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Correlations in one-dimensional fully developed chaos.

T Tsuchiya¹, A Ichimura, Y Nagai

¹Biophysics Division, Natural Science Laboratory, Advance Co. Ltd., Tokyo, Japan.

Cell Biophysics

|December 1, 1987

Summary

Higher-order correlations are crucial for understanding chaos, as second-order functions alone are insufficient. A new fixed-separation correlation integral captures comprehensive correlational behaviors in chaotic systems.

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Area of Science:

Dynamical Systems
Chaos Theory
Statistical Mechanics

Background:

One-dimensional chaotic maps like the baker and tent maps exhibit complex correlations.
Traditional correlation functions may not fully capture the intricacies of fully developed chaos.

Purpose of the Study:

To investigate the limitations of second-order correlation functions in chaotic systems.
To introduce a novel method for quantifying higher-order correlations in chaos.

Main Methods:

Analysis of symbolic dynamical systems derived from the baker and tent maps.
Development and application of a fixed-separation correlation integral.
Comparison with standard correlation integrals used for fractal dimension calculations.

Main Results:

The vanishing second-order correlation function does not guarantee uncorrelatedness in these chaotic systems.
Higher-order correlation functions, particularly third-order, are essential for a complete description.
The proposed fixed-separation correlation integral effectively encapsulates all even-order autocorrelation functions.

Conclusions:

Standard correlation measures are insufficient for fully characterizing chaos.
Higher-order correlations are vital for understanding chaotic dynamics.
The fixed-separation correlation integral offers a more comprehensive tool for analyzing chaotic systems.

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