Ott-Grebogi-Yorke Research Articles

Dynamic Behavior and Bifurcation Analysis of a Modified Reduced Lorenz Model

This study introduces a newly modified Lorenz model capable of demonstrating bifurcation within a specified range of parameters. The model demonstrates various bifurcation behaviors, which are depicted as distinct structures in the diagram. The study aims to discover and analyze the existence and stability of fixed points in the model. To achieve this, the center manifold theorem and bifurcation theory are employed to identify the requirements for pitchfork bifurcation, period-doubling bifurcation, and Neimark–Sacker bifurcation. In addition to theoretical findings, numerical simulations, including bifurcation diagrams, phase pictures, and maximum Lyapunov exponents, showcase the nuanced, complex, and diverse dynamics. Finally, the study applies the Ott–Grebogi–Yorke (OGY) method to control the chaos observed in the reduced modified Lorenz model.

Dynamic Behaviors in a Discrete Model for Predator–Prey Interactions Involving Hibernating Vertebrates

This paper presents a discrete predator–prey interaction model involving hibernating vertebrates, with detailed analysis and simulation. Hibernation contributes to the survival and reproduction of organisms and species in the ecosystem as a whole. In addition, it also constitutes a wise sharing of time, space, and resources with others. We have created a new predator–prey model by integrating the two species, Holling-III and Holling-I, which have a bifurcation within a specified parameter range. We discovered that this system possesses the stability of fixed points as well as several bifurcation behaviors. To accomplish this, the center manifold theorem and bifurcation theory are applied to create existence conditions for period-doubling bifurcations and Neimark–Sacker bifurcations, which are depicted in the graph as distinct structures. Examples of numerical simulations include bifurcation diagrams, maximum Lyapunov exponents, and phase portraits, which demonstrate not just the validity of theoretical analysis but also complex dynamical behaviors and biological processes. Finally, the Ott–Grebogi–Yorke (OGY) method and phases of chaos control bifurcation were used to control the chaos of predator–prey model in hibernating vertebrates.

Nonlinear Analysis and Attitude Control of a Gyrostat Satellite with Chaotic Dynamics Using Discrete‐Time LQR‐OGY

AbstractQuasi‐periodic and chaotic behavior, along with the control of chaos for a Gyrostat satellite (GS), is investigated in this work. The quaternion‐based dynamical model of the GS is first derived, and then the influences of the reaction wheels in the GS structure, under the gravity gradient perturbation that causes a route to chaos through quasi‐periodicity mechanism, is investigated. For the suppression of chaos in the system, a chaos control system with the quaternion feedback is designed for the GS based on the extension of the Ott‐Grebogi‐Yorke (OGY) method using the linearization of the Poincaré map. In the extended OGY controller, the Poincaré map is estimated using the Least Square Support Vector Machine (LSSVM) technique. After linearization of the Poincaré map, the Discrete‐time Linear Quadratic Regulator (DLQR) is applied on the linearized Poincaré map, making the DLQR‐OGY controller for chaos. The DLQR‐OGY control system stabilizes the orbits to the fixed points providing a small control input signal, which leads to a decrease in the control effort and energy consumption in the GS system.

Bifurcations and chaos in the semi-passive bipedal dynamic walking model under a modified OGY-based control approach

This paper aims at investigating the semi-passive dynamic walking of a torso-driven biped robot under the Ott–Grebogi–Yorke (OGY) control approach as it goes down inclined planes. For the control, we used the desired torso angle as the accessible control parameter. Compared with our work in Gritli et al. (Nonlinear Dyn 79(2):1363–1384, 2015), a modified design of the OGY-based controller is proposed in this paper. Such controller is obtained by linearizing the impulsive hybrid nonlinear dynamics of the biped robot around a desired one-periodic hybrid limit cycle. Both the differential equation and the algebraic equation are linearized. As a result, we develop a simple mathematical expression of a controlled hybrid Poincare map. Determination of its fixed point and its Jacobian matrix requires only the knowledge of the nominal impact instant. We show efficiency of the designed OGY controller for the control of chaos in the impulsive hybrid nonlinear dynamics for some desired nominal values of the slope, and the desired torso angle. Furthermore, we analyzed via bifurcation diagrams the displayed behaviors in the controlled semi-passive biped robot as the slope parameter varies. We show the appearance of a Neimark–Sacker bifurcation and a cyclic-fold bifurcation, and also the exhibition of chaos. Our analysis of the controlled semi-passive gait is achieved also by means of the spectrum of Lyapunov exponents. Such study is realized via the controlled hybrid Poincare map where a reduction of its dimension is achieved.

