Paul's Papers

+ Detecting Unstable Periodic Orbits in Chaotic Experimental Data

Paul So, Edward Ott, Steven J. Schiff, Daniel T. Kaplan, Tim Sauer, and Celso Grebogi


A new method is proposed for detecting unstable periodic orbits and their linear stability properties from chaotic experimental time series. Illustrative examples are presented for both numerically and experimentally generated time series. The statistical significance of the results is assessed using surrogate data.

This paper is published in Physical Review Letters, 4705 (June 1996).

To down load a copy of this paper click here (fixprl.tar.Z) (Text in Latex & Figures in PostScript)

+ Spectral Statistics for Quantum Chaos with Ray Splitting

R. N. Oerter, E. Ott, T. M. Antonsen, Jr., and P. So


We investigate the behavior of ray trajectories and solutions of the wave equation of two dimensional billiard-like systems with ray splitting. By ``ray splitting'' we mean the phenomenon whereby a ray incident on a sharp boundary leads to two or more rays traveling away from the boundary (e.g. a transmitted ray and a reflected ray). Billiard systems with the same overall shape, but with and without ray splitting boundaries present are examined and compared. It is found that, for the configurations considered, the level spacing distribution and the spectral rigidity for the case without ray splitting are intermediate between Poisson and Gaussian Orthogonal Ensemble (GOE) statistics, while the behavior with ray splitting is very close to GOE.

This paper appears in Physics Letters A (June 1996).

To down load a copy of this paper click here (raysplit.tex) (Text in LaTex only)

+ Quantum Chaos Experiments with and without Time Reversal Symmetry

Paul So, Steven M. Anlage, Edward Ott, and Robert N. Oerter


It hsas been predicted that in the semi-classical regime, the level statistics of a classically chaotic system corresponds to that of the Gaussian Unitary Ensemble (GUE) of random matrices when time reversal symmetry is broken. This paper presents the first experimental test of this prediction. The system employed is a microwave cavity containing a thin ferrite strip adjacent to one of the walls. When a sufficiently large magnetic field is applied to the ferrite (thus breaking the time-reversal symmetry) good agreement with GUE statistics is obtained. The transition from GOE (which applies in the absence of the applied field) to GUE is also investigated.

This paper appears in Physical Review Letters (May 1995).

To down load a copy of this paper click here (qpaper.tar.Z) (Text and Figures in PostScript)

+ Controlling Chaos Using Time Delay Coordinates via Stabilization of Periodic Orbits

Paul So and Edward Ott


Using time delay coordinates, we propose a method that stabilizes a desired periodic orbit with an arbitrary number of unstable directions. Similar to the original OGY control algorithm, the stabilization is done via small time dependent perturbations of an accessible control parameter. However, since the system in time delay coordinates will in general depend on past parameteric variations, our parameteric control law is constructed based on the combined dynamics of the "state-plus-parameter" system.

This paper is published in Physical Review E, vol 51, 2955 (1995).

To down load a copy of this paper click here (zcontrol.tar.Z) (Text in Latex & Figures in PostScript)


Observing Chaos: Deducing and Tracking the State of a Chaotic System from Limited Observation

Paul So, Edward Ott, and W.P. Dayawansa


A method is proposed whereby the full state vector of a chaotic system can be reconstructed and tracked using only the time series of a single observed scalar. It is assumed that an accurate mathematical description of the system is available. Noise effects on the procedure are investigated using as an example a kicked mechanical system which results in a four-dimensional dissipative map.

This paper appears in Physical Review E, vol. 49, 2650, (1994).

A short version appears in Physics Letters A, vol. 176, 421, (1993).

To down load a copy of this paper click here (observer.tar.Z) (Text and Figures in PostScript)

+ Detecting Dynamical Interdependence and Generalized Synchrony through Mutual Prediction in a Neural Ensemble

Steven J. Schiff, Paul So, Taeun Chang, Robert E. Burke, and Tim Sauer


A method to characterize dynamical interdependence among nonlinear systems is derived based on mutual nonlinear prediction. Systems with nonlinear correlation will show mutual nonlinear prediction when standard analysis with linear cross correlation might fail. Mutual nonlinear prediction also provides information on the directionality of the coupling between systems. Furthermore, the existence of bidirectional mutual nonlinear prediction in unidirectionally coupled systems implies generalized synchrony. Numerical examples studied include three classes of unidirectionally coupled systems: systems with identical parameters, nonidentical parameters, and stochastic driving of a nonlinear system. This technique is then applied to the activity of motoneurons within a spinal cord motoneuron pool. The interrelationships examined include single neuron unit firing, the total number of neurons discharging at one time as measured by the integrated monosynaptic reflex, and intracellular measurements of integrated excitatory postsynaptic potentials (EPSPs). Dynamical interdependence, perhaps generalized synchrony, was identified in this neuronal network between simultaneous single unit firings, between units and the population, and between units and intracellular EPSPs.

+ Extracting Unstable Periodic Orbits from Chaotic Time Series Data

Paul So, Edward Ott, Time Sauer, Bruce J. Gluckman, Celso Grebogi, and Steven J. Schiff


A general nonlinear method to extract unstable periodic orbits from chaotic time series is proposed. By utilizing the estimated local dynamics along a trajectory, we devise a transformation of the time series data such that the transformed data is concentrated on the periodic orbits. Thus, one can extract unstable periodic orbits from a chaotic time series by simply looking for peaks in a finite grid approximation of the distribution function of the transformed data. Our method is demonstrated using data from both numerical and experimental examples, including neuronal ensemble data from mammalian brain slices. The statistical significance of the results in the presence of noise is assessed using surrogate data.

This paper is published in Physical Review E, vol. 55, pg. 5398 (May 1997).

To down load a copy of this paper
here (polong.pdf ~ 790K)