Revisiting Hammel et al. (1987): Does the shadowing property hold for modern computers?

TL;DR

This paper reevaluates the shadowing property for logistic map simulations on modern computers, revealing discrepancies with classical results and highlighting limitations in numerical reliability for small iteration counts.

Contribution

It revisits Hammel et al.'s 1987 theorem using modern computational techniques, demonstrating that shadowing may not hold for fewer than 100 iterations on current hardware.

Findings

Shadowing property does not hold for less than 100 iterations on modern computers.

Numerical errors exceed $10^{-8}$ in fewer than 100 iterations.

Classical shadowing results may not apply to current computational practices.

Abstract

Computational techniques are extensively applied in nonlinear science. However, while the use of computers for research has been expressive, the evaluation of numerical results does not grow in the same pace. Hammel et al. (Journal of Complexity, 1987, 3(2), 136--145) were pioneers in the numerical reliability field and have proved a theorem that a pseudo-orbit of a logistic map is shadowed by a true orbit within a distance of $10^{-8}$ for $10^{7}$ iterates. But the simulation of the logistic map with less than 100 iterates presents an error greater than $10^{-8}$ in a modern computer, performing a test based on the concept of multiple pseudo-orbits and symbolic computing.