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

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 ∈ [0, Tc], together with an explicit estimation of the critical time Tc. 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.