Kevin Dolan

Counting unstable periodic orbits in noisy chaotic systems: A scaling relation connecting experiment with theory.

The experimental detection of unstable periodic orbits in dynamical systems, especially those which yield short, noisy or nonstationary data sets, is a current topic of interest in many research areas. Unfortunately, for such data sets, only a few of the lowest order periods can be detected with quantifiable statistical accuracy. The primary observable is the number of encounters the general trajectory has with a particular orbit. Here we show that, in the limit of large period, this quantity scales exponentially with the period, and that this scaling is robust to dynamical noise. (c) 1998 American Institute of Physics.

Chapter 4 Detecting unstable periodic orbits in biological systems

Publisher Summary This chapter detects unstable periodic orbits (UPOs) in biological systems. In both driven and autonomous chaotic systems, the motion (or orbit) of the system in phase space can at times be almost periodic. Unstable periodic orbits can also be used for the anticontrol (or maintenance) of chaos. By analyzing the UPOs in the region of either transition, small perturbations can be used to lock the systems dynamics into one basin of attraction. The study of UPOs has enabled researchers in the field of nonlinear dynamics to work with short and noisy sets of nonlinear data. This allows the study of biological systems, which noise and nonstationarity had previously made intractable. The methods described in the chapter are very robust to the presence of dynamical noise and can operate on extremely short data files. They are also demonstrated to be capable of distinguishing between dynamical data and colored noise, something that is difficult for more traditional nonlinear techniques even when very long data files are available.

Spatiotemporal Distributions of Unstable Periodic Orbits in Noisy Coupled Chaotic Systems

Techniques for detecting encounters with unstable periodic orbits (UPOs) have been very successful in the analysis of noisy, experimental time series. We present here a technique for applying the topological recurrence method of UPO detection to spatially extended systems. This approach is tested on a network of diffusively coupled chaotic Rossler systems, with both symmetric and asymmetric coupling schemes. We demonstrate how to extract encounters with UPOs from such data, and present a preliminary method for analyzing the results and extracting dynamical information from the data, based on a linear correlation analysis of the spatiotemporal occurrence of encounters with these low period UPOs. This analysis can provide an insight into the coupling structure of such a spatially extended system.

A New Surrogate for Experimental Data Analysis

We show here that the commonly used surrogate generating techniques based on phase randomization produce a population of surrogates that are not consistent with the null hypothesis that they are designed to test. We present a new surrogate generating method, based on digital filtering techniques. This surrogate algorithm has significant advantages over the most commonly used techniques, in that it provides a more robust statistical test by producing an entire population of surrogates that are consistent with the null hypothesis. The new surrogate is tested against an autoregression process and the Rossler system.