Chaos Control Method Research Articles

Modification of Nicholson–Bailey model under refuge effects with stability, bifurcation, and chaos control

In this article, a modification is proposed for the classical Nicholson–Bailey model. It is assumed that the modified model follows all axioms of Nicholson–Bailey model except that in every generation a fraction of the hosts have a safe refuge from attack of parasitoids. It is investigated that under this assumption the modified model has stable coexistence. Furthermore, Neimark–Sacker bifurcation is interrogated at positive steady-state of modified model by implementing the normal forms theory of bifurcation. Chaos control methods based on perturbation of parameter and state feedback strategy are implemented to escape the trajectories from bifurcating and chaotic behavior. Furthermore, numerical simulations are carried out for illustration of theoretical discussion. Finally, all theoretical discussion is illustrated by taking into account real observed field data of host–parasitoid interaction.

Monetary policy rules: An approach based on the theory of chaos control

This article explores the relationship between Taylor rules for monetary policy and those derived from chaos control methods. A similar structure of both rule types would theoretically support the stabilizing role of the Taylor rule for the control of inflation, which until now has been based on an empirical framework. This link between monetary policy and chaos control rules is illustrated using the OGY method of chaos control, resulting in a control rule that is applied to a monetary model that presents chaotic solutions and becomes stable at an objective equilibrium point with a stable inflation rate.

Intervention Time in Target-Oriented Chaos Control

The timing of interventions plays a central role in managing and exploiting biological populations. However, few studies in the literature have addressed its effect on population stability. The Seno equation is a discrete-time equation that describes the dynamics of single-species populations harvested according to the proportional feedback method at any moment between two consecutive censuses. Here we study a discrete-time equation that generalizes the Seno equation by considering the management and exploitation of populations through the target-oriented chaos control method. We investigate the combined effect of timing, targeting, and control on population stability, focusing on global stability. We prove that high enough control values create a positive equilibrium that attracts all positive solutions. We also prove that it is possible to determine parameter values to stabilize the controlled populations at any preset population size. Finally, we investigate the parameter combinations for which the management and exploitation are optimized in different scenarios.

Neimark–Sacker Bifurcation and Controlling Chaos in a Three-Species Food Chain Model through the OGY Method

The dynamics behavior of a discrete-time three-species food chain model is investigated. By using bifurcation theory, it is shown that the equilibrium point of the system loses its stability, and the system undergoes Neimark–Sacker bifurcation, which leads to chaos as the parameter changes. The chaotic motion is controlled on the stable periodic period-1 orbit using the implementation of the hybrid control strategy. The factor affecting the control time of chaos is also studied. Numerical simulations are consistent with the theoretical analysis. The results of this research prove that the chaos control method can be extended to the higher-dimensional biological model and can be realized.

FPGA Realization of the Parameter-Switching Method in the Chen Oscillator and Application in Image Transmission

In this paper, the parameter-switching technique was applied to control chaos in the Chen oscillator and as a decryption mechanism in a secure transmission system, to transmit RGB and grayscale images. In the past few decades, considerable efforts have been put into the study of the stabilization of chaotic dynamical systems. Most of the well-known chaos control methods, such as Ott, Grebogi, and Yorke (OGY), Pyragas, and open-loop methods, force an unstable periodic orbit into a stable one while distorting the original attractor. On the other hand, the parameter-switching technique is an elegant method that can synthesize an already-existing stable orbit, thereby preserving the underlying attractor. Consequently, the main contributions of this work were the FPGA realizations of the parameter-switching method and a secure image transmission system using a synchronized master and slave topology. The results of the parameter-switching technique and synchronization were verified using phase plots and time series. The chaos-encrypted image from the image transmission system, verified using correlation, showed no relativity with the original image, while the recovery of the decrypted image has no loss of quality. The encryption and decryption system was symmetric, whereby the key was private. In this work, co-simulations were performed in Active-HDL with MATLAB/Simulink, while the target FPGA board was the Xilinx’s Artix-7 AC701.

