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A relation on round-off error, attractor size and its dynamics in driven or coupled logistic map system.

Pengliang Shi¹

¹Institute of Theoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100080, People's Republic of China. pshi@itp.ac.cn

Chaos (Woodbury, N.Y.)

|April 2, 2008

Summary

Numerical artifacts like period-like and chaos-like behaviors emerge in extended logistic maps due to computational errors. High-precision simulations reveal these artifacts transform into real attractors, dependent on system dynamics and precision.

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Area of Science:

Numerical Analysis
Dynamical Systems
Computational Mathematics

Background:

Extended logistic map systems exhibit complex behaviors.
Numerical computations can introduce artificial artifacts.
Round-off errors significantly impact simulation accuracy.

Purpose of the Study:

Investigate numerical artifacts in an extended logistic map.
Analyze the origin and nature of period-like, strange-nonchaotic-attractor-like, and chaos-like behaviors.
Determine the conditions under which these artifacts emerge.

Main Methods:

Simulations using varying precision levels (double and high-precision).
Analysis of dynamical processes and round-off truncation errors.
Introduction of a quantity beta related to the local Lyapunov exponent to measure dynamical capability.

Main Results:

Identified three types of numerical artifacts: period-like, strange-nonchaotic-attractor-like, and chaos-like.
Observed that artifacts in double precision transform into real attractors in high-precision simulations.
Established a condition for artifact emergence: alphabeta < gamma, where alpha is computational precision, beta measures dynamical capability, and gamma is attractor size.

Conclusions:

Numerical artifacts in extended logistic maps are a result of system dynamics and computational precision.
The proposed quantity beta and the condition alphabeta < gamma provide a framework for understanding artifact generation.
High-precision computations are crucial for accurately representing the true dynamics of such systems.