Limits of time-delayed feedback control

General features of stability domains for time-delayed feedback control exist, which can be predicted analytically. We clarify, why the control scheme with a single delay term can only stabilise orbits with short periods or small Lyapunov exponents, and derive a quantitative estimate. The limitation can be relaxed by employing multiple delay terms. In particular, the extended time delay autosynchronisation method is investigated in detail, Analytic calculations are in good agreement with results of numerical simulations and with experimental data from a nonlinear diode resonator, (C) 1999 Elsevier Science B.V.