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Simulations using low-precision numbers in chaotic systems can lead to shorter, problematic periodic orbits. Stochastic rounding and increased system complexity, not just precision, affect simulation fidelity.

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Area of Science:

  • Computational Physics
  • Dynamical Systems Theory
  • Numerical Analysis

Background:

  • Chaotic dynamical systems exhibit non-periodic solutions crucial for their behavior.
  • Finite-precision simulations, especially with low-precision formats, can introduce artificial periodicity.
  • The fidelity of chaotic system simulations is questioned with the trend towards low-precision computing.

Purpose of the Study:

  • To investigate the impact of various number formats and precisions on chaotic system simulations.
  • To analyze the fidelity of low-precision simulations in chaotic dynamical systems.
  • To understand the interplay between precision, system complexity, and simulation accuracy.

Main Methods:

  • Simulations of the logistic map and generalized Bernoulli map using floats, posits, and logarithmic fixed-point formats.
  • Analysis of orbit periodicity and fidelity across different numerical precisions.
  • Investigation of the Lorenz 1996 system with varying numbers of variables to assess complexity effects.

Main Results:

  • Higher precision improves simulations, but stochastic rounding can prevent periodic orbits even at low precision.
  • In larger systems like Lorenz 1996, orbit periods grow exponentially with the number of variables.
  • Increased system complexity (more variables) improves invariant measure approximation more than increased precision.

Conclusions:

  • Low-precision simulations of chaotic systems may yield problematic periodic orbits, challenging simulation fidelity.
  • System complexity significantly influences orbit lengths and invariant measure approximation in chaotic systems.
  • For large-scale models, artificial periodicity is unlikely to be a primary concern due to computational limits, but deviations from continuum solutions remain an open question.