control are calulated in dependence on the feedback strength, delay time (cavity round trip time), memory parameter (mirror reflectivity), latency time, feedback phase, and bandpass filtering, Due to the optical feedback, multistable behavior can al…
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 report on an ultra-high-frequency (> 1 GHz), piecewise-linear chaotic system designed from low-cost, commercially available electronic components. The system is composed of two electronic time-delayed feedback loops: A primary analog loop with a variable gain that produces multi-mode oscillations centered around 2 GHz and a secondary loop that switches the variable gain between two different values by means of a digital-like signal. We demonstrate experimentally and numerically that such an approach allows for the simultaneous generation of analog and digital chaos, where the digital chaos …
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 et al. to an unstable focus. This system represents a generic model of an unstable steady state which can be found for instance in a 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 p…
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 r…
Lyapunov exponents are a widely used tool for studying dynamical systems. When calculating Lyapunov exponents for piecewise-smooth systems with time-delayed arguments one faces a lack of continuity in the variational problem. This paper studies how to build a variational equation for the efficient construction of Jacobians along trajectories of the delayed nonsmooth system. Trajectories of the piecewise-smooth system may encounter a so-called grazing event where the trajectory approaches a discontinuity surface in the state space in a non-transversal manner. For this event we develop a grazing…
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 Pyragus. A recent paper by Fiedler et al 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 sys…
The chaotic dynamics of directly modulated semiconductor lasers with delayed optoelectronic feedback is studied numerically. The effects of positive and negative delayed optoelectronic feedback in producing chaotic outputs from such lasers with nonlinear gain reduction in its optimum value range is investigated using bifurcation diagrams. The results are confirmed by calculating the Lyapunov exponents. A negative delayed optoelectronic feedback configuration is found to be more effective in inducing chaotic dynamics to such systems with nonlinear gain reduction factor in the practical value ra…
Delayed feedback laser dynamics is described by means of Lang-Kobayashi equation model. Since a lot of initial states asymptotically approach to periodic attractor in the phase space, only periodic steady-state regimes have been studied here. Lyapunov transformation allows us to reduce problem to the differential equation of the first order whereas the spectrum of laser oscillation is governed by the appropriate eigen value problem. Using the symmetry proves that there is the dual laser system with anticipated feedback which has the same dynamic characteristics as laser with delayed feedback.
We demonstrate a way to generate a two-dimensional rogue waves in two types of broad area nonlinear optical systems subject to time-delayed feedback: in the generic Lugiato-Lefever model and in model of a broad-area surface-emitting laser with saturable absorber. The delayed feedback is found to induce a spontaneous formation of rogue waves. In the absence of delayed feedback, spatial pulses are stationary. The rogue waves are exited and controlled by the delay feedback. We characterize their formation by computing the probability distribution of the pulse height. The long-tailed statistical c…
We demonstrate that time-delayed feedback control can be improved by adaptively tuning the feedback gain. This adaptive controller is applied to the stabilization of an unstable fixed point and an unstable periodic orbit embedded in a chaotic attractor. The adaptation algorithm is constructed using the speed-gradient method of control theory. Our computer simulations show that the adaptation algorithm can find an appropriate value of the feedback gain for single and multiple delays. Furthermore, we show that our method is robust to noise and different initial conditions.
A theoretical study of delayed feedback in spin-torque nano-oscillators is presented. A macrospin geometry is considered, where self-sustained oscillations are made possible by spin transfer torques associated with spin currents flowing perpendicular to the film plane. By tuning the delay and amplification of the self-injected signal, we identify dynamical regimes in this system such as chaos, switching between precession modes with complex transients, and oscillator death. Such delayed feedback schemes open up a new field of exploration for such oscillators, where the complex transient states…
Experiments with catalytic CO oxidation on Pt(110) show that chemical turbulence in this system can be suppressed by application of appropriate global delayed feedbacks. Different spatiotemporal patterns, seen near a transition from turbulence to uniform oscillations, are investigated. Using a method based on the Hilbert transform, spatial distributions of local phase and amplitude in such patterns are reconstructed from the experimental data. The observed phenomena are reproduced in simulations using a theoretical model of the reaction.
