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INFLUENCE OF THE IMPULSE LENGTH IN POINCARE-BASED CHAOS CONTROL

A new stability analysis of the two elementary Poincaŕ e-based chaos control schemes, OGY and difference control, 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, similar to that of [Just et al. 1997] for the time-continuous Pyragas scheme. Here the expansion dynamics in vicinity of the orbit is assumed to be homogeneous along the orbit, which is of course a simplification to have a model situation to obtain qualitative results. As an unexpected result, while for OGY control the influence of the impulse length is marginal, difference control is shown to fail when the impulse length is taken longer than one half of the orbit length [Claussen 1998; Claussen 2002; Claussen 2003]. This impulse length limitation reproduces the experiments by [Mausbach 1998]. The linear model approach is extended to the case where the local Ljapunov exponent varies along the orbit to be stabilized; the dynamics of the system with applied OGY control can be piecewise integrated as before to show that the qualitative result survives if the simplifying assumption of a constant local Ljapunov exponent is relaxed.