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.

Under the influence of crowding effects: Stability, bifurcation and chaos control for a discrete-time predator–prey model

The interaction between predators and preys exhibits more complicated behavior under the influence of crowding effects. By taking into account the crowding effects, the qualitative behavior of a prey–predator model is investigated. Particularly, we examine the boundedness as well as existence and uniqueness of positive steady-state and stability analysis of the unique positive steady-state. Moreover, it is also proved that the system undergoes Hopf bifurcation and flip bifurcation with the help of bifurcation theory. Moreover, a chaos control technique is proposed for controlling chaos under the influence of bifurcations. Finally, numerical simulations are provided to illustrate the theoretical results. These results of numerical simulations demonstrate chaotic long-term behavior over a broad range of parameters. The presence of chaotic behavior in the model is confirmed by computing maximum Lyapunov exponents.

Control of Pyragas Applied to a Coupled System with Unstable Periodic Orbits

We apply a Pyragas-type control in order to synchronize the solutions of a glycolytic model that exhibits an aperiodic behavior. This delay control is used to stabilize the orbits of ordinary differential nonlinear equations systems. Inspired by several works that studied the chaotic behavior of diverse systems for the enzymatic reactions in the presence of feedbacks, the control to two of these models is analyzed.

Detecting unstable periodic orbits in chaotic time series using synchronization

An alternative approach of detecting unstable periodic orbits in chaotic time series is proposed using synchronization techniques. A master-slave synchronization scheme is developed, in which the chaotic system drives a system of harmonic oscillators through a proper coupling condition. The proposed scheme is designed so that the power of the coupling signal exhibits notches that drop to zero once the system approaches an unstable orbit yielding an explicit indication of the presence of a periodic motion. The results shows that the proposed approach is particularly suitable in practical situations, where the time series is short and noisy, or it is obtained from high-dimensional chaotic systems.

On fuzzy sampled-data control of chaotic systems via a time-dependent Lyapunov functional approach

In this paper, a novel approach to fuzzy sampled-data control of chaotic systems is presented by using a time-dependent Lyapunov functional. The advantage of the new method is that the Lyapunov functional is continuous at sampling times but not necessarily positive definite inside the sampling intervals. Compared with the existing works, the constructed Lyapunov functional makes full use of the information on the piecewise constant input and the actual sampling pattern. In terms of a new parameterized linear matrix inequality (LMI) technique, a less conservative stabilization condition is derived to guarantee the exponential stability for the closed-loop fuzzy sampled-data system. By solving a set of LMIs, the fuzzy sampled-data controller can be easily obtained. Finally, the chaotic Lorenz system and Rössler's system are employed to illustrate the feasibility and effectiveness of the proposed method.

Phase-reduction-theory-based treatment of extended delayed feedback control algorithm in the presence of a small time delay mismatch

The delayed feedback control (DFC) methods are noninvasive, which means that the control signal vanishes if the delay time is adjusted to be equal to the period of a target unstable periodic orbit (UPO). If the delay time differs slightly from the UPO period, a nonvanishing periodic control signal is observed. We derive an analytical expression for this period for a general class of multiple-input multiple-output systems controlled by an extended DFC algorithm. Our approach is based on the phase-reduction theory adapted to systems with time delay. The analytical results are supported by numerical simulations of the controlled Rössler system.

Control of chaos: methods and applications in mechanics

A survey of the field related to control of chaotic systems is presented. Several major branches of research that are discussed are feed-forward ('non-feedback') control (based on periodic excitation of the system), the 'Ott-Grebogi-Yorke method' (based on the linearization of the Poincaré map), the 'Pyragas method' (based on a time-delayed feedback), traditional for control-engineering methods including linear, nonlinear and adaptive control. Other areas of research such as control of distributed (spatio-temporal and delayed) systems, chaotic mixing are outlined. Applications to control of chaotic mechanical systems are discussed.

Controlling chaos with weak periodic signals optimized by a genetic algorithm

In the present study we develop a relatively novel and effective chaos control approach with a multimode periodic disturbance applied as a control signal and perform an in-depth analysis on this nonfeedback chaos control strategy. Different from previous chaos control schemes, the present method is of two characteristic features: (1) the parameters of the controlling signal are optimized by a genetic algorithm (GA) with the largest Lyapunov exponent used as an index of the stability, and (2) the optimization is justified by a fitness function defined with the target Lyapunov exponent and the controlling power. This novel method is then tested on the noted Rössler and Lorenz systems with and without the presence of noise. The results disclosed that, compared to the existing chaos control methods, the present GA-based control needs only significantly reduced signal power and a shorter transient stage to achieve the preset control goal. The switching control ability and the robustness of the proposed method for cases with sudden change in a system parameter and/or with the presence of noise environment are also demonstrated.