Stochastic tracking in nonlinear dynamical systems

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Abstract
In a previous paper [Phys. Rev. A 46, 7439 (1992)] we have introduced an alternative continuation method which does not require an analytical model, but only an experimental time series. Using a predictor-corrector technique, the method tracks a given unstable orbit through different bifurcation regimes by varying an accessible system parameter. In this method, the continuation parameter was varied deterministically. That is, the location of the parameter is chosen by the experimenter. In this paper we introduce a similar algorithm, but now the parameter is varied randomly. A correction procedure is introduced so that control of an unstable orbit is not lost as the parameter changes. Moreover, we show that the small-amplitude feedback-control technique used for correction allows large-amplitude bursts in the parameter. These features are useful to experimentalists for canceling drift in experiments, which is inevitable at some level.