A hybrid adaptive Kriging-based single loop approach for complex reliability-based design optimization problems

The study proposed a hybrid adaptive Kriging-based single loop approach (HAK-SLA) to prevent the conventional single loop approach (SLA) from being less efficient, converging to an inaccurate solution or encountering significant difficulty in convergence when dealing with complex RBDO problems with highly nonlinear constraints. The procedural strategy here is to apply surrogate modeling approach in SLA, in which the constraints were surrogated specifically by Kriging models updated using most probable points (MPPs) in each loop. Since the MPPs obtained directly from the Kriging-based SLA may be insufficiently accurate, two different enhanced chaos control methods are developed here to obtain robustly accurate MPPs. Nonetheless, searching for accurate MPPs would increase considerable computational cost; thus, a criterion combining Karush-Kuhn-Tucker (KKT) optimality condition of performance measure approach (PMA) with Kriging model is utilized to determine whether it is necessary to update the Kriging models using accurate MPPs. When the criterion is satisfied, the accurate MPPs are then used to update Kriging models. In this study, performance of the proposed HAK-SLA is demonstrated by comparing with the other conventional RBDO methods through five illustrative RBDO examples. The results exhibit that the proposed approach is of high efficiency and robustness in solving complex RBDO problems.

Piezoelectric autoparametric vibration energy harvesting with chaos control feature

To extend the working bandwidth for piezoelectric vibration energy harvesting, autoparametric internal resonant system is proposed. Unstable quasiperiodic motion and chaos, however, are accompanied by broadband characteristic in nonlinear energy conversion. In order to control unwanted irregular motions for power management, we presented a new autoparametric vibration energy harvester coupled by nonlinear energy sink. Mathematical model considering the electromechanical coupling and modal coupling for the high-order nonlinear energy harvesting system is established. Nonlinear analyses via power spectrum, phase portrait and Poincaré map illustrate the effectiveness of chaos control using nonlinear energy sink. In addition, average power of the lower boundary of the harvesting band is enhanced by increasing the stiffness and damping of nonlinear energy sink. Average power of the upper boundary of the harvesting band is improved by increase of the nonlinear energy sink mass and substructural natural frequency ratio. The maximum average power of the proposed piezoelectric autoparametric vibration energy harvester is enhanced 200%, compared to the power harvested by the system without the nonlinear energy sink. This study provides a chaos control method for stable broadband vibration energy harvesting. With simultaneous energy harvesting and chaos control, such system can fit in with more applications.

Bi-directional impulse chaos control in crystal growth.

Mono-silicon crystals, free of defects, are essential for the integrated circuit industry. Chaotic swing in the flexible shaft rotating-lifting (FSRL) system of the mono-silicon crystal puller causes harm to the quality of the crystal and must be suppressed in the crystal growth procedure. From the control system viewpoint, the constraints of the FSRL system can be summarized as not having measurable state variables for state feedback control, and only one parameter is available to be manipulated, namely, the rotation speed. From the application side, an additional constraint is that the control should affect the crystallization physical growth process as little as possible. These constraints make the chaos suppression in the FSRL system a challenging task. In this work, the analytical periodic solution of the swing in the FSRL system is derived using perturbation analysis. A bi-directional impulse control method is then proposed for suppressing chaos. This control method does not alter the average rotation speed. It is thus optimum regarding the crystallization process as compared with the single direction impulse control. The effectiveness and the robustness of the proposed chaos control method to parameter uncertainties are validated by the simulations.

Asymptotic Stability Analysis Applied in Two and Three-Dimensional Discrete Systems to Control Chaos

Asymptotic stability analysis applied to stabilize unstable fixed points and to control chaotic motions in two and three-dimensional discrete dynamical systems. A new set of parameter values obtained which stabilizes an unstable fixed point and control the chaotic evolution to regularity. The output of the considered model and that of the adjustable system continuously compared by a typical feedback and the difference used by the adaptation mechanism to modify the parameters. Suitable numerical simulation which are used thoroughly discussed and parameter values are adjusted. The findings are significant and interesting. This strategy has some advantages over many other chaos control methods in discrete systems but, however it can be applied within some limitations.

