Adaptive control of unknown unstable steady states of dynamical systems

Adaptive delayed feedback control for stabilizing unstable steady states

Delayed feedback control is a commonly used control method for stabilizing unstable periodic orbits and unstable steady states. The present paper proposes an adaptive tuning delay time rule for delayed feedback control focused on stabilizing unstable steady states. The rule is designed to slowly vary the delay time, increasing the difference between the past and current states of dynamical systems, which induces the delay time to automatically fall into the stability region. We numerically confirm that the tuning rule works well for the Stuart-Landau oscillator, FitzHugh-Nagumo model, and Lorenz system.

Revival and death of oscillation under mean-field coupling: Interplay of intrinsic and extrinsic filtering

Mean-field diffusive coupling was known to induce the phenomenon of quenching of oscillations even in identical systems, where the standard diffusive coupling (without mean-field) fails to do so [Phys. Rev. E 89, 052912 (2014)PLEEE81539-375510.1103/PhysRevE.89.052912]. In particular, the mean-field diffusive coupling facilitates the transition from amplitude to oscillation death states and the onset of a nontrivial amplitude death state via a subcritical pitchfork bifurcation. In this paper, we show that an adaptive coupling using a low-pass filter in both the intrinsic and extrinsic variables in the coupling is capable of inducing the counterintuitive phenomenon of reviving of oscillations from the death states induced by the mean-field coupling. In particular, even a weak filtering of the extrinsic (intrinsic) variable in the mean-field coupling facilitates the onset of revival (quenching) of oscillations, whereas a strong filtering of the extrinsic (intrinsic) variable results in quenching (revival) of oscillations. Our results reveal that the degree of filtering plays a predominant role in determining the effect of filtering in the extrinsic or intrinsic variables, thereby engineering the dynamics as desired. We also extend the analysis to networks of mean-field coupled limit-cycle and chaotic oscillators along with the low-pass filters to illustrate the generic nature of our results. Finally, we demonstrate the observed dynamical transition experimentally to elucidate the robustness of our results despite the presence of inherent parameter fluctuations and noise.

Quenching and revival of oscillations induced by coupling through adaptive variables

An adaptive coupling based on a low-pass filter (LPF) is proposed to manipulate dynamic activity of diffusively coupled dynamical systems. A theoretical analysis shows that tracking either the external or internal signal in the coupling via a LPF gives rise to distinctly different ways of regulating the rhythmicity of the coupled systems. When the external signals of the coupling are attenuated by a LPF, the macroscopic oscillations of the coupled system are quenched due to the emergence of amplitude or oscillation death. If the internal signals of the coupling are further filtered by a LPF, amplitude and oscillation deaths are effectively revoked to restore dynamic behaviors. The applicability of this approach is demonstrated in laboratory experiments of coupled oscillatory electrochemical reactions by inducing coupling through LPFs. Our study provides additional insight into (ar)rhythmogenesis in diffusively coupled systems.

Revoking amplitude and oscillation deaths by low-pass filter in coupled oscillators

When in an ensemble of oscillatory units the interaction occurs through a diffusion-like manner, the intrinsic oscillations can be quenched through two structurally different scenarios: amplitude death (AD) and oscillation death (OD). Unveiling the underlying principles of stable rhythmic activity against AD and OD is a challenging issue of substantial practical significance. Here, by developing a low-pass filter (LPF) to track the output signals of the local system in the coupling, we show that it can revoke both AD and OD, and even the AD to OD transition, thereby giving rise to oscillations in coupled nonlinear oscillators under diverse death scenarios. The effectiveness of the local LPF is proven to be valid in an arbitrary network of coupled oscillators with distributed propagation delays. The constructive role of the local LPF in revoking deaths provides a potential dynamic mechanism of sustaining a reliable rhythmicity in real-world systems.

Detection meeting control: Unstable steady states in high-dimensional nonlinear dynamical systems

We articulate an adaptive and reference-free framework based on the principle of random switching to detect and control unstable steady states in high-dimensional nonlinear dynamical systems, without requiring any a priori information about the system or about the target steady state. Starting from an arbitrary initial condition, a proper control signal finds the nearest unstable steady state adaptively and drives the system to it in finite time, regardless of the type of the steady state. We develop a mathematical analysis based on fast-slow manifold separation and Markov chain theory to validate the framework. Numerical demonstration of the control and detection principle using both classic chaotic systems and models of biological and physical significance is provided.

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.

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.

