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

Class-C semiconductor lasers with time-delayed optical feedback

We perform a linear stability analysis and numerical bifurcation diagrams of a class-C laser with time-delayed optical feedback. We employ a rate equation system based on the Maxwell-Bloch equations, and study the influence of the dephasing time on the laser dynamics. We find a stabilizing effect of an intermediate dephasing time, i.e. when moving from a class-B to a class-C laser. At long dephasing times, a destabilization of the laser solution occurs by a feedback-induced unlocking of Rabi oscillations at the second laser threshold. We predict an optimum resistance to time-delayed optical feedback for dephasing times close to the photon cavity lifetime. This article is part of the theme issue 'Nonlinear dynamics of delay systems'.

Amplitude death in a pair of one-dimensional complex Ginzburg-Landau systems coupled by diffusive connections

This paper shows that, in a pair of one-dimensional complex Ginzburg-Landau (CGL) systems, diffusive connections can induce amplitude death. Stability analysis of a spatially uniform steady state in coupled CGL systems reveals that amplitude death never occurs in a pair of identical CGL systems coupled by no-delay connection, but can occur in the case of delay connection. Moreover, amplitude death never occurs in coupled identical CGL systems with zero nominal frequency. Based on these analytical results, we propose a procedure for designing the connection delay time and the coupling strength to induce spatial-robust stabilization, that is, a stabilization of the steady state for any system size and any boundary condition. Numerical simulations are performed to confirm the analytical results.

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.

Effect of autaptic activity on the response of a Hodgkin-Huxley neuron

An autapse is a special synapse that connects a neuron to itself. In this study, we investigated the effect of an autapse on the responses of a Hodgkin-Huxley neuron to different forms of external stimuli. When the neuron was subjected to a DC stimulus, the firing frequencies and the interspike interval distributions of the output spike trains showed periodic behaviors as the autaptic delay time increased. When the input was a synaptic pulse-like train with random interspike intervals, we observed low-pass and band-pass filtering behaviors. Moreover, the region over which the output ISIs are distributed and the mean firing frequency display periodic behaviors with increasing autaptic delay time. When specific autaptic parameters were chosen, most of the input ISIs could be filtered, and the response spike trains were nearly regular, even with a highly random input. The background mechanism of these observed dynamics has been analyzed based on the phase response curve method. We also found that the information entropy of the output spike train could be modified by the autapse. These results also suggest that the autapse can serve as a regulator of information response in the nervous system.

Accurate control of hyperbolic trajectories in any dimension

The unsteady (nonautonomous) analog of a hyperbolic fixed point is a hyperbolic trajectory, whose importance is underscored by its attached stable and unstable manifolds, which have relevance in fluid flow barriers, chaotic basin boundaries, and the long-term behavior of the system. We develop a method for obtaining the unsteady control velocity which forces a hyperbolic trajectory to follow a user-prescribed variation with time. Our method is applicable in any dimension, and accuracy to any order is achievable. We demonstrate and validate our method by (1) controlling the fixed point at the origin of the Lorenz system, for example, obtaining a user-defined nonautonomous attractor, and (2) the saddle points in a droplet flow, using localized control which generates global transport.

Feedback control of unstable periodic orbits in equivariant Hopf bifurcation problems

Symmetry-breaking Hopf bifurcation problems arise naturally in studies of pattern formation. These equivariant Hopf bifurcations may generically result in multiple solution branches bifurcating simultaneously from a fully symmetric equilibrium state. The equivariant Hopf bifurcation theorem classifies these solution branches in terms of their symmetries, which may involve a combination of spatial transformations and temporal shifts. In this paper, we exploit these spatio-temporal symmetries to design non-invasive feedback controls to select and stabilize a targeted solution branch, in the event that it bifurcates unstably. The approach is an extension of the Pyragas delayed feedback method, as it was developed for the generic subcritical Hopf bifurcation problem. Restrictions on the types of groups where the proposed method works are given. After addition of the appropriately optimized feedback term, we are able to compute the stability of the targeted solution using standard bifurcation theory, and give an account of the parameter regimes in which stabilization is possible. We conclude by demonstrating our results with a numerical example involving symmetrically coupled identical nonlinear oscillators.

A dynamical systems approach to the control of chaotic dynamics in a spatiotemporal jet flow

We present a strategy for control of chaos in open flows and provide its experimental validation in the near field of a transitional jet flow system. The low-dimensional chaotic dynamics studied here results from vortex ring formation and their pairings over a spatially extended region of the flow that was excited by low level periodic forcing of the primary instability. The control method utilizes unstable periodic orbits (UPO) embedded within the chaotic attractor. Since hydrodynamic instabilities in the open flow system are convective, both monitoring and control can be implemented at a few locations, resulting in a simple and effective control algorithm. Experiments were performed in an incompressible, initially laminar, 4 cm diameter circular air jet, at a Reynolds number of 23,000, housed in a low-noise, large anechoic chamber. Distinct trajectory bundles surrounding the dominant UPOs were found from experimentally derived, time-delayed embedding of the chaotic attractor. Velocity traces from a pair of probes placed at the jet flow exit and farther downstream were used to empirically model the UPOs and compute control perturbations to be applied at the jet nozzle lip. Open loop control was used to sustain several nearly periodic states.

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.

