Influences of artificial numerical noise on statistics and qualitative properties of chaotic system

Abstract

Taking the nonlinear Schrödinger equation (NLSE) as an example, we provide from a mathematical viewpoint, rigorous evidence that numerical noise of a chaotic system as tiny artificial stochastic disturbances can increase exponentially to a macro-level. As a result, numerical simulations given by traditional algorithms in double precision may rapidly become badly polluted leading to huge deviations from the ‘true’ solution not only in trajectory but also, sometimes, even in statistics and/or some qualitative properties. Small physical disturbances in time and space are unavoidable in practice, which are often much larger than artificial numerical noise. So, from a physical viewpoint, it is wrong to neglect small spatio-temporal disturbances of a chaotic system: chaos should not be described by deterministic equations.