Narrow Results

Combining a modified Independent Component Analysis (ICA) and a feedback cancellation of nonlinear terms, the approach of engineering control can efficiently govern a noisy chaotic system. The methodology is easy to comprehend and to implement, but previous knowledge of the system dynamics is needed.

Delayed feedback control (DFC) is a powerful method for stabilizing unstable periodic orbits embedded in chaotic attractors, which uses a small control input fed by the difference between the current state and the delayed state. One drawback of the DFC is known as the odd number limitation; that is, DFC can never stabilize a target unstable fixed point of a chaotic discrete-time system, if the Jacobian of its linearized system around the unstable fixed point has an odd number of real eigenvalues greater than unity. To overcome it, in this paper we propose a dynamic DFC method using output measurements of the chaotic systems. The proposed dynamic DFC is realized by using an output feedback controller with a minimal-order observer that has the least order for estimating the state of the chaotic system from the control input and the output measurements. In addition to the design procedure of the controller, we derive a necessary and sufficient condition for the existence of such controllers.

In this paper, we propose a general method for controlling chaos in a nonlinear dynamical system containing a state-dependent switch. The pole assignment for the corresponding discrete system derived from such a nonsmooth system via Poincaré mapping works effectively. As an illustrative example, we consider controlling the chaos in the Rayleigh-type oscillator with a state-dependent switch, which is changed by the hysteresis comparator. The unstable one- and two-periodic orbits in the chaotic attractor are stabilized in both numerical and experimental simulations.

This paper investigates the complex dynamics, synchronization and control of chaos in a system of strongly connected Wilson–Cowan neural oscillators. Some typical synchronized periodic solutions are analyzed by using the Poincaré mapping method, for which bifurcation diagrams are obtained. It is shown that topological change of the synchronization mode is mainly caused and carried out by the Neimark–Sacker bifurcation. Finally, a simple feedback control method is presented for stabilizing an in-phase synchronizing periodic solution embedded in the chaotic attractor of a higher-dimensional model of such coupled neural oscillators.

We study chaotic maps with multiple coexisting strange attractors and show how such systems can be controlled. To this end, a control scheme is proposed which is capable of stabilizing a desired motion within one strange attractor as well as taking the system dynamics from one strange attractor to another. To demonstrate the given control scheme, several examples are considered.

In this paper, we present a flatness based control approach for the stabilization and tracking problem, for the well-known Chua chaotic circuit, that includes an additional input. We introduce two feedback controller design options for the set-point stabilization and the trajectory tracking problem: a direct pole placement approach, and Generalized Proportional Integral (GPI) approach based only on measured inputs and outputs.

In this paper, a criteria of suppressing chaos for a kind of nonlinear oscillators is established by the theory of the strange attractor. The oscillators considered include Duffing, van der Pol, Duffing–van der Pol and pendulum. According to this criteria, we analyze the phase effect using two methods, one by adding the second external force term and the other by adding parametric excitation, both of which may be used to suppress chaos in the systems. Some examples are used to illustrate the validity of the criteria and the importance of phase effect in suppressing chaos.

In this letter, we reconsider the problem of controlling chaos in the well-known Lorenz system. Firstly, the difficulty in controlling the Lorenz system is discussed in the general strict-feedback form. Then, singularity-free adaptive control is presented for the Lorenz system with three key parameters unknown by exploiting the physical property of the system using decoupled backstepping design. The proposed controller guarantees the asymptotic convergence of the output and the boundedness of all the signals in the closed-loop system. Simulation results are conducted to show the effectiveness of the approach.

Departing from the OGY method reported in 1990, many methodologies for chaos control have been proposed. A major criticism is that most of them are merely straightforward applications of methodologies borrowed from feedback control theory. In fact, an authentic chaos control methodology should rely on the underlying structure (e.g. topology, dynamics, etc.) of chaotic behavior in order to provide simple and successful feedback control algorithms. This brief paper focuses on this objective; namely, to exploit chaos structure to design feedback controllers aimed to eliminate chaos. The well-known Duffing oscillator is used to show how the mechanisms leading to chaotic behavior can be destroyed by means of a simple feedback control. Specifically, it is shown that injected damping, via feedback control, is able to eliminate the transverse homoclinic orbit responsible for the chaotic behavior of the Duffing oscillator.

This paper presents a simple numerical scheme for estimating the attraction region of a fixed point in one-dimensional discrete-time chaotic systems controlled by the delayed-feedback method. This scheme employs the well-known linear matrix inequality approach. A systematic procedure for estimating the region is provided, and numerical examples are used to validate the results.

In this paper, we are concerned with the chaotic behavior of a class of control systems described by partial differential equations. By means of finite dimensional output feedback control, it is shown that the flow on the global attractor is topologically equivalent to that of a finite dimensional difference system. In addition, an example is given to illustrate that the chaotic behavior of the flow on the global attractor can be determined by computing the Lyapunov exponent of an associated finite dimensional difference system.

