Floquet Stability Analysis of Ott-Grebogi-Yorke and Difference Control

Stabilization of instable periodic orbits of nonlinear dynamical systems has been a widely explored field theoretically and in applications. The techniques can be grouped in time-continuous control schemes based on Pyragas, and the two Poincar\'e-based chaos control schemes, Ott-Gebogi-Yorke (OGY) and difference control. Here a new stability analysis of these two Poincar\'e-based chaos control schemes is given by means of Floquet theory. This approach allows to calculate exactly the stability restrictions occuring for small measurement delays and for an impulse length shorter than the length of the orbit. This is of practical experimental relevance; to avoid a selection of the relative impulse length by trial and error, it is advised to investigate whether the used control scheme itself shows systematic limitations on the choice of the impulse length. To investigate this point, a Floquet analysis is performed. For OGY control the influence of the impulse length is marginal. As an unexpected result, difference control fails when the impulse length is taken longer than a maximal value that is approximately one half of the orbit length for small Ljapunov numbers and decreases with the Ljapunov number.