What good are numerical simulations of chaotic dynamical systems?
A number of factors which influence numerical simulations of chaotic dynamical systems are discussed. After an introduction in which the meaning of some concepts like ill-conditioning and stability are made clear, the author considers models with four levels of abstraction (physical reality, continuous mathematical problem, discrete numerical approximation, floating point approximation) and points out how a change of level may introduce or destroy chaos. In section 2 some remarkable features of chaos in discrete models are illustrated by considering the behaviour of the well-known Gauss map. Several shadowing type results are presented to show its interest to study the backward error in floating point computations. In section 3 continuous dynamical systems governed by differential equations are considered. In particular the author claims that a control of the defect of an interpolant for a discrete solution of an initial value problem may be used as a backward error control. A modification of the Matlab code computing the defect of Euler's method in connection with the Lorentz equations is presented. However the general use of the defect in chaotic problems is unclear. The paper ends with the remark that a numerical simulation of a dynamical system can be considered as the exact solution of a nearby problem. However the main difficulty consists in checking whether this perturbed problem retains the same chaotic behaviour as the original one.
scientific article; zbMATH DE number 699057
scientific article