Physical Review E 61, 5045-5056 (2000) Abstract of the article

Influence of stable Floquet exponents on time-delayed feedback control

Wolfram Just 1 Ekkehard Reibold 2 Krzysztof Kacperski 2,3 Piotr Fronczak 2,4 Janusz A. Hołyst 3,4 and Hartmut Benner 2
1 School of Mathematical Sciences, Queen Mary & Westfield College, Mile End Road, London E1 4NS, United Kingdom
2 Institut für Festkörperphysik, Technische Universität Darmstadt, Hochschulstrasse 6, D-64289 Darmstadt, Germany
3 Max Planck Institute for Physics of Complex Systems, Nöthnitzer Strasse 38, D-01187 Dresden, Germany
4 Faculty of Physics, Warsaw University of Technology, Koszykowa 75, PL-00-662 Warsaw, Poland

The performance of time-delayed feedback control is studied by linear stability analysis. Analytical approximations for the resulting eigenvalue spectrum are proposed. Our investigations demonstrate that eigenbranches that develop from the stable Lyapunov exponents of the free system also have a strong influence on the control properties, either by hybridization or by a crossing of branches which interchanges the role of the leading eigenvalue. Our findings are confirmed by numerical analysis of two particular examples, the Toda and the Rössler models. More important is the verification by actual electronic circuit experiments. Here, the observed reduction of control domains can be attributed to these additional eigenvalue branches. The investigations lead to a thorough analytical understanding of the stability properties in time-delayed feedback systems.