Effects of synchronization in a system of two coupled oscillators with time-delayed feedback are investigated. Phase space of a system with time delay is infinite-dimensional. Thus, the picture of synchronization in such systems acquires many new features not inherent to finite-dimensional ones. A picture of oscillation modes in cases of identical and non-identical coupled oscillators is studied in detail. Periodical structure of amplitude death and "broadband synchronization" zones is investigated. Such a behavior occurs due to the resonances between different modes of the infinite-dimensiona…
Using the model of a generalized Van der Pol oscillator in the regime of subcritical Hopf bifurcation we investigate the influence of time delay on noise-induced oscillations. It is shown that for appropriate choices of time delay either suppression or enhancement of coherence resonance can de achieved. Analytical calculations are combined with numerical simulations and experiments on an electronic circuit.
For controlling periodic orbits with delayed feedback methods the periodicity has to be known a priori. We propose a simple scheme, how to detect the period of orbits from properties of the control signal, at least if a periodic but nonvanishing signal is observed. We analytically derive a simple expression relating the delay, the control amplitude, and the unknown period. Thus, the latter can be computed from experimentally accessible quantities. Our findings are confirmed by numerical simulations and electronic circuit experiments
The foremost aim of this study is to investigate the influence of time-delayed feedback on extreme events in a non-polynomial system with velocity dependent potential. To begin, we investigate the effect of this feedback on extreme events for four different values of the external forcing parameter. Among these four values, in the absence of time-delayed feedback, for two values, the system does not exhibit extreme events and for the other two values, the system exhibits extreme events. On the introduction of time-delayed feedback and varying the feedback strength, we found that extreme events …
We study the stability of unstable steady states in scalar retarded time-delayed systems subjected to a variable-delay feedback control. The important aspect of such a control problem is that time-delayed systems are already infinite-dimensional before the delayed feedback control is turned on. When the frequency of the modulation is large compared to the system's dynamics, the analytic approach consists of relating the stability properties of the resulting variable-delay system with those of an analogous distributed delay system. Otherwise, the stability domains are obtained by a numerical in…
The Pyragas method for controlling chaos is investigated in detail from the experimental as well as theoretical point of view. We show by an analytical stability analysis that the revolution around an unstable periodic orbit governs the success of the control scheme. Our predictions concerning the transient behaviour of the control signal are confirmed by numerical simulations and an electronic circuit experiment.
Using the Poincaré section technique, we study in detail the dynamical behaviors of delay differential system and find a new type of solutions $S_i$ in short-time delay feedback. Our numerical results remind us to deny the opinion that there are no complex phenomena in short-time delay case. Many similarities between foundamental solution and the new type of solutions are found. We demonstrate that the scales of $S_i$ increase with exponential growth via $i$ in the direction of $μ$, while decrease with exponential decays in the direction of $x$ or delay time $t_R$.
We study the stability of the fixed-point solution of an array of mutually coupled logistic maps, focusing on the influence of the delay times, $τ_{ij}$, of the interaction between the $i$th and $j$th maps. Two of us recently reported [Phys. Rev. Lett. {\bf 94}, 134102 (2005)] that if $τ_{ij}$ are random enough the array synchronizes in a spatially homogeneous steady state. Here we study this behavior by comparing the dynamics of a map of an array of $N$ delayed-coupled maps with the dynamics of a map with $N$ self-feedback delayed loops. If $N$ is sufficiently large, the dynamics of a map of …
We study the synchronization and stability of power grids within the Kuramoto phase oscillator model with inertia with a bimodal frequency distribution representing the generators and the loads. We identify critical nodes through solitary frequency deviations and Lyapunov vectors corresponding to unstable Lyapunov exponents. To cure dangerous deviations from synchronization we propose time-delayed feedback control, which is an efficient control concept in nonlinear dynamic systems. Different control strategies are tested and compared with respect to the minimum number of controlled nodes requi…
Dynamical systems with complex delayed interactions arise commonly when propagation times are significant, yielding complicated oscillatory instabilities. In this Letter, we introduce a class of systems with multiple, hierarchically long time delays, and using a suitable space-time representation we uncover features otherwise hidden in their temporal dynamics. The behaviour in the case of two delays is shown to ''encode'' two-dimensional spiral defects and defects turbulence. A multiple scale analysis sets the equivalence to a complex Ginzburg-Landau equation, and a novel criterium for the att…
In this paper, we propose a dynamic delayed feedback control approach for desynchronization of chaotic-bursting synchronous activities in an ensemble of globally coupled neuronal oscillators. We demonstrate that the difference signal between an ensemble's mean field and its time delayed state, filtered and fed back to the ensemble, can suppress the self-synchronization in the ensemble. These individual units are decoupled and stabilized at the desired desynchronized states while the stimulation signal reduces to the noise level. The effectiveness of the method is illustrated by examples of two…
