On the Mechanism of Time--Delayed Feedback Control

The Pyragas method for controlling chaos is investigated in detail from the experimental as well as theoretical point of view. We show by an analytical stability analysis that the revolution around an unstable periodic orbit governs the success of the control scheme. Our predictions concerning the transient behaviour of the control signal are confirmed by numerical simulations and an electronic circuit experiment.Comment: 4 pages, REVTeX, 4 eps-figures included Phys. Rev. Lett., in press also available at http://athene.fkp.physik.th-darmstadt.de/public/wolfram.htm

Restricted feedback control of one-dimensional maps

Dynamical control of biological systems is often restricted by the practical constraint of unidirectional parameter perturbations. We show that such a restriction introduces surprising complexity to the stability of one-dimensional map systems and can actually improve controllability. We present experimental cardiac control results that support these analyses. Finally, we develop new control algorithms that exploit the structure of the restricted-control stability zones to automatically adapt the control feedback parameter and thereby achieve improved robustness to noise and drifting system parameters.Comment: 29 pages, 9 embedded figure

Approaching nonlinear dynamics by studying the motion of a pendulum I: Observing trajectories in state space

.-A phenomenological introduction into concepts of nonlinear dynamics is given. A driven pendulum is used as an appropriate model to demonstrate characteristic features of the dynamics of nonlinear systems. The mechanical and electronic setup of the pendulum is described and quantified in detail, and different techniques to visualize the motion in state space are discussed. Finally, a 'road map' in parameter space is calculated which provides a tool to find special scenarios by choosing appropriate driving amplitudes and frequencies. 1 Introduction Nonlinear systems have been a topic of general physics ever since, but progress in this field was slow, as in most cases analytical access is nearly impossible or is restricted to very special systems and parameter sets. Powerful mathematical tools such as the superposition principle can not be applied to these systems and so nonlinear systems have -- apart from some pioneering work of Poincar'e and others [Poincar'e, 1899]-- mainly been st..