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A New Pathology in the Simulation of Chaotic Dynamical Systems on Digital Computers
Bruce M Boghosian1, Peter V Coveney2,3, Hongyan Wang1
1Department of Mathematics Tufts University Medford MA 02155 USA.
Advanced Theory and Simulations
|September 16, 2021
Summary
Standard computer calculations introduce systematic errors in chaotic dynamical systems. These errors significantly distort statistical properties, impacting simulations in fields like turbulence and molecular dynamics.
Area of Science:
- Computational Physics
- Numerical Analysis
- Chaos Theory
Background:
- Chaotic dynamical systems exhibit complex behaviors.
- Digital computers use IEEE floating-point numbers for simulations.
- Accurate statistical properties are crucial for understanding chaotic systems.
Purpose of the Study:
- To investigate systematic distortions in chaotic systems simulated using standard floating-point numbers.
- To analyze the impact of these distortions on statistical properties.
Main Methods:
- Studied a generalized Bernoulli map, a model chaotic system with a known parameter β.
- Compared exact properties with results from IEEE floating-point simulations.
- Analyzed errors for integer and non-integer values of β.
Main Results:
- Floating-point representation causes significant loss of dynamical system structure.
- Long-term behavior is inaccurate for even integer β, including known anomalies.
- Relative errors in observables reach 14% for non-integer β.
- Errors for odd integer β are larger than expected roundoff errors.
Conclusions:
- Systematic distortions in chaotic system computations are a deeper problem.
- Increasing floating-point precision does not mitigate these errors.
- Warns against uncritical use of numerical methods in chaotic system studies.
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