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

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.