Surrogates for finding unstable periodic orbits in noisy data sets

Recently, searches for unstable periodic orbits in biological and medical applications have become of interest. The motivations for this research range, in order of ascending complexity, from efforts to understand the dynamics of simple sensory neurons, through speculations regarding neural coding, to the hopeful development of new diagnostic and/or control techniques for cardiac and epileptic pathologies. Biological and medical data are, however, noisy and nonstationary. Findings of unstable periodic orbits in such data thus require convincing assessments of their statistical significance. Such tests are accomplished by comparison with surrogate data files designed to test an appropriate null hypothesis. In this paper we test surrogates generated by three different algorithms against correlated noise as well as stable periodic orbits. One of the surrogates is new, and has been specifically designed to preserve the shape of the attractor. We discuss the suitability of these surrogates and argue that the simple shuffled one correctly tests the appropriate null hypothesis.