On the clean numerical simulation (CNS) of chaotic dynamic systems

Abstract

According to Lorenz, chaotic dynamic systems have sensitive dependence on initial conditions (SDIC), i.e., the butterfly-effect: a tiny difference on initial conditions might lead to huge difference of computer-generated simulations after a long time. Thus, computer-generated chaotic results given by traditional algorithms in double precision are a kind of mixture of “true” (convergent) solution and numerical noises at the same level. Today, this defect can be overcome by means of the “clean numerical simulation” (CNS) with negligible numerical noises in a long enough interval of time. The CNS is based on the Taylor series method at high enough order and data in the multiple precision with large enough number of digits, plus a convergence check using an additional simulation with even smaller numerical noises. In theory, convergent (reliable) chaotic solutions can be obtained in an arbitrary long (but finite) interval of time by means of the CNS. The CNS can reduce numerical noises to such a level even much smaller than micro-level uncertainty of physical quantities that propagation of these physical micro-level uncertainties can be precisely investigated. In this paper, we briefly introduce the basic ideas of the CNS, and its applications in long-term reliable simulations of Lorenz equation, three-body problem and Rayleigh-Bénard turbulent flows. Using the CNS, it is found that a chaotic three-body system with symmetry might disrupt without any external disturbance, say, its symmetry-breaking and system-disruption are “self-excited”, i.e., out-of-nothing. In addition, by means of the CNS, we can provide a rigorous theoretical evidence that the micro-level thermal fluctuation is the origin of macroscopic randomness of turbulent flows. Naturally, much more precise than traditional algorithms in double precision, the CNS can provide us a new way to more accurately investigate chaotic dynamic systems.