Applying the method of surrogate data to cyclic time series

The surrogate data methodology is used to test a given time series for membership of specific classes of dynamical systems. Currently, there are three algorithms that are widely applied in the literature. The most general of these tests the hypothesis of nonlinearly scaled linearly filtered noise. However, these tests and the many extensions of them that have been suggested are inappropriate for data exhibiting strong cyclic components. For such data it is more natural to ask if there exist any long term (of period longer than the data cycle length) determinism. In this paper we discuss existing techniques that attempt to address this hypothesis and introduce a new approach. This new approach generates surrogates that are constrained (i.e., they look like the data) and for cyclic time series tests the null hypothesis of a periodic orbit with uncorrelated noise. We examine various alternative implementations of this algorithm, applying it to a variety of known test systems and experimental time series with unknown dynamics.