Stabilizing saddles

A synergetic control technique for stabilizing a priori unknown saddle steady states of dynamical systems is described. The method involves an unstable filter technique combined with a derivative feedback. The cut-off frequency of the filter is not limited by the damping of the system, and therefore can be set relatively high. This essentially increases the rate of convergence to the steady state. The synergetic technique is robust to the influence of unknown external forces, which change the coordinates of the steady state in the phase space.

Enhanced control of saddle steady states of dynamical systems

An adaptive feedback technique for stabilizing a priori unknown saddle steady states of dynamical systems is described. The method is based on an unstable low-pass filter combined with a stable low-pass filter. The cutoff frequencies of both filters can be set relatively high. This allows considerable increase in the rate of convergence to the steady state. We demonstrate numerically and experimentally that the technique is robust to the influence of unknown external forces, which change the position of the steady state in the phase space. Experiments have been performed using electrical circuits imitating the damped Duffing-Holmes and chaotic Lindberg systems.

Frustration induced oscillator death on networks

An array of identical maps with Ising symmetry, with both positive and negative couplings, is studied. We divide the maps into two groups, with positive intra-group couplings and negative inter-group couplings. This leads to antisynchronization between the two groups which have the same stability properties as the synchronized state. Introducing a certain degree of randomness in signs of these couplings destabilizes the anti-synchronized state. Further increasing the randomness in signs of these couplings leads to oscillator death. This is essentially a frustration induced phenomenon. We explain the observed results using the theory of random matrices with nonzero mean. We briefly discuss applications to coupled differential equations.

Electrochemical Oscillations of Nickel Electrodissolution in an Epoxy-Based Microchip Flow Cell

We investigate the nonlinear dynamics of transpassive electrodissolution of nickel in sulfuric acid in an epoxy-based microchip flow cell. We observed bistability, smooth, relaxation, and period-2 waveform current oscillations with external resistance attached to the electrode in the microfabricated electrochemical cell with 0.05 mm diameter Ni wire under potentiostatic control. Experiments with 1mm × 0.1 mm Ni electrode show spontaneous oscillations without attached external resistance; similar surface area electrode in macrocell does not exhibit spontaneous oscillations. Combined experimental and numerical studies show that spontaneous oscillation with the on-chip fabricated electrochemical cell occurs because of the unusually large ohmic potential drop due to the constrained current in the narrow flow channel. This large IR potential drop is expected to have an important role in destabilizing negative differential resistance electrochemical (e.g., metal dissolution and electrocatalytic) systems in on-chip integrated microfludic flow cells. The proposed experimental setup can be extendend to multi-electrode configurations; the epoxy-based substrate procedure thus holds promise in electroanalytical applications that require collector-generator multi-electrodes wires with various electrode sizes, compositions, and spacings as well as controlled flow conditions.

Topology-free stability of a steady state in network systems with dynamic connections

This paper investigates the robust stability of a steady state in network systems with dynamic connections. A linear stability analysis reveals that the odd number property holds for any network topology. It is shown that the stability of the steady state in network systems with topological uncertainty is governed by the interval family of real characteristic polynomials. A parametric approach, which plays an important role in the field of robust control theory, allows us to analyze the robust stability against the network topological uncertainty. The stability of the steady state, which is valid for any arbitrary topology, can be evaluated by examining at most four characteristic polynomials. These analytical results are applied to coupled Rössler oscillators on a small-world network.

Stabilization of saddle steady states of conservative and weakly damped dissipative dynamical systems

An adaptive feedback method for tracking and stabilizing unknown and/or slowly varying saddle-type steady states of conservative and weakly damped dissipative dynamical systems is proposed. We demonstrate that a conservative saddle point can be stabilized with neither unstable nor stable filter technique. The proposed controller involves both filters working in parallel. As a specific example, the Lagrange point L2 of the Sun-Earth system is discussed and the second-order saddle model is considered. Analog simulations have been performed using an inclusive nonlinear electrical circuit, imitating dynamics of a body along the Sun-Earth line. External chaotic perturbations have been used to check the robustness of the control technique.

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.

Washout filter aided mean field feedback desynchronization in an ensemble of globally coupled neural oscillators

We propose an approach for desynchronization in an ensemble of globally coupled neural oscillators. The impact of washout filter aided mean field feedback on population synchronization process is investigated. By blocking the Hopf bifurcation of the mean field, the controller desynchronizes the ensemble. The technique is generally demand-controlled. It is robust and can be easily implemented practically. We suggest it for effective deep brain stimulation in neurological diseases characterized by pathological synchronization.