Chaos control in attitude dynamics of a gyrostat satellite based on linearised Poincare' map estimation by support vector machine

Chaos control of an apparent-type gyrostat satellite (GS) is investigated in this work. The GS under study consists of a main platform along with the three reaction wheels. The mathematical model of the GS is first derived using the quaternion-based kinematic and Euler-based kinetic equations of motion under the gravity gradient perturbation. Chaotic dynamics of the open-loop system without a feedback is then analyzed by the use of the numerical simulation in the phase portrait trajectories, Poincare' section, and time series responses. The existence of chaos is also demonstrated using the Lyapunov exponent criterion. In order to suppress chaos in the GS, a quaternion feedback controller is designed by the modification of Ott–Grebogi–Yorke (OGY) algorithm based on the linearisation of Poincare' map. In the controller strategy, the Poincare' map is estimated by the least square-support vector machine technique. Then, the discrete-time proportional-integral-derivative (PID) controller is applied on the linearised Poincare' map. The discrete-time PID-OGY control system rejects the chaotic behaviours in the attitude orientation of GS with the generation of a small control input leading to a decrease in the control effort and energy consumption.

A multiparameter chaos control method applied to maps

Chaos is a kind of nonlinear system response that has a dense set of unstable periodic orbits (UPOs) embedded in a chaotic attractor. The idea of the chaos control is to explore the UPO stabilization obtaining dynamical systems that may quickly react to some new situation, changing conditions and their response. The OGY (Ott-Grebogi-Yorke) method achieves system stabilization by using small perturbations promoted in the neighborhood of the desired orbit when the trajectory crosses a specific surface, such as a Poincare section. This contribution proposes a multiparameter (MP) method based on OGY approach in order to control chaotic behavior using different control parameters. 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 maps using two control parameters. The two-dimensional Henon and Ikeda maps are of concern. Results show that the proposed procedure can be a good alternative for chaos control since it provides a more effective UPO stabilization than the classical single-parameter OGY approach.

Stabilization of Unstable Fixed Points of Chaotic Fractional Order Systems by a State Fractional PI Controller

This paper presents a new method to control chaos in fractional order systems based on the fractional control theory. The proposed controller is a fractional PI (PI α ) controller and can locally stabilize unstable equilibrium points of a class of chaotic fractional order systems. Using the ideas available in the chaos control methods such as Ott-Grebogi-Yorke (OGY), this local stabilization can be extended to the global stabilization. The controller has simple structure and its parameters can be determined by pole placement technique. To illustrate its capability, the proposed controller is applied to control chaos in the fractional order unified system. Numerical simulations confirm the ability of the controller to reject external constant disturbances as well as its robustness against the variation of system parameters.

Applications of chaos control techniques to a three-species food chain

We achieve control of deterministic chaos in an ecosystem model, involving three first-order nonlinear differential equations with a control parameter, recently proposed by Hastings and Powell (HP) in order to describe the dynamical behavior of a three-species food chain. After identifying a chaotic attractor corresponding to a particular value of the parameter of this ecological model, we locate periodic saddle orbits embedded in it. By applying the Ott–Grebogi–Yorke (OGY) method of controlling chaos, which introduces small time-dependent perturbations on the system parameter, we stabilize two of the saddle orbits. Furthermore, we check the versatility of the OGY method, as the system behavior is allowed to switch between ‘no control’ and ‘control’ about one or other of different stabilized periodic orbits.

