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The resistive–capacitive–inductive-shunted (RCL-shunted) Josephson junction (RCLSJJ) shows chaotic behavior under some parameter conditions. This paper studies chaos control in the RCLSJJ based on the delayed feedback theory. Numerical results demonstrate that chaotic states in this junction can be controlled into stable periodic states by appropriately adjusting the feedback intensity and delay time.

We propose an application of state predictor to delayed feedback control (DFC) for continuous-time dynamical systems. First, we define the DFC for continuous-time systems clearly. Next, on the basis of this definition, we propose a DFC method with state-predictor to stabilize unstable equilibrium points in autonomous systems and derive a sufficient condition for stabilization. Then we extend this method to stabilizing unstable periodic orbits and also derive a sufficient condition for stabilization. Finally, we briefly discuss the relation between the conditions for stabilization of our methods and those of the dynamic DFC and the prediction-based feedback control for discrete-time systems.

This paper presents a study on the problem of Hopf bifurcation control of time delayed systems with weak nonlinearity via delayed feedback control. It focusses on two control objectives: one is to annihilate the periodic solution, namely to perform a linear delayed feedback control so that the trivial equilibrium is asymptotically stable, and the other is to obtain an asymptotically stable periodic solution with given amplitude via linear or nonlinear delayed feedback control. On the basis of the averaging method and the center manifold reduction for delayed differential equations, an effective method is developed for this problem. It has been shown that a linear delayed feedback can always stabilize the unstable trivial equilibrium of the system, and a linear or nonlinear delayed feedback control can always achieve an asymptotically stable periodic solution with desired amplitude. The illustrative example shows that the theoretical prediction is in very good agreement with the simulation results, and that the method is valid with high accuracy not only for delayed systems with weak nonlinearity and via weak feedback control, but also for those when the nonlinearity and feedback control are not small.

We propose a periodic feedback scheme for the stabilization of periodic orbits for discrete time chaotic systems. We first consider one-dimensional discrete time systems and obtain some stability results. Then we extend these results to higher dimensional discrete time systems. The proposed scheme is quite simple and we show that any hyperbolic periodic orbit can be stabilized with this scheme. We also present some simulation results.

We consider noise-induced charge density dynamics in a semiconductor superlattice. The parameters are fixed in the regime below the Hopf bifurcation that gives birth to spatio-temporal oscillations, where in the absence of noise of the system rests at a fixed point. It is shown that in this case noise can induce in the superlattice quite coherent oscillations of the current through the device. While the regularity of these oscillations depends on the noise intensity, their dominant frequency remains almost constant with variation of the noise level in the system. Further, we demonstrate that a time-delayed feedback scheme that was previously used to control purely temporal oscillations induced by noise, can not only enhance or deteriorate the regularity of stochastic spatio-temporal patterns but also allow for the manipulation of the system's time scales with varying time delay.

Synchronization processes may severely impair brain function, for instance, in Parkinson's disease, essential tremor or epilepsies. We present three different effectively desynchronizing stimulation techniques which have been developed with methods from nonlinear dynamics and statistical physics. These techniques exploit either stochastic phase resetting principles or complex delayed feedback mechanisms. We explain how these methods work and how they can be applied to therapeutic brain stimulation.

A novel control method for desynchronization of strongly synchronized populations of interacting oscillators is described. We show that an ensemble's mean field, nonlinearly processed and fed back into the ensemble, causes an effective desynchronization. The method is mild, demand controlled, and robust against system and stimulation parameter variations. The desynchronization and decoupling effects of the method are illustrated by examples of one and two interacting populations of limit-cycle oscillators. We suggest our method for mild and effective deep brain stimulation in neurological diseases characterized by pathological cerebral synchronization.

We present a detailed bifurcation analysis of desynchronization transitions in a system of two coupled phase oscillators with delay. The coupling between the oscillators combines a delayed self-feedback of each oscillator with an instantaneous mutual interaction. The delayed self-feedback leads to a rich variety of dynamical regimes, ranging from phase-locked and periodically modulated synchronized states to chaotic phase synchronization and desynchronization. We show that an increase of the coupling strength between oscillators may lead to a loss of synchronization. Intriguingly, the delay has a twofold influence on the oscillations: synchronizing for small and intermediate coupling strength and desynchronizing if the coupling strength exceeds a certain threshold value. We show that the desynchronization transition has the form of a crisis bifurcation of a chaotic attractor of chaotic phase synchronization. This study contributes to a better understanding of the impact of time delay on interacting oscillators.

