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

2016/12/07 by B. C. Silva, F. Milani, Silva, B. C. +7
Computer Science · Physics and Astronomy · #Chaos control and synchronization #Evolutionary Algorithms and Applications #FOS: Mathematics #Neural Networks and Applications #Numerical Analysis (math.NA)

paper · pdf · doi:10.48550/arxiv.1612.02153

openalex publication_date 2016/12/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 10⁷ 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.

Citations

• Deterministic Nonperiodic Flow

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