OGY Method Research Articles

Chaotic dynamics of a three-species food chain model

An analysis of a discrete-time food chain model is presented in this paper to investigate the effects of strong prey pressure on the chain. A bifurcation and stability theory is used to analyze equilibrium formation in system. It is shown by numerical simulations that chaos occurs as a result of the bifurcation of invariant curves following Neimark-Sacker (NS) bifurcation rules. The OGY method on stable periodic orbits of period 1 further controls chaos. Stability is achieved through iterations for chaotic motion controlled by different regulator poles. Numerical simulations are performed to examine the effects of the control method along with the results of our theoretical analysis.

Controlling Chaotic Fluctuations through Monetary Policy

Abstract This paper applies the chaos control method (the OGY method) proposed by Ott, E., C. Grebogi, and J. A. Yorke. (1990. “Controlling Chaos.” Physical Review Letters 64: 1196–9) to policy-making in macroeconomics. This paper demonstrates that the monetary equilibrium paths in a discrete-time, two-dimensional overlapping generations model exhibit chaotic fluctuations depending on the money supply rate and the elasticity of substitution between capital and labor under the assumption of the constant elasticity of substitution (CES) production function. We also show that the chaotic fluctuations can be stabilized by controlling the money supply rate by using the OGY method and that even when the OGY method does not work due to periodic attractors, adding moderate stochastic shocks to the model can successfully stabilize the economy.

On the analysis of stability, bifurcation, and chaos control of discrete-time predator-prey model with Allee effect on predator

In this paper, a discrete predator-prey model with Allee effect which is obtained by the forward Euler method has been investigated. The local stability conditions of the model at the fixed point have been discussed. In addition, it is shown that the model undergoes Neimark-Sacker bifurcation by using bifurcation theory. Then, the direction of Neimark-Sacker bifurcation has been given. The OGY method is applied in order to control chaos in considered model due to emergence of Neimark-Sacker bifurcation. Some numerical simulations such as phase portraits and bifurcation figures have been presented to support the theoretical results. Also, the chaotic features are justified numerically by computing Lyapunov exponents. Because of consistency with the biological facts, the parameter values have been taken from literature [Controlling chaos and Neimark-Sacker bifurcation discrete-time predator-prey system, Hacet. J. Math. Stat. 49 (5), 1761-1776, 2020].

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.

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.

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.

Chaotic Dynamics and Chaos Control of Hassell-Type Recruitment Population Model

For certain parameters, the mapping of a Hassell-type recruitment population model has a chaotic attractor. The control parameter is disturbed slightly with time by the improvement OGY method. When the mapping point wanders to the neighborhood of the periodic point, the control parameter is perturbed. The chaotic motion is controlled on the stable periodic period-1 point and period-2 orbits, and the influence of different control parameter ranges on the control average time is analyzed. When the selected regulator poles are different, the number of iterations used to control chaotic motion on a stable periodic orbit is different. Numerical simulations are presented to illustrate our results with the theoretical analysis and show the effect of the control method.

Controlling Chaos of the Ricker Population Model by OGY Method

The chaotic dynamics of the Ricker mapping are studied. Controlling chaos of the Ricker population model is searched by OGY control method. The dynamic behavior in the Ricker mapping is very complex in different values of a. When the value of a is changed from 0.455 to 90, the mapping goes through doubling bifurcation to Neimark-Sacker bifurcation. Several strange attractors coexist at last. Through the numerical simulation and analysis of the bifurcation and phase diagrams of the mapping, it is consistent with the theoretical analysis. Studies have shown that ecological balance can be achieved by appropriately adjusting the birth rate a, survival rate b.

Bifurcation and Controlling Chaos in a Discrete-Time Biological System

The dynamics of a discrete-time food chain model with strong pressure on preys is investigated. The types of equilibria of the system are analyzed using stability theory and bifurcation theory. The route to chaos via Neimark-Sacker bifurcation followed by period-doubling bifurcations of invariant curves is found for some parameter values through numerical simulation. Moreover, the chaos is controlled on the stable periodic period-1 orbit by the improvement of OGY method. It is shown that the number of iterations used to control chaotic motion on a stable periodic orbit is difference, when the selected regulator poles are different. Numerical simulations are presented to illustrate our results based on the theoretical analysis and show the effect of the control method.

