{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/on-the-numerical-simulation-of-propagation-of","title":"On the numerical simulation of propagation of micro-level inherent uncertainty for chaotic dynamic systems","arxiv_id":"1109.0130","date":"2011-09-01","proceeding":null,"authors":["S. J. Liao"],"abstract":"In this paper, an extremely accurate numerical algorithm, namely the \"clean numerical simulation\" (CNS), is proposed to accurately simulate the propagation of micro-level inherent physical uncertainty of chaotic dynamic systems. The chaotic Hamiltonian H\\'{e}non-Heiles system for motion of stars orbiting in a plane about the galactic center is used as an example to show its basic ideas and validity. Based on Taylor expansion at rather high-order and MP (multiple precision) data in very high accuracy, the CNS approach can provide reliable trajectories of the chaotic system in a finite interval $t\\in[0,T_c]$, together with an explicit estimation of the critical time $T_c$. Besides, the residual and round-off errors are verified and estimated carefully by means of different time-step $\\Delta t$, different precision of data, and different order $M$ of Taylor expansion. In this way, the numerical noises of the CNS can be reduced to a required level, i.e. the CNS is a rigorous algorithm. It is illustrated that, for the considered problem, the truncation and round-off errors of the CNS can be reduced even to the level of $10^{-1244}$ and $10^{-1000}$, respectively, so that the micro-level inherent physical uncertainty of the initial condition (in the level of $10^{-60}$) of the H\\'{e}non-Heiles system can be investigated accurately. It is found that, due to the sensitive dependence on initial condition (SDIC) of chaos, the micro-level inherent physical uncertainty of the position and velocity of a star transfers into the macroscopic randomness of motion. Thus, chaos might be a bridge from the micro-level inherent physical uncertainty to the macroscopic randomness in nature. This might provide us a new explanation to the SDIC of chaos from the physical viewpoint.","url_abs":"https://arxiv.org/abs/1109.0130v4","url_pdf":"https://arxiv.org/pdf/1109.0130v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"on-the-numerical-simulation-of-propagation-of","repo_url":"https://github.com/m4r35n357/ODE-Playground","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}