Periodic Orbits Analysis


Unstable periodic orbits
Scholarpedia 2(2):1353 (2007)
Paul So

Periodic Orbits: A New Language for Neuronal Dynamics
Biophysical Journal 74, 2776-2785 (1998).
Paul So, Joseph T. Francis, Theoden I. Netoff, Bruce J. Gluckman, and Steven J. Schiff

A new nonlinear dynamical analysis is applied to complex behavior from neuronal systems. The conceptual foundation of this analysis is the abstraction of observed neuronal activities into a dynamical landscape characterized by a hierarchy of "unstable periodic orbits" (UPOs). UPOs are rigorously identified in data sets representative of three different levels of organization in mammalian brain. An analysis based on UPOs affords a novel alternative to decode, predict, and control these neuronal systems.

Related Article: Mastering the Nonlinear Brain [James Glanz, Science, 277, 1758 (1997)]

To down load a copy of this paper click here (bioj74_2776.pdf)(457K) (Text and Figures in pdf format)

To down load only the color figures from this paper click here (bioj74_figs.pdf)(640K) (Color figures in pdf format)

To down load the chinese translation of this paper published in Advances in Mechanics, 29 121 (1999) by Yang Qian click here (ch_upo.pdf) (1.2M)


Extracting Unstable Periodic Orbits from Chaotic Time Series Data
Physical Review E 55, 5398 (1997).
Paul So, Edward Ott, Tim 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.

 

To down load a copy of this paper click here (pre55_5398.pdf)(790K) (Text & Figures in pdf format)


Detecting Unstable Periodic Orbits in Chaotic Experimental Data
Phys. Rev. Lett. 76, no.25, 4705 (1996).
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.

 

To down load a copy of this paper click here (prl76_4705.pdf)(173K) (Text & Figures in pdf format)


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