Combination of Improved OGY and Guiding Orbit Method for Chaos Control

Chaotic systems have gathered much attention. When the OGY method is applied to control a chaotic system, chaos can be contained and target signals can be traced with satisfactory accuracy. However, the traditional control method have a low convergence speed, which may hamper the performance of the whole system. To solve this problem, the cuckoo search algorithm is used to guide the orbits of chaotic systems. Moreover, the OGY method is improved so that a chaotic system can be stabilized for different target points. Finally, the effectiveness of the proposed method is verified through several typical chaotic systems. The simulation results indicate that the modified method has a faster convergence speed and yields better performance than the traditional OGY control method.

Qualitative analysis and chaos control in a density-dependent host–parasitoid system

We study the comprehensive dynamics of a density-dependent host–parasitoid system with the Hassell growth function for the host population. Particularly, we investigate the dynamical properties related to boundedness, local asymptotic stability of boundary equilibrium, existence and uniqueness of positive equilibrium point and global asymptotic stability of the positive equilibrium point of this modified host–parasitoid model. Moreover, it is also proved both populations endure period-doubling bifurcation and Neimark–Sacker bifurcation at positive steady-state with suitable choice of parametric values. OGY method is implemented in order to controlling chaos in host–parasitoid model. Finally, numerical simulations are provided to illustrate theoretical results. These results of numerical simulations demonstrate chaotic long-term behavior over a broad range of parameters. The computation of the maximum Lyapunov exponents confirm the presence of chaotic behavior in the model.

Bifurcation analysis and chaos control in a host‐parasitoid model

We investigate the qualitative behavior of a host‐parasitoid model with a strong Allee effect on the host. More precisely, we discuss the boundedness, existence and uniqueness of positive equilibrium, local asymptotic stability of positive equilibrium and existence of Neimark–Sacker bifurcation for the given system by using bifurcation theory. In order to control Neimark–Sacker bifurcation, we apply pole‐placement technique that is a modification of OGY method. Moreover, the hybrid control methodology is implemented in order to control Neimark–Sacker bifurcation. Numerical simulations are provided to illustrate theoretical discussion. Copyright © 2017 John Wiley & Sons, Ltd.

Neimark-Sacker bifurcation and chaos control in Hassell-Varley model

We investigate the complex behaviour of a modified Nicholson–Bailey model. The modification is proposed by Hassel and Varley taking into account that interaction between parasitoids is taken in such a way that the searching area per parasitoid is inversely proportional to the m-th power of the population density of parasitoids. Under certain parametric conditions the unique positive equilibrium point of system is locally asymptotically stable. Moreover, it is proved that system undergoes Neimark-Sacker bifurcation for small range of parameters by using standard mathematical techniques of bifurcation theory. In order to control Neimark-Sacker bifurcation, we apply simple feedback control strategy and pole-placement technique which is a modification of OGY method. Moreover, the hybrid control methodology is also implemented for chaos controlling. Numerical simulations are provided to illustrate theoretical discussion.

Detection of Unstable Periodic Orbits and Chaos Control in a Passive Biped Model

Dynamic analysis of passive biped models plays a significant role both in understanding human locomotion and in developing humanoid robots. In this investigation, two chaos control algorithms based on linearization of Poincare map (OGY method) and artificial neural networks (ANNs) are utilized to control the motion of a passive biped model. To this end, the chaotic characteristics of the system are analyzed using several nonlinear dynamics tools such as Poincare map, bifurcation diagram, and Lyapunov exponents, and then, unstable periodic orbits (UPOs) of the system are detected. Detection of these orbits helps to extract a desired walking pattern and also is utilized for chaos elimination of the system. The robustness of the proposed ANN-based control algorithm is verified by applying toe-off impulses to the biped during the gait. Furthermore, the effect of network parameters on the biped walking performance is investigated to get design guidelines for the ANN-based controller.

Takagi-Sugeno fuzzy logic controller for Liu-Chen four-scroll chaotic system

Chaos modelling, chaos control and chaos synchronisation are active areas in chaos theory, while research in these areas has potential applications in many fields like nonlinear optics, oscillations, chemical reactions, biology, neurology, secure data encryption, secure communications, robotics, neural networks, fractals and nonlinear circuits design. Especially, chaos control is an important research area, and it has a wide range of applications like control of turbulence, dynamics of lasers, plasma, chemical reactions such as Belousov-Zhabotinsky reaction, ecological systems and cardiac arrhythmias. Until now, many methods for chaos control such as open-loop control based on periodic system excitation, closed-loop control such as OGY method and time-delayed feedback control such as Pyragas method have been used. In this paper, a fuzzy logic control using fuzzy sets and fuzzy logic for the control of chaotic systems has been proposed. In more detail, this work presents a fuzzy logic controller design of a chaotic system using Takagi-Sugeno fuzzy model. The stability analysis of the fuzzy logic controller system is carried out using Barbashin-Krasovskii theorem. For the proposed fuzzy logic control, the Liu-Chen four-scroll chaotic system was used. Finally, the simulation results of the proposed control method present the effectiveness of this method.

