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arXiv 1109.0130nlin.CD
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混沌动力学系统微观不确定性传播的数值模拟

On the numerical simulation of propagation of micro-level inherent uncertainty for chaotic dynamic systems

S. J. Liao

备注：
24 pages, 5 Figures, 7 Tables
发表信息：
Chaos, Solitons & Fractals 47 (2013) 1-12
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摘要：

本文提出了一种非常精确的数值算法，即“干净数值模拟”（CNS），用于精确模拟混沌动力学系统微观层次固有物理不确定性的传播。以平面内恒星绕银心运动的混沌Hamilton Hénon-Heiles系统为例，说明其基本思想和有效性.基于高阶Taylor展开和高精度MP数据，CNS方法可以给出混沌系统在有限区间$t\in[0,T_c]$内的可靠轨迹，并能精确估计临界时刻$T_c$ .此外，还对不同时间步长$Δt$、不同数据精度和不同阶数的泰勒展开式$M$的残差和舍入误差进行了详细的检验和估计。这样，CNS的数值噪声可以被降低到所需的水平，即CNS是严格的算法。结果表明，对于所考虑的问题，CNS的截断误差和舍入误差可以分别减小到$10^{-1244}$和$10^{-1000}$的水平，从而可以精确地研究Hénon-Heiles系统初始条件（在$10^{-60}$水平）的微观固有物理不确定性.研究发现，由于混沌对初始条件的敏感依赖性（SDIC），恒星的位置和速度的微观物理不确定性转化为运动的宏观随机性。因此，混沌可能是一个桥梁，从微观层次的固有的物理不确定性的宏观随机性的性质。这可能为混沌的SDIC提供一种新的物理解释。

英文摘要：

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 Hénon-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 $Δt$, 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 Hénon-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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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