Parameter impulse control of chaos in crystal growth process

Chaos occurs in the crystal growth process as an irregular swing phenomenon in the flexible shaft rotating-lifting (FSRL) system. Chaos may lead to the failure of mono-silicon crystal production. Therefore, it should be suppressed. Many chaos control methods have been proposed theoretically and some of them have been used in applications. For a practical plant displaying harmful chaos, engineers from a specific area usually face with the challenge to identifying chaos and to suppress it using a proper method. However, despite of the existing methods, chaos control method for the FSRL system is not a trivial task. For example, the seminal chaos control method proposed by Ott-Grebogi-Yorke (OGY method) requires a proper practical adjustable parameter, which cannot be identified for the FSRL system. In this work, an impulsive control method is being proposed to suppress chaos in the system. The merits of the method lie in, first, it manipulates the rotation speed of the motor, which is the only and easily accessible parameter; second, it does not need the state feedback, which is unavailable in the system; third, it is robust against the bounded system parameter uncertainty, as a plant requirement. The control parameter precept is obtained by using the Melnikov method. Simulation results verify the correctness of our theoretical analysis and the effectiveness of the proposed chaos control method.

A chaos control method of single-phase H-bridge photovoltaic inverter

There are complex nonlinear behaviors such as bifurcation and chaos in a single-phase H-bridge photovoltaic inverter under proportional integral control, which will increase the harmonic content of the output current greatly and reduce the stability of system operation and reliability of power supply. There are the problems suffering the complicated modeling and difficulty in determining the control coefficients in existing chaos control methods. The exponential delay feedback control is a further development of the delay feedback control, which has the advantages of requiring no precise mathematical model of the system and simple implementation. However, our research shows that when the exponential delay feedback control is directly applied to the system, the feedback intensity cannot be controlled, which will bring too big a disturbance to the system. Based on it, an improved exponential delayed feedback control method is proposed in this paper. Firstly, a feedback signal is formed by the difference between the output current of the system and its own delay, then the feedback signal is used to obtain the control signal through an exponential link, a subtraction link and an proportion link, and the control signal is applied to the controlled system in the form of a feedback. At the same time, the discrete mapping model of the system is established and its Jacobian matrix expression is determined. Finally, the limiting conditions of the feedback control coefficient of the control signal are derived based on the stability criterion, and the control is applied to the system. In order to verify the control effect of this method, a lot of simulation experiments are conducted. The results show that the problems that the exponential delay feedback cannot control the feedback strength and causes excessive disturbance to the system will be effectively solved by this method. When the bifurcation parameters vary greatly, the chaos behaviors in the system will be suppressed effectively, the stable operating domain of the system will be expanded greatly and the harmonic content of the output current will be reduced.

Research on Dynamic and Complexity of Duopoly Price Strategies between Low‐Carbon and Nonlow‐Carbon Products under Cap‐and‐Trade Policies

This paper studies the complexity of the pricing system for the production of low‐carbon and nonlow‐carbon products in a market composed of duopoly manufacturers under the cap‐and‐trade policies. Through nonlinear system theory and numerical derivation and simulation, it considers the influence of different market power structures, carbon trading prices, consumer environmental awareness, and other factors on price decisions, carbon emission decisions, profits, and system stability. The influence of price adjustment parameters and unit product carbon emission decision adjustment parameters on the complexity of the pricing system under different market power structures is analyzed. And, it was found that compared with the variable feedback chaos control method, the parameter adjustment chaos control method is more effective in controlling the pricing system in this paper. Our research provides management implications for market competition and operational decision‐making for low‐carbon and nonlow‐carbon products.