Control of chaos in the vicinity of the Earth‐Moon L5 Lagrangian point to keep a spacecraft in orbit

AbstractThe concept of Space Manifold Dynamics is a new method of space research. We have applied it along with the basic idea of the method of Ott, Grebogi, and York (OGY method) to stabilize the motion of a spacecraft around the triangular Lagrange point L5 of the Earth‐Moon system. We have determined the escape rate of the trajectories in the general three‐ and four‐body problem and estimated the average lifetime of the particles. Integrating the two models we mapped in detail the phase space around the L5 point of the Earth‐Moon system. Using the phase space portrait our next goal was to apply a modified OGY method to keep a spacecraft close to the vicinity of L5. We modified the equation of motions with the addition of a time dependent force to the motion of the spacecraft. In our orbit‐keeping procedure there are three free parameters: (i) the magnitude of the thrust, (ii) the start time, and (iii) the length of the control. Based on our numerical experiments we were able to determine possible values for these parameters and successfully apply a control phase to a spacecraft to keep it on orbit around L5. (© 2015 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

OGY-based control of chaos in semi-passive dynamic walking of a torso-driven biped robot

This paper aims at controlling chaos exhibited in the semi-passive dynamic walking of a torso-driven biped robot as it goes down an inclined surface. Our control approach is based on the OGY method. The proposed biped robot is a three-degrees-of-freedom planar biped having an impulsive hybrid nonlinear dynamics. For this walker, we use only one torque between the stance leg and the torso in order to control the torso at some desired position and then in order to generate a semi-passive gait. The desired torso angle is considered as the control parameter in our OGY-based control approach. We develop a reduced simple impulsive hybrid linear model by linearizing the impulsive hybrid nonlinear dynamics around a desired period-1 hybrid limit cycle. This conducts to determine an explicit expression of a constrained controlled Poincare map. A linearization of the controlled Poincare map around its fixed point permits to looking for the gain matrix of the stabilizing control law. We show that application of the developed OGY-based control parameter law has controlled the chaotic semi-passive gait.

State space reconstruction applied to a multiparameter chaos control method

The idea of the chaos control is the stabilization of unstable periodic orbits (UPOs) embedded in chaotic attractors. The OGY method achieves system stabilization by using small perturbations promoted in the neighborhood of the desired orbit when the trajectory crosses a Poincare section. A generalization of this method considers multiple actuations of parameters and sections, known as semi-continuous multiparameter method. This paper investigates the state space reconstruction applied to this general method, allowing chaotic behavior control of systems with non-observable states using multiple control parameters from time series analysis, avoiding the use of governing equations. As an application of the proposed multiparameter general formulation it is presented an uncoupled approach where the control parameters do not influence the system dynamics when they are not active. This method is applied to control chaos in a nonlinear pendulum using delay coordinates to perform state space reconstruction. Results show that the proposed procedure can be applied together with delay coordinates providing UPO stabilization.

Controlling Chaos of Hybrid Systems by Variable Threshold Values

We try to stabilize unstable periodic orbits embedded in a given chaotic hybrid dynamical system by a perturbation of a threshold value. In conventional chaos control methods, a control input is designed by state-feedback, which is proportional to the difference between the target orbit and the current state, and it is applied to a specific system parameter or the state as a small perturbation. During a transition state, the control system consumes a certain control energy given by the integration of such perturbations. In our method, we change the threshold value dynamically to control the chaotic orbit. Unlike the OGY method and the delayed feedback control, no actual control input is added into the system. The state-feedback is utilized only to determine the dynamic threshold value, thus the orbit starting from the current threshold value reaches the next controlled threshold value without any control energy. We obtain the variation of the threshold value from the composite Poincaré map, and the controller is designed by the linear feedback theory with this variation. We demonstrate this method in simple hybrid chaotic systems and show its control performances by evaluating basins of attraction.

Chaos Control of Lozi Mapping

In this paper, the chaos control of Lozi mapping is carried out through the improved OGY method. By establishing linear mapping and applying the pole placement technique of the linear control theory, small time-dependent perturbations of a control parameter are selected. Unstable periodic orbits embedding in the chaos attractor are examined firstly, and then the period one orbit is chosen as a control target. When map point wanders to the neighborhood of the periodic point, system control parameter is perturbed. The unstable period one point is controlled to be the stable periodic orbit. At the same time, the different choice of the regulator poles is analyzed. Using numerical simulation, the effectiveness of the method is demonstrated.