Tracking controlled chaos: Theoretical foundations and applications

Targeting fixed-point solutions in nonlinear oscillators through linear augmentation

We propose a general strategy to stabilize the fixed points of nonlinear oscillators with augmented dynamics. By using this scheme, either the unstable fixed points of the oscillatory system or a new fixed point of the augmented system can be stabilized. The Lyapunov exponents are used to study the dynamical properties. This scheme is illustrated with a chaotic Lorenz oscillator coupled through an external linear dynamical system. The experimental demonstration of the proposed scheme to stabilize the fixed points is also presented.

Converging towards the optimal path to extinction

Extinction appears ubiquitously in many fields, including chemical reactions, population biology, evolution and epidemiology. Even though extinction as a random process is a rare event, its occurrence is observed in large finite populations. Extinction occurs when fluctuations owing to random transitions act as an effective force that drives one or more components or species to vanish. Although there are many random paths to an extinct state, there is an optimal path that maximizes the probability to extinction. In this paper, we show that the optimal path is associated with the dynamical systems idea of having maximum sensitive dependence to initial conditions. Using the equivalence between the sensitive dependence and the path to extinction, we show that the dynamical systems picture of extinction evolves naturally towards the optimal path in several stochastic models of epidemics.

Control of chaos: methods and applications in mechanics

A survey of the field related to control of chaotic systems is presented. Several major branches of research that are discussed are feed-forward ('non-feedback') control (based on periodic excitation of the system), the 'Ott-Grebogi-Yorke method' (based on the linearization of the Poincaré map), the 'Pyragas method' (based on a time-delayed feedback), traditional for control-engineering methods including linear, nonlinear and adaptive control. Other areas of research such as control of distributed (spatio-temporal and delayed) systems, chaotic mixing are outlined. Applications to control of chaotic mechanical systems are discussed.

Tracking unstable steady states and periodic orbits of oscillatory and chaotic electrochemical systems using delayed feedback control

Experimental results are presented on successful application of delayed-feedback control algorithms for tracking unstable steady states and periodic orbits of electrochemical dissolution systems. Time-delay autosynchronization and delay optimization with a descent gradient method were applied for stationary states and periodic orbits, respectively. These tracking algorithms are utilized in constructing experimental bifurcation diagrams of the studied electrochemical systems in which Hopf, saddle-node, saddle-loop, and period-doubling bifurcations take place.

Tracking sustained chaos: A segmentation method

We introduce a segmentation control method to sustain chaotic transients in dynamical systems. The sustained transient can be tracked as a system parameter is substantially varied, allowing sustained chaotic transients far away from crisis parameter values. The method is applied to a chaotic CO2 laser as well as to a hyperchaotic continuum mechanics model.

Quality factor control in a lasing microcavity model

We consider a dynamics model of lasing microcavities, a class of optical resonators (1-10 &mgr;m in diameter) used in microlasers and for optical coupling of optical fibers. Inside such a cavity light circulates around the perimeter and is trapped by internal reflection. This is known as "whispering gallery" or high-Q modes. The cavity is a deformable cylindrical (or spherical) dielectric and at certain deformations light can escape by refraction. The quality of the resonator or Q factor, is defined as Q=omegatau, where tau is the escape time and omega is the frequency of light. We show that by appropriately deforming the cavity, the Q factor can be controlled by prolonging or shortening the average length of time spent by light trajectories inside the cavity.

On a simple recursive control algorithm automated and applied to an electrochemical experiment

We review a simple recursive proportional feedback (RPF) control strategy for stabilizing unstable periodic orbits found in chaotic attractors. The method is generally applicable to high-dimensional systems and stabilizes periodic orbits even if they are completely unstable, i.e., have no stable manifolds. The goal of the control scheme is the fixed point itself rather than a stable manifold and the controlled system reaches the fixed point in d+1 steps, where d is the dimension of the state space of the Poincare map. We provide a geometrical interpretation of the control method based on an extended phase space. Controllability conditions or special symmetries that limit the possibility of using a single control parameter to control multiply unstable periodic orbits are discussed. An automated adaptive learning algorithm is described for the application of the control method to an experimental system with no previous knowledge about its dynamics. The automated control system is used to stabilize a period-one orbit in an experimental system involving electrodissolution of copper. (c) 1997 American Institute of Physics.