In this paper we consider the stabilization problem of unstable periodic orbits of discrete time chaotic systems by using a scalar input. We use a simple periodic delayed feedback law and present some stability results. These results show that all hyperbolic periodic orbits as well as some nonhyperbolic periodic orbits can be stabilized with the proposed method by using a scalar input, provided that some controllability or observability conditions are satisfied. The stability proofs also lead to the possible feedback gains which achieve stabilization. We will present some simulation results as well.

We study the effect of a time-delayed feedback within a generic model for a saddle-node bifurcation on a limit cycle. Without delay the only attractor below this global bifurcation is a stable node. Delay renders the phase space infinite-dimensional and creates multistability of periodic orbits and the fixed point. Homoclinic bifurcations, period-doubling and saddle-node bifurcations of limit cycles are found in accordance with Shilnikov's theorems.

In this paper, we consider the stabilization problem of unstable periodic orbits of one-dimensional discrete time chaotic systems. We propose a novel generalization of the classical delayed feedback law and present some stability results. These results show that for period 1 all hyperbolic periodic orbits can be stabilized by the proposed method; for higher order periods the proposed scheme possesses some inherent limitations. However, some more improvement over the classical delayed feedback scheme can be achieved with the proposed scheme. The stability proofs also give the possible feedback gains which achieve stabilization. We will also present some simulation results.

In this paper, a time-delayed feedback controller is applied to a fluid flow model of Internet congestion control system in order to control Hopf bifurcation. It has been shown that the system without control loses stability and a Hopf bifurcation occurs when the bifurcation parameter, which is the communication delay of the model, passes through a critical value. Therefore, a control approach based on delayed feedback is proposed to postpone the onset of undesired Hopf bifurcation. Theoretical analysis and numerical simulations confirm that the controller is efficient in bifurcation control of the Internet congestion control system.

In this paper, we have studied an important class of first order differential equations with nonlinear delayed feedback. The nonlocal dynamics of such equations was investigated by analytical and numeric–analytical methods. An approach based on asymptotic analysis made it possible to study attractors consisting of impulse type solutions. We obtained some numerical characteristics of irregular oscillations and studied their dependence on the delay.

The dynamic feedback control of the cardiac pacing interval has been widely used to suppress alternans. In this paper, temporally and spatially suppressing the alternans for cardiac tissue consisting of a one-dimensional chain of cardiac units is investigated. The model employed is a nonlinear partial difference equation. The model's fixed points and their stability conditions are determined, and bifurcations and chaos phenomenon have been studied by numerical simulations. The main objective of this paper is to stabilize the unstable fixed point of the model. The proposed approach is nonlinear spatiotemporal delayed feedback, and the appropriate interval for controller feedback gain is calculated using the linear stability analysis. It is proven that the proposed approach is robust with respect to all bifurcation parameter variations. Also, set point tracking is achieved by employing delayed feedback with an integrator. Finally, simulation results are provided to show the effectiveness of the proposed methodology.

A diffusive photosensitive CDIMA system with delayed feedback subject to Neumann boundary conditions is considered. We derive the conditions of the occurrence of Turing instability. We also investigate the influence of delay on the stability of the positive equilibrium of the system, and prove that delay induces the occurrence of Hopf bifurcation. By computing the normal form on the center manifold, we give the formulas determining the properties of the Hopf bifurcation. Finally, we give some numerical simulations to support and strengthen the theoretical results. Our study shows that diffusion and delayed feedback can effect the stability of the equilibrium of the system.

Explosive death in coupled nonlinear oscillators has been an active area of extensive research in nonlinear dynamics in the recent decades. Depending on proper choice of network topology, coupling scenarios, and feedback strength, explosive death can be revealed. In this work, for the first time, we report the effect of delayed feedback on the death behavior in an ensemble of identical mean-field coupled van der Pol oscillators. In both systems with or without time delay, the normalized amplitude exhibits an abrupt transition between the oscillatory state and the death state. Intriguingly, the presence of time delay in the coupling may induce the normalized amplitude of all oscillators in the network to experience a step-like descent with small jumps in approaching the death state, pulling back the forward and backward transition points. The backward transition point has been explicitly obtained, which is confirmed by the numerical results.

