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Automatic Detection of Highly Organized Theta Oscillations in the Murine EEG

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Detecting unstable periodic orbits from transient chaotic time series

Dhamala¹, Lai, Kostelich

¹Department of Physics and Astronomy, University of Kansas, Lawrence, Kansas 66045 and Department of Mathematics, Arizona State University, Tempe, Arizona 85287, USA.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

|November 23, 2000

Summary

This study detects unstable periodic orbits in chaotic time series using recurrence times. The method reliably finds low-period orbits, even with noise, but high-period orbits remain undetectable.

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Area of Science:

Nonlinear dynamics
Chaos theory
Time series analysis

Background:

Transient chaotic time series are common in experimental data.
Detecting unstable periodic orbits (UPOs) is crucial for understanding chaotic systems.
Traditional methods struggle with noisy, finite experimental data.

Purpose of the Study:

To develop a method for detecting UPOs from experimental transient chaotic time series.
To analyze the probability of detecting UPOs and derive a scaling law.
To determine the limits of UPO detection in chaotic systems.

Main Methods:

Reconstruction of the state space from time series data.
Analysis of trajectory recurrence times within the reconstructed space.
Development and application of a probability scaling law.

Main Results:

The recurrence time method successfully detects low-period UPOs.
The method is robust to the presence of noise in the time series.
A scaling law was derived, quantifying the probability of UPO detection.
High-period UPOs were found to be practically undetectable.

Conclusions:

Recurrence time analysis is a viable strategy for UPO detection in experimental chaos.
The detectability of UPOs is strongly dependent on their period.
The derived scaling law provides a theoretical limit for UPO discovery from transient chaos.