The effect of numerical noises on statistics of a chaotic dynamical system

Abstract

It is well known that chaotic dynamical systems have the sensitive dependence on initial conditions (SDIC). Unfortunately, numerical noises (such as truncation error and round-off error) always exist in practice. Thus, due to the SDIC, long-term accurate prediction of chaotic dynamical systems is practically impossible. In this paper, using the so-called “clean numerical simulation” (CNS) whose numerical noises might be much smaller even than micro-level physical uncertainty and thus are negligible, we gain accurate prediction of a chaotic dynamical system in a long enough interval of time. Then, based on these reliable simulations, the influence of numerical noises on statistic computations is investigated. It is found that the influence of numerical noises is negligible when statistic results are time-independent. Unfortunately, when the statistics of a chaotic dynamical system are time-dependent, numerical noises have a great influence even on statistic computations.