Applications of chaos control techniques to a three-species food chain

We achieve control of deterministic chaos in an ecosystem model, involving three first-order nonlinear differential equations with a control parameter, recently proposed by Hastings and Powell (HP) in order to describe the dynamical behavior of a three-species food chain. After identifying a chaotic attractor corresponding to a particular value of the parameter of this ecological model, we locate periodic saddle orbits embedded in it. By applying the Ott–Grebogi–Yorke (OGY) method of controlling chaos, which introduces small time-dependent perturbations on the system parameter, we stabilize two of the saddle orbits. Furthermore, we check the versatility of the OGY method, as the system behavior is allowed to switch between ‘no control’ and ‘control’ about one or other of different stabilized periodic orbits.

Control of three-synchronous-generator infinite-bus system by OGY method

Stability of a power system has been investigated in terms of chaos and bifurcation. In this paper, the OGY (Ott–Grebogi–Yorke) method for controlling chaos of three machines operating onto an infinite-bus system is investigated by computer simulations. The swing equation with the controlling input u is used. The OGY method is extended to the form in the six-dimensional space. The eight equilibrium points are obtained. The swing equation is normalized and transformed into a discrete-time state equation from which the control input is calculated. The time series of the phase angles of generators without the control input show the chaotic irregular motion and the step-out. The time series of the phase angle of generators with the control inputs by the OGY method show the stable motion. The phase angles are successfully controlled onto the unstable equilibrium points with the three unstable manifolds and the three stable manifolds. © 2005 Wiley Periodicals, Inc. Electr Eng Jpn, 151(2): 32–39, 2005; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/eej.20008

On the OGY Control of an Impact-Friction Oscillator

An earlier paper by the same authors examined the dynamic behavior of an impact-friction oscil lator. It was shown that a variety of highly complex behavior was possible including chaos, and the results of numerical simulation were verified experimentally Because friction is so common in mechanical applica tions and has a relatively profound effect on behavior, it is logical to consider whether friction forces can be used to advantage in such engineering systems. The Ott-Grebogi-Yorke (OGY) method is used here to control the impact oscillator using friction force as the control parameter.

COMBINATORIAL CONTROL OF GLOBAL DYNAMICS IN A CHAOTIC DIFFERENTIAL EQUATION

Controlling chaos has been an extremely active area of research in applied dynamical systems, following the introduction of the Ott, Grebogi, Yorke (OGY) technique in 1990 [Ott et al., 1990], but most of this research based on parametric feedback control uses local techniques. Associated with a dynamical system which pushes forward initial conditions in time, transfer operators, including the Frobenius–Perron operator, are associated dynamical systems which push forward ensemble distributions of initial conditions. In [Bollt, 2000a, 2000b; Bollt & Kostelich, 1998], we have shown that such global representations of a discrete dynamical system are useful in controlling certain aspects of a chaotic dynamical system which could only be accessible through such a global representation. Such aspects include invariant measure targeting, as well as orbit targeting. In this paper, we develop techniques to show that our previously discrete time techniques are accessible also to a differential equation. We focus on the Duffing oscillator as an example. We also show that a recent extension of our techniques by Góra and Boyarsky [1999] can be further simplified and represented in a convenient and compact way by using a tensor product.

Chaos control of Bonhoeffer–van der Pol oscillator using neural networks

This paper addresses the control of chaos using a neural network for a continuous time dynamical system. The neural network is trained on both the Ott–Grebogi–Yorke (OGY) control algorithm and the Pyragas's delayed feedback control algorithm. The system considered for this study is a Bonhoeffer–van der Pol (BVP) oscillator. A feed-forward backpropagating neural network is used for the control application. It is found that the control effected by the neural network trained on the OGY control algorithm results in smaller control transients than when the control is effected directly by the OGY algorithm itself. The control transients are of the same order in the case of the Pyragas method.

Unstable equilibrium point in chaotic domain-wall motion and Ott–Grebogi–Yorke control

A method for finding the unstable equilibrium points in Bloch wall motion is proposed, which is important for controlling the chaotic domain-wall motion by using the Ott–Grebogi–Yorke (OGY) method. The dynamics of Bloch wall motion are expressed by a nonlinear differential equation with the terms of inertia, damping, restoring, and an external magnetic drive force. An equation is transformed into the difference equations by following the OGY method, approximating linearly around an unstable equilibrium point (a saddle point), and adding a controlling input. The unstable equilibrium points are obtained by using the return map and the condition of hyperbolic fixed point. The time series of domain-wall motion successfully controlled on the unstable equilibrium points by the OGY method is shown.