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.

Adaptive steady-state stabilization for nonlinear dynamical systems

By means of LaSalle's invariance principle, we propose an adaptive controller with the aim of stabilizing an unstable steady state for a wide class of nonlinear dynamical systems. The control technique does not require analytical knowledge of the system dynamics and operates without any explicit knowledge of the desired steady-state position. The control input is achieved using only system states with no computer analysis of the dynamics. The proposed strategy is tested on Lorentz, van der Pol, and pendulum equations.

Switching from stable to unknown unstable steady states of dynamical systems

We demonstrate that a dynamical system can be switched from a stable steady state to a previously unknown unstable (saddle) steady state using proportional feedback coupling to an auxiliary unstable system. The simplest one-dimensional nonlinear model is treated analytically, the more complicated two-dimensional pendulum is considered numerically, while the damped Duffing-Holmes oscillator is investigated analytically, numerically, and experimentally. Experiments have been performed using a simplified version of the electronic Young-Silva circuit imitating the dynamical behavior of the Duffing-Holmes system. The physical mechanism behind the switching effect is discussed.

Feedback suppression of neural synchrony in two interacting populations by vanishing stimulation

We discuss the suppression of collective synchrony in a system of two interacting oscillatory networks. It is assumed that the first network can be affected by the stimulation, whereas the activity of the second one can be monitored. The study is motivated by ongoing attempts to develop efficient techniques for the manipulation of pathological brain rhythms. The suppression mechanism we consider is related to the classical problem of interaction of active and passive systems. The main idea is to connect a specially designed linear oscillator to the active system to be controlled. We demonstrate that the feedback loop, organized in this way, provides an efficient suppression. We support the discussion of our approach by a theoretical treatment of model equations for the collective modes of both networks, as well as by the numerical simulation of two coupled populations of neurons. The main advantage of our approach is that it provides a vanishing-stimulation control, i.e., the stimulation reduces to the noise level as soon as the goal is achieved.

Design of model-based controllers for a class of nonlinear chaotic systems using a single output feedback and state observers

A model-based control methodology for suppressing chaos for nonlinear systems is introduced. The proposed methodology generates new periodic orbits or steady states, which are not the solutions of the free system, using output feedback and state observers. The design uses the certainty equivalence principle to construct a linear closed loop system that can follow desired outputs with zero steady state offsets via using a pole-placement-like approach. The combined dynamics of both the controller and the state observer are carefully studied using the well-known Rössler system. The similarities and differences between the proposed control technique and both the double-notch filter feedback and the time-delay autosynchronization are investigated. The effect of parameter uncertainties is studied and robustness of the proposed controller is analyzed.

Conversion of stability in systems close to a Hopf bifurcation by time-delayed coupling

We propose a control method with time delayed coupling which makes it possible to convert the stability features of systems close to a Hopf bifurcation. We consider two delay-coupled normal forms for Hopf bifurcation and demonstrate the conversion of stability, i.e., an interchange between the sub- and supercritical Hopf bifurcation. The control system provides us with an unified method for stabilizing both the unstable periodic orbit and the unstable steady state and reveals typical effects like amplitude death and phase locking. The main method and the results are applicable to a wide class of systems showing Hopf bifurcations, for example, the Van der Pol oscillator. The analytical theory is supported by numerical simulations of two delay-coupled Van der Pol oscillators, which show good agreement with the theory.

Feedback suppression of neural synchrony by vanishing stimulation

We suggest a method for suppression of collective synchrony in an ensemble of all-to-all interacting units. The suppression is achieved by organizing an interaction between the ensemble and a passive oscillator. Technically, this can be easily implemented by a simple feedback scheme. The important feature of our approach is that the feedback signal vanishes as soon as the control is successful. The technique is illustrated by the simulation of a model of an isolated population of neurons. We discuss the possible application of the technique in neuroscience.

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.

Control of unstable steady states by long delay feedback

We present an asymptotic analysis of time-delayed feedback control of steady states for large delay time. By scaling arguments, and a detailed comparison with exact solutions, we establish the parameter ranges for successful stabilization of an unstable fixed point of focus type. Insight into the control mechanism is gained by analyzing the eigenvalue spectrum, which consists of a pseudocontinuous spectrum and up to two strongly unstable eigenvalues. Although the standard control scheme generally fails for large delay, we find that if the uncontrolled system is sufficiently close to its instability threshold, control does work even for relatively large delay times.

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.