FINITE DIMENSIONAL STRUCTURE OF THE GPI DISCHARGE IN PATIENTS WITH PARKINSON'S DISEASE

Stochastic systems are infinitely dimensional and deterministic systems are low dimensional, while real systems lie somewhere between these two limit cases. If the calculation of a low (finite) dimension is in fact possible, one could conclude that the system under study is not purely random. In the present work we calculate the maximal Lyapunov exponent from interspike intervals time series recorded from the internal segment of the Globus Pallidusfrom patients with Parkinson's disease. We show the convergence of the maximal Lyapunov exponent at a dimension equal to 7 or 8, which is therefore our estimation of the embedding dimension for the system. For dimensions below 7 the observed behavior is what would be expected from a stochastic system or a complex system projecting onto lower dimensional spaces. The maximal Lyapunov exponent did not show any differences between tremor and akineto-rigid forms of the disease. However, it did decay with the value of motor Unified Parkinson's Disease Rating Scale -OFF scores. Patients with a more severe disease (higher UPDRS-OFF score) showed a lower value of the maximal Lyapunov exponent. Taken together, both indexes (the maximal Lyapunov exponent and the embedding dimension) remark the importance of taking into consideration the system's non-linear properties for a better understanding of the information transmission in the basal ganglia.

Towards easier realization of time-delayed feedback control of odd-number orbits

We develop generalized time-delayed feedback schemes for the stabilization of periodic orbits with an odd number of positive Floquet exponents, which are particularly well suited for experimental realization. We construct the parameter regimes of successful control and validate these by numerical simulations and numerical continuation methods. In particular, it is shown how periodic orbits can be stabilized with symmetric feedback matrices by introducing an additional latency time in the control loop. Finally, we show using normal form analysis and numerical simulations how our results could be implemented in a laser setup using optoelectronic feedback.

Odd-number theorem: optical feedback control at a subcritical Hopf bifurcation in a semiconductor laser

A subcritical Hopf bifurcation is prepared in a multisection semiconductor laser. In the free-running state, hysteresis is absent due to noise-induced escape processes. The missing branches are recovered by stabilizing them against noise through application of phase-sensitive noninvasive delayed optical feedback control. The same type of control is successfully used to stabilize the unstable pulsations born in the Hopf bifurcation. This experimental finding represents an optical counterexample to the so-called odd-number limitation of delayed feedback control. However, as a leftover of the limitation, the domains of control are extremely small.

Stabilization of a steady state in network oscillators by using diffusive connections with two long time delays

The present study shows that diffusive connections with two long-time delays can induce the stabilization of a steady state in network oscillators. A linear stability analysis shows that, if the two delay times retain a proportional relation with a certain bias, the stabilization can be achieved independent of the delay times. Furthermore, a simple systematic procedure for designing the coupling strength and the delay times in the connections is proposed. The procedure has the following two advantages: one can employ time delays as long as one wants and the stabilization can be achieved independently of its network topology. Our analytical results are applied to the well-known double-scroll circuit model on a small-world network.

Variable-delay feedback control of unstable steady states in retarded time-delayed systems

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 integration of the linearized variable-delay system. The analysis shows that the control domains are significantly larger than those in the usual time-delayed feedback control, and that the complexity of the domain structure depends on the form and the frequency of the delay modulation.

Delay and periodicity

Systems with time delay play an important role in modeling of many physical and biological processes. In this paper we describe generic properties of systems with time delay, which are related to the appearance and stability of periodic solutions. In particular, we show that delay systems generically have families of periodic solutions, which are reappearing for infinitely many delay times. As delay increases, the solution families overlap leading to increasing coexistence of multiple stable as well as unstable solutions. We also consider stability issue of periodic solutions with large delay by explaining asymptotic properties of the spectrum of characteristic multipliers. We show that the spectrum of multipliers can be split into two parts: pseudocontinuous and strongly unstable. The pseudocontinuous part of the spectrum mediates destabilization of periodic solutions.

Hopf bifurcation control in a congestion control model via dynamic delayed feedback

A typical objective of bifurcation control is to delay the onset of undesirable bifurcation. In this paper, the problem of Hopf bifurcation control in a second-order congestion control model is considered. In particular, a suitable Hopf bifurcation is created at a desired location with preferred properties and a dynamic delayed feedback controller is developed for the creation of the Hopf bifurcation. With this controller, one can increase the critical value of the communication delay, and thus guarantee a stationary data sending rate for larger delay. Furthermore, explicit formulae to determine the period and the direction of periodic solutions bifurcating from the equilibrium are obtained by applying perturbation approach. Finally, numerical simulation results are presented to show that the dynamic delayed feedback controller is efficient in controlling Hopf bifurcation.

Delay stabilization of rotating waves near fold bifurcation and application to all-optical control of a semiconductor laser

We consider the delayed feedback control method for stabilization of unstable rotating waves near a fold bifurcation. Theoretical analysis of a generic model and numerical bifurcation analysis of the rate-equations model demonstrate that such orbits can always be stabilized by a proper choice of control parameters. Our paper confirms the recently discovered invalidity of the so-called "odd-number limitation" of delayed feedback control. Previous results have been restricted to the vicinity of a subcritical Hopf bifurcation. We now refute such a limitation for rotating waves near a fold bifurcation. We include an application to all-optical realization of the control in three-section semiconductor lasers.