Control and synchronization of fractional‐order chaotic satellite systems using feedback and adaptive control techniques

SummaryIn this article, we present the basic concepts of fractional calculus and control and synchronization of fractional‐order chaotic satellite system . Existence and uniqueness solutions of fractional‐order satellite system are discussed and local stability of the system at the equilibrium points are studied. The lowest dimension of chaotic attractor of satellite system is 2.88 which is obtained through utilization of the fractional dynamics and computational simulation. Lyapunov exponents and bifurcation diagrams are drawn to measure the existence of chaos in the system. Feedback control method for chaos control in the fractional‐order satellite system is achieved. Synchronization of two identical noninteger order chaotic satellite systems are achieved through adaptive control methodology.

Fractional Grassi–Miller Map Based on the Caputo h-Difference Operator: Linear Methods for Chaos Control and Synchronization

Investigating dynamic properties of discrete chaotic systems with fractional order has been receiving much attention recently. This paper provides a contribution to the topic by presenting a novel version of the fractional Grassi–Miller map, along with improved schemes for controlling and synchronizing its dynamics. By exploiting the Caputo h-difference operator, at first, the chaotic dynamics of the map are analyzed via bifurcation diagrams and phase plots. Then, a novel theorem is proved in order to stabilize the dynamics of the map at the origin by linear control laws. Additionally, two chaotic fractional Grassi–Miller maps are synchronized via linear controllers by utilizing a novel theorem based on a suitable Lyapunov function. Finally, simulation results are reported to show the effectiveness of the approach developed herein.

Inertial Neural Networks with Unpredictable Oscillations

In this paper, inertial neural networks are under investigation, that is, the second order differential equations. The recently introduced new type of motions, unpredictable oscillations, are considered for the models. The motions continue a line of periodic and almost periodic oscillations. The research is of very strong importance for neuroscience, since the existence of unpredictable solutions proves Poincaré chaos. Sufficient conditions have been determined for the existence, uniqueness, and exponential stability of unpredictable solutions. The results can significantly extend the role of oscillations for artificial neural networks exploitation, since they provide strong new theoretical and practical opportunities for implementation of methods of chaos extension, synchronization, stabilization, and control of periodic motions in various types of neural networks. Numerical simulations are presented to demonstrate the validity of the theoretical results.

Stabilizing Circadian Rhythms in Bipolar Disorder by Chaos Control Methods

Bipolar disorder (BD), which involves mood swings between mania and depression, is associated with multiple relapses during long-term treatment and high suicide and morbidity rates. In BD, the circadian rhythms, which are measured by daily mood scores and actigraphic records, are disturbed. Chronotherapy has emerged as a potential treatment for BD because it stabilizes the disturbed circadian rhythms and improves BD symptoms. Concrete treatments include light therapy and combination therapy (light therapy and drugs). However, some patients have difficulty adjusting to light therapy; inappropriate light and duration of treatment increase risks for inducing mixed states and the emergence of conditions, such as hypomania and autonomic hyperactivation. Therefore, it is important to devise methods for optimizing chronotherapy for BD. We aimed to develop feedback signals for the frontal cortex, which were based on the delayed feedback method as one of the chaos control methods, to stabilize the disturbed circadian rhythms of BD. Concrete procedures of this study are indicated as follows: first, circadian rhythms of BD are reproduced using the frontal cortex and hypothalamus neural system, which has been previously proposed. Second, the delayed feedback signal is developed by using bifurcation analysis. Third, the effect of delayed feedback signal is evaluated by index for complexity, and power spectrum under the condition with/without stochastic noise in feedback term. We found that application of the delayed feedback signal to the frontal cortical neural activity induces the periodic state of circadian rhythms from the disturbed complex and is feasible for treating BD. However, when increasing the influence of noise in feedback term, the stabilizing effect is diminished. In conclusion, we developed a stabilizing method for disturbed circadian rhythms of BD using the circadian neural systems. The present study highlights the potential usefulness of the chaos control method for treating BD.