The restoration and maintenance of oscillatory rhythms are integrated to the regular operation of biological and physical processes. In this paper, we explore the revival of oscillations, specifically focusing on the phenomenon of explosive Nontrivial Amplitude Death (NAD) that occurs within N van der Pol oscillators coupled through mean-field attractive–repulsive interactions. By incorporating discrete and distributed delays into the repulsive mean-field, respectively, we provide theoretical and numerical evidence that the explosive NAD can be effectively destabilized, leading to the restoration of the oscillatory behavior. By utilizing Lyapunov stability, we theoretically determine the boundaries of the oscillation region recovered from NAD. For discrete delay, adjusting the magnitude of delay can result in a substantial reduction of the NAD region within the parameter space. Under a specific delay, the NAD region completely becomes an oscillation state. Moreover, in the scenario of uniformly distributed delay, the presence of delay also contributes to a decrease of the NAD area, and as the distribution width increases, the revived oscillation region gradually contracts compared to the case with discrete delay. Importantly, the results from the numerical simulations are in agreement with the theoretical analysis. Our framework establishes a theoretical foundation for comprehending the regeneration of oscillation in mean-field attractive–repulsive coupled systems.

Problem solvers employ strategies in searching for solutions to given problem instances. Strategies have traditionally been designed by experts using prior knowledge and refined manually using trial and error. Recent attempts to automate these processes have produced strategy-learning systems. This paper shows how various issues in strategy learning are affected by the nature of performance tasks, problem solvers, and learning environments. Surveyed learning systems are grouped by the commonality of their approaches into four general architectures.

We consider a class of spiking neuronal models, defined by a set of conditions typical for basic threshold-type models, such as the leaky integrate-and-fire or the binding neuron model and also for some artificial neurons. A neuron is fed with a Poisson process. Each output impulse is applied to the neuron itself after a finite delay $\mathrm{\Delta }$. This impulse acts as being delivered through a fast Cl-type inhibitory synapse. We derive a general relation which allows calculating exactly the probability density function (pdf) $p(t)$ of output interspike intervals of a neuron with feedback based on known pdf $p^0(t)$ for the same neuron without feedback and on the properties of the feedback line (the $\mathrm{\Delta }$ value). Similar relations between corresponding moments are derived.

Furthermore, we prove that the initial segment of pdf $p^0(t)$ for a neuron with a fixed threshold level is the same for any neuron satisfying the imposed conditions and is completely determined by the input stream. For the Poisson input stream, we calculate that initial segment exactly and, based on it, obtain exactly the initial segment of pdf $p(t)$ for a neuron with feedback. That is the initial segment of $p(t)$ is model-independent as well. The obtained expressions are checked by means of Monte Carlo simulation. The course of $p(t)$ has a pronounced peculiarity, which makes it impossible to approximate $p(t)$ by Poisson or another simple stochastic process.

A class of spiking neuronal models with threshold 2 is considered. It is defined by a set of conditions typical for basic threshold-type models, such as the leaky integrate-and-fire (LIF) or the binding neuron model, and also for some artificial neurons. A neuron is stimulated with a Poisson stream of excitatory impulses. Each output impulse is conveyed through the feedback line to the neuron input after finite delay $\mathrm{\Delta }$. This impulse is identical to those delivered from the input stream. We have obtained a general relation allowing calculating exactly the probability density function (PDF) $p(t)$ for distribution of the first passage time of crossing the threshold, which is the distribution of output interspike intervals (ISI) values for this neuron. The calculation is based on known PDF $p^0(t)$ for that same neuron without feedback, intensity of the input stream $\lambda$ and properties of the feedback line. Also, we derive exact relation for calculating the moments of $p(t)$ based on known moments of $p^0(t)$.

The obtained general expression for $p(t)$ is checked numerically using Monte Carlo simulation for the case of LIF model. The course of $p(t)$ has a $\delta$-function-type peculiarity. This fact contributes to the discussion about the possibility to model neuronal activity with Poisson process, supporting the “no” answer.