Revisiting Hammel et al. (1987): Does the shadowing property hold for modern computers?
For researchers using numerical simulations in nonlinear dynamics, this work highlights that classical reliability guarantees may not hold on modern hardware, raising concerns about the validity of long-term simulations.
The paper revisits a 1987 theorem by Hammel et al. about the shadowing property of logistic maps and finds that on modern computers, the error exceeds the claimed bound of 10^{-8} even for fewer than 100 iterates, contradicting the original result.
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.