Tracking unstable steady states by large periodic modulation of a control parameter in a nonlinear system

Phantom instabilities in adiabatically driven systems: dynamical sensitivity to computational precision

We study the robustness of dynamical phenomena in adiabatically driven nonlinear mappings with skew-product structure. Deviations from true orbits are observed when computations are performed with inadequate numerical precision for monotone, periodic, or quasiperiodic driving. The effect of slow modulation is to "freeze" orbits in long intervals of purely contracting or purely expanding dynamics in the phase space. When computations are carried out with low precision, numerical errors build up phantom instabilities which ultimately force trajectories to depart from the true motion. Thus, the dynamics observed with finite precision computation shows sensitivity to numerical precision: the minimum accuracy required to obtain "true" trajectories is proportional to an internal timescale that can be defined for the adiabatic system.

Chaos suppression in the parametrically driven Lorenz system

We predict theoretically and verify experimentally the suppression of chaos in the Lorenz system driven by a high-frequency periodic or stochastic parametric force. We derive the theoretical criteria for chaos suppression and verify that they are in a good agreement with the results of numerical simulations and the experimental data obtained for an analog electronic circuit.

Stabilization of unstable fixed points in the dynamics of a laser with feedback

We report theoretical and experimental results on the stabilization of unstable steady states in a CO2 laser with feedback. Periodic and chaotic oscillations have been suppressed by means of a control loop consisting of a high-pass filter, known as washout filter. Although the filter characteristics are determined on the basis of linear analysis, taking into account the problem of robustness with respect to parameter changes, the present control strategy provides a large attractive domain in the phase space.

Tracking controlled chaos: Theoretical foundations and applications

Tracking controlled states over a large range of accessible parameters is a process which allows for the experimental continuation of unstable states in both chaotic and non-chaotic parameter regions of interest. In algorithmic form, tracking allows experimentalists to examine many of the unstable states responsible for much of the observed nonlinear dynamic phenomena. Here we present a theoretical foundation for tracking controlled states from both dynamical systems as well as control theoretic viewpoints. The theory is constructive and shows explicitly how to track a curve of unstable states as a parameter is changed. Applications of the theory to various forms of control currently used in dynamical system experiments are discussed. Examples from both numerical and physical experiments are given to illustrate the wide range of tracking applications. (c) 1997 American Institute of Physics.