Stability borders of feedback control of delayed measured systems

When stabilization of unstable periodic orbits or fixed points by the method given by Ott, Grebogi, and Yorke (OGY) must be based on a measurement delayed by τ orbit lengths, the performance of unmodified OGY method is expected to decline. For experimental considerations, it is desired to know the range of stability with minimal knowledge of the system. We find that unmodified OGY control fails beyond a maximal Lyapunov number of λ_max=1+(1∕τ). In this paper the area of stability is investigated both for OGY control of known fixed points and for difference control of unknown or inaccurately known fixed points. An estimated value of the control gain is given. Finally we outline what extensions must be considered if one wants to stabilize fixed points with Lyapunov numbers above λ_max.