In this paper we investigate several methods for controlling chaos in Aihara's chaotic neuron model. We first discuss the stability of exponential feedback control method for this model. To obviate predetermining the unstable periodic orbits of the system, two other methods are developed. We analyze why the conventional delayed feedback control method cannot be employed here, and then give a modified form for recursive delayed feedback control and apply it to control chaos in this model. To obtain high-periodic orbits more easily, a delayed exponential feedback control method is proposed, by which we can obtain different periodic orbits by changing parameters. Computer simulations show good control effects and robustness against noise.

This paper studies stabilization of low-period unstable periodic orbits (UPOs) in a simplified model of a current mode H-bridge inverter. The switching of the inverter is controlled by pulse-width modulation signal depending on the sampled inductor current. The inverter can exhibit rich nonlinear phenomena including period doubling bifurcation and chaos. Our control method is realized by instantaneous opening of inductor at a zero-crossing moment of an objective UPO and can stabilize the UPO instantaneously as far as the UPO crosses zero in principle. Typical system operations can be confirmed by numerical experiments.

We have demonstrated that the chaotic circuit with a switching delay is modeled by a return map, and a controller for the suppression of chaos is proposed. A circuit representing a controller stabilizing a period-1 unstable periodic orbit in an interrupted electric circuit with a certain switching delay is also discussed.

Chaotic scattering in open Hamiltonian systems is relevant for different problems in physics. Particles in such kind of systems can exhibit both bounded or unbounded motions for which escapes from the scattering region can take place. This paper analyzes how to control the escape of the particles from the scattering region in the presence of noise. For that purpose, we apply the partial control technique to the Hénon–Heiles system, which implies that we need to use a control smaller than the noise present in the system. The main finding of our work is the successful control of the particles in the scattering region with a control smaller than noise. We have also analyzed and compared the escapes time of orbits in the scattering region for different situations. Finally, we believe that our results might contribute to a better understanding of both chaotic scattering phenomena and the application of the partial control technique to continuous dynamical systems.

We report on the tendency of chaotic systems to be controlled onto their unstable periodic orbits in such a way that these orbits are stabilized. The resulting orbits are known as cupolets and collectively provide a rich source of qualitative information on the associated chaotic dynamical system. We show that pairs of interacting cupolets may be induced into a state of mutually sustained stabilization that requires no external intervention in order to be maintained and is thus considered bound or entangled. A number of properties of this sort of entanglement are discussed. For instance, should the interaction be disturbed, then the chaotic entanglement would be broken. Based on certain properties of chaotic systems and on examples which we present, there is further potential for chaotic entanglement to be naturally occurring. A discussion of this and of the implications of chaotic entanglement in future research investigations is also presented.

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.

Based on the theory of symbolic dynamical systems, we propose a novel computation method to locate and stabilize the unstable periodic points (UPPs) in a two-dimensional dynamical system with a Smale horseshoe. This method directly implies a new framework for controlling chaos. By introducing the subset based correspondence between a planar dynamical system and a symbolic dynamical system, we locate regions sectioned by stable and unstable manifolds comprehensively and identify the specified region containing a UPP with the particular period. Then Newton’s method compensates the accurate location of the UPP with the regional information as an initial estimation. On the other hand, the external force control (EFC) is known as an effective method to stabilize the UPPs. By applying the EFC to the located UPPs, robust controlling chaos is realized. In this framework, we never use ad hoc approaches to find target UPPs in the given chaotic set. Moreover, the method can stabilize UPPs with the specified period regardless of the situation where the targeted chaotic set is attractive. As illustrative numerical experiments, we locate and stabilize UPPs and the corresponding unstable periodic orbits in a horseshoe structure of the Duffing equation. In spite of the strong instability of UPPs, the controlled orbit is robust and the control input retains being tiny in magnitude.

The close relationship between chaos and dynamical systems leads to naturally consider the iteration processes that are related to dynamical systems of fixed point theory. From this natural relationship, the control of chaos that occurs in fixed point iteration dynamics will be the main focus of the article. To achieve this goal, analytical solutions are obtained and used to control chaos that occurs at unstable fixed points of multistep iteration process. Later, we show an effective regime for the parameters of multistep iteration. To illustrate this claim, well-known special cases of multistep iteration process by Noor, Ishikawa, Mann, Krasnoselskij, Picard iteration processes are introduced. In particular, among these iterations, the Noor iteration process is studied in detail in terms of controlled chaos. The Lyapunov exponent is used to estimate the stability and unstability of fixed points and periods that generate chaos in iteration processes. Finally, with the help of MATLAB program, all these results are shown on logistic and cubic equations with chaotic properties.