Controlling chaos of nonlinear domain-wall motion

The two-step Ott–Grebogi–Yorke (OGY) method and the prediction OGY method for controlling chaos of magnetic domain-wall motion are proposed to improve the long settling time in the original OGY method. In the two-step OGY method, a magnetic domain wall is first moved on a periodic orbit and the OGY method is used when the orbit approaches a saddle point. In the prediction OGY method, the motion of the domain wall is predicted before the OGY method is applied. An attractor in the state space can be reconstructed by using the time series of the domain-wall motion. The near future can be predicted even in the chaotic system, because the short time developments of the neighborhood system of a predictee in the attractor are not so different from each other. The settling time of the improved OGY methods is 1/5–1/30 times as long as that of the original OGY method.

On the optimality of the Ott-Grebogi-Yorke control scheme

Some of the characteristics of the Ott-Grebogi-Yorke (OGY) control technique are presented as applied to nonlinear flows, as distinct from nonlinear maps. Specifically, we consider the case where the magnitude of the control parameter varies in time within each control cycle in proportion to a given function referred to as a basis function. The choice of the basis function is shown to influence the basin of convergence for a given level of parameter variation in the OGY controller. An algorithm for designing the optimal basis function is presented. The optimal basis function is shown to be defined by a step function with potentially several jumps, thus revealing the intrinsic power of a standard OGY technique that uses a single step function as a basis function. Two numerical applications of the optimal design technique to a Duffing oscillator are also presented to show that the standard OGY technique may be significantly improved by making an optimal choice of a basis function. Copyright

System identification for the Ott-Grebogi-Yorke controller design

A technique to compute the linearized Poincar\'e map and the sensitivity vector required for the implementation of an Ott-Grebogi-Yorke (OGY) controller from experimental data is presented. Unlike previous methods, the linearized map and the sensitivity vector are computed using only data collected over a single period of the limit cycle. The main advantage of the present method is the fact that it eliminates the long waiting period required to design an OGY controller using previously known methods. Numerical examples are presented using the Duffing oscillator with random perturbations to simulate noisy experimental data.

Controlling chaotic dynamical systems

We review the major ideas involved in the control of chaos. We present the Ott-Grebogi-Yorke (OGY) method of controlling chaos, which is a particular case of the pole placement technique, but which is the one leading to the shortest time to achieve the control of chaotic systems. Implementation using only measured time series in experimental settings is also described.

Quasicontinuous control of a bronze ribbon experiment using time-delay coordinates

We investigate quasicontinuous versions of the Ott-Grebogi-Yorke (OGY) control method in a bronze ribbon experiment using time-delay coordinates for the reconstruction of the attractor. We apply as quasicontinuous control methods the local control method and the minimal expected deviation method. As is known for the original OGY method with time-delay coordinates, values of the control parameter at previous times appear in the linearized dynamics. We discuss two possible ways to derive from this linearization feedback control formulas. These are the extended state space approach and a modified control requirement that demands stabilization not in the next time step but after w+1 time steps, where w is the number of preceding parameter dependences of the linearized dynamics. We show theoretically and demonstrate in the experiment that for the quasicontinuous control methods considered in this paper only the modified control requirement shows satisfying results in the control experiment.

Controlling chaos in high dimensions

We review the major ideas involved in the control of chaos by considering higher dimensional dynamics. We present the Ott-Grebogi-Yorke (OGY) method of controlling chaos to achieve time periodic motion by utilizing only small feedback control. The time periodic motion results from the stabilization of unstable periodic orbits embedded in the chaotic attractor. We demonstrate that the OGY method, also applicable to high dimensions, is a particular case of the pole placement technique, and we argue that it is the one leading to shortest time to achieve control. Implementation using only a measured time series in experimental situations is described.