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

Although computational techniques have been extensively applied in nonlinear science, the reliability evaluation of numerical results does not grow in the same pace. Hammel et al. (Journal of Complexity, 1987, 3(2), 136–145) 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 iterates. We checked this theorem in a modern computational platform, and on the contrary, we found the simulation of the logistic map with less than 10 iterates presents an error greater than 10−8. keywords: Modeling, Numerical Simulation and Optimization, Nonlinear Dynamics and Complex Systems, Chaos and Global Nonlinear Dynamics, Pseudo-Orbits, Nonhyperbolic Maps.