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Morphology-Based Distinction Between Healthy and Pathological Cells Utilizing Fourier Transforms and Self-Organizing Maps

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A note on chaotic unimodal maps and applications.

C T Zhou¹, X T He, M Y Yu

¹Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, Beijing 100088, People's Republic of China.

Chaos (Woodbury, N.Y.)

|October 4, 2006

Summary

This study links chaotic kneading sequences from symbolic dynamics to linear maximum-length shift-register sequences. It provides evidence that maximum-length sequences are a subset of chaotic unimodal map sequences, enabling controlled binary sequence generation.

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Area of Science:

Dynamical Systems
Information Theory
Chaos Theory

Background:

Symbolic dynamics provides tools to analyze complex systems.
One-dimensional unimodal maps exhibit chaotic behavior.
Maximum-length shift-register sequences are fundamental in digital communications.

Purpose of the Study:

Investigate the relationship between chaotic kneading sequences and linear maximum-length shift-register sequences.
Determine if maximum-length sequences are a subset of chaotic map sequences.
Develop methods for controlling the generation of long binary sequences.

Main Methods:

Utilizing the word-lift technique from symbolic dynamics.
Applying theoretical analysis and numerical simulations.
Stabilizing unstable periodic orbits onto superstable periodic orbits.

Main Results:

Established a theoretical and numerical link between chaotic kneading sequences and maximum-length shift-register sequences.
Demonstrated that maximum-length shift-register sequences form a subset of universal sequences in one-dimensional chaotic unimodal maps.
Developed techniques for controlled generation of long binary sequences.

Conclusions:

The study confirms a subset relationship between maximum-length shift-register sequences and chaotic unimodal map sequences.
The findings offer new insights into the structure of chaotic sequences.
The developed techniques facilitate the controlled generation of binary sequences for potential applications.