International Journal of Bifurcation and Chaos

Bifurcation control deals with modification of bifurcation characteristics of a parameterized nonlinear system by a designed control input. Typical bifurcation control objectives include delaying the onset of an inherent bifurcation, stabilizing a bifurcated solution or branch, changing the parameter value of an existing bifurcation point, modifying the shape or type of a bifurcation chain, introducing a new bifurcation at a preferable parameter value, monitoring the multiplicity, amplitude, and/or frequency of some limit cycles emerging from bifurcation, optimizing the system performance near a bifurcation point, or a combination of some of these objectives. This article offers an overview of this emerging, challenging, stimulating, and yet promising field of research, putting the main subject of bifurcation control into perspective.

In this paper, the problem of making a stable nonlinear autonomous system chaotic or enhancing the existing chaos of an originally chaotic system by using a small-amplitude feedback controller is studied. The designed controller is a linear feedback controller composed with a nonlinear modulo or sawtooth function, which can lead to uniformly bounded state vectors of the controlled system with positive Lyapunov exponents, thereby yielding chaotic dynamics. We mathematically prove that the controlled system is indeed chaotic in the sense of Li and Yorke. A few potential applications of the new chaotification algorithm are briefly discussed.

Tracking unstable periodic states first introduced in [Schwartz & Triandaf, 1992] is the process of continuing unstable solutions as a systems parameter is varied in experiments. The tracked dynamical objects have been periodic saddles of well-defined finite periods. However, other saddles, such as chaotic saddles, have not been successfully "tracked," or continued. In this paper, we introduce a new yet simple method which can be used to track chaotic saddles in dynamical systems, which allows an experimentalist to sustain chaotic transients far away from crisis parameter values. The method is illustrated on a periodically driven CO₂ laser.

A ball bouncing on a vibrating plate is surely one of the simplest physical systems that can produce chaotic motion, yet it is a prototype for a large class of mechanical systems. We wish to design a control algorithm for the plate motion so that the ball, starting at rest on the plate, can be bounced up to a prescribed periodic orbit, and maintained bouncing there. Keeping the amplitude of the vibrating plate fixed, we use the frequency of the plate as a control input. We examine two approaches for designing the controller. The first is classical LQR design and the second is a variant of a "greedy" method which only looks a short distance ahead in time. Using both of these methods we attempt control by linearizing an analytic model, the so-called high-bounce approximation of the ball map, and also by using a data-derived approximation to the true system. These approximate controllers are applied to a more accurate continuous system model of the bouncing ball. Encouragingly, both controllers perform well in spite of the approximations in their construction. The greedy control method appears to be more robust in certain situations.

Chaos control has been applied to control atrial fibrillation in humans. Results are presented on the application and evaluation of chaos control for slowing and regularizing local electrical activation of the right atrium of humans during induced atrial fibrillation.

A simple numerical experiment on Fitzhugh–Nagumo equations demonstrates that time-delay feedback methods can stabilize regular behavior in a paced, excitable oscillator without using large applied pulses. The method is robust against slow variations in the pacing rate and can be used to track an orbit into a parameter regime where it is unstable. Moreover, the method requires no information about the desired orbit other than period, which is determined directly by the observation of the pacing signal. Results are presented for feedback stabilization of a nonspiking orbit in the rapid pacing regime.

A novel adaptive time-delayed control method is proposed for stabilizing inherent unstable periodic orbits (UPOs) in chaotic systems with unknown parameters. We first explore the inherent properties of chaotic systems and use the system state and time-delayed system state to form an asymptotically stable invariant manifold so that when the system state enters the manifold and stays in it thereafter, the resulting motion enables the stabilization of the desired UPOs. We then use the model following concept to construct an identifier for the estimation of the uncertain system parameters. We shall prove that under the developed scheme, the system parameter estimates will converge to their true values. The effectiveness of the method is confirmed by computer simulations.

The problem of adaptive output feedback control aimed at stabilization of a (periodic or chaotic) goal trajectory is considered. Advantages and drawbacks of chaos control method based on linearization of Poincaré map (first proposed by Ott, Grebogi and Yorke in 1990) are discussed. It is suggested that the recurrence of the goal trajectory is the key property for applicability of approach. Algorithms of adaptive control based on linearization of controlled Poincaré map and method of goal inequalities are proposed. It is shown that stabilization of recurrent trajectories is possible under additional controllability-like and observability-like conditions. Examples of stabilization of periodic and chaotic trajectories for forced brusselator and Rössler systems are studied by computer simulations.

In this paper the problem of local exponential stability of periodic orbits in a general class of forced nonlinear systems is considered. Some lower bounds for the degree of local exponential stability of a given periodic solution are provided by mixing results concerning the analysis of linear time-varying systems and the real parametric stability margin of uncertain linear time-invariant systems. Although conservative with respect to the degree of stability obtainable via the Floquet-based approach, such lower bounds can be efficiently computed also in cases where the periodic solution is not exactly known and the design of a controller ensuring a satisfactory transient behavior is the main concern. The main features of the developed approach are illustrated via two application examples.

Pattern and turbulence control and synchronization are investigated by considering the model of the complex Ginzburg–Landau equation. It is shown that in fully-developed turbulence region, when a large number of positive Lyapunov exponents exists, in the presence of a large gradient force, we can drive the system to various desired spatiotemporal patterns, and control and synchronize violent turbulence by applying convenient boundary control technique.

Let X_i = F (X_i-1), where F is chaotic on a domain Ω of Rⁿ. If F becomes insensitive, i.e. gives the same response, H, in an open subset of D of Ω, then F engenders locally stable periodic orbits. We discussed the relationship between F, D, H and the periods of these orbits. The idea of local defect might be of use in discussing chaos in health and disease.

Several control design techniques applied to nonlinear dynamics and chaos assume that a model of the uncontrolled dynamics is available. The extent to which the controller performance is influenced by the model largely depends on many factors such as control effort and the control objective. It is suggested that model quality and control gains, which are strongly related to the control effort, define an uncertainty region in state space around the goal dynamics. For the unrealistic, but frequently assumed case in which model accuracy and admissible control effort are unlimited and there is no noise, the uncertainty region reduces to zero. But what happens when the model is realistically inaccurate, the maximum control effort is limited and the system state is observed with some noise? The objective of this paper is to discuss some issues related to model-based control techniques for nonlinear dynamical systems and to try to shed some light on questions like: How does model accuracy influence control performance? How can model inaccuracies be compensated for in the design? Which limits exist for such compensation? Extensive simulations are carried out and the results suggest that the requirements on nonlinear models might be different if the final objective is the reconstruction of nonlinear dynamics or controller design, and that even linear models can be used in some control design problems.

A natural process can be deterministically modeled if solutions from its mathematical model stay close to the ones produced by nature. The mathematical model, however, is not exact due to imperfections of the natural system. We describe, in this paper, that there exists a class of models of chaotic processes, for which severe obstruction to deterministic modeling may arise. In particular, such obstruction may occur when unstable periodic orbits embedded in the chaotic invariant set have a distinct number of unstable directions, a type of nonhyperbolicity called unstable-dimension variability. We make these ideas concrete by investigating a class of deterministic models: chaotic systems with an invariant subspace such as systems of coupled chaotic oscillators. We show that unstable-dimension variability can occur in wide parameter regimes of these systems. The implications of our results to scientific modeling are discussed.