Discretization, bifurcation analysis and chaos control for Schnakenberg model

Schnakenberg model is a system showing sustained oscillations for a simple model of glycolysis in which a metabolic process that converts glucose to provide energy for metabolism. Euler approximation is implemented to obtain discrete version of Schnakenberg model. It is proved that discrete-time system via Euler approximation undergoes Neimark–Sacker bifurcation as well as period-doubling bifurcation is also examined at its unique positive steady-state. Keeping in view the dynamical consistency for continuous models, a nonstandard finite difference scheme is proposed for Schnakenberg model. It is proved that continuous system undergoes Hopf bifurcation at its interior equilibrium, whereas discrete-time system via nonstandard finite difference scheme undergoes Neimark–Sacker bifurcation at its interior fixed point. Some chaos and bifurcation control methods are implemented to both discrete-time models. Numerical simulation is provided to strengthen our theoretical discussion.

Finite-time function projective synchronization control method for chaotic wind power systems

Wind power is a rapidly growing renewable energy source that plays an increasingly important role in power systems. However, its dynamic behaviors are complex because of the instability of wind speed, strong coupling, highly nonlinear subsystems, and mass grid connections. Chaotic oscillation is one of the most typical complex dynamic behaviors of wind power systems. This behavior can seriously influence the stable operation of wind power systems and cause great harm. This study proposed a control method for finite-time function projective synchronization on the basis of the chaos synchronization principle and finite-time theory for wind power systems. Wind power systems with or without uncertain parameters were considered. First, we built a high-dimensional wind power system and analyzed its chaotic behaviors. Lyapunov exponents were derived to prove the existence of chaos and bifurcation. Second, we aimed to increase the robustness of the controller by adding parameter observers to the controller. The result helped solve the problem of unknown parameters. If a wind power system comprises unknown parameters, the unknown parameters can be defined as the system's states. An unknown parameter observer was designed to realize the identification of unknown parameters. Third, we proposed a control method for finite-time function projective synchronization. Finite-time stability theory and function projective synchronization theory were used to construct the controller. These theories can ensure the quick synchronization of a chaotic wind power system with a stable wind power system in finite time. Finally, the mathematical proof of the stability theorem was derived, and a corresponding numerical simulation was performed to validate the chaos control method for wind power systems.

Time-delay control for stabilization of the Shapovalov mid-size firm model

Control and stabilization of irregular and unstable behavior of dynamic systems (including chaotic processes) are interdisciplinary problems of interest to a variety of scientific fields and applications. Using the control methods allows improvements in forecasting the dynamics of unstable economic processes and offers opportunities for governments, central banks, and other policy makers to modify the behaviour of the economic system to achieve its best performance. One effective method for control of chaos and computation of unstable periodic orbits (UPOs) is the unstable delay feedback control (UDFC) approach, suggested by K. Pyragas. This paper proposes the application of the Pyragas’ method within framework of economic models. We consider this method through the example of the Shapovalov model, by describing the dynamics of a mid-size firm. The results demonstrate that suppressing chaos is capable in the Shapovalov model, using the UDFC method.

A class of discrete predator–prey interaction with bifurcation analysis and chaos control

The interaction between prey and predator is well-known within natural ecosystems. Due to their multifariousness and strong link population dynamics, predators contain distinct features of ecological communities. Keeping in view the Nicholson-Bailey framework for host-parasitoid interaction, a discrete-time predator–prey system is formulated and studied with implementation of type-II functional response and logistic prey growth in form of the Beverton-Holt map. Persistence of solutions and existence of equilibria are discussed. Moreover, stability analysis of equilibria is carried out for predator–prey model. With implementation of bifurcation theory of normal forms and center manifold theorem, it is proved that system undergoes transcritical bifurcation around its boundary equilibrium. On the other hand, if growth rate of consumers is taken as bifurcation parameter, then system undergoes Neimark-Sacker bifurcation around its positive equilibrium point. Methods of chaos control are introduced to avoid the populations from unpredictable behavior. Numerical simulation is provided to strengthen our theoretical discussion.