Statistical properties of some spatially discretized dynamical systems

Abstract

Computer simulations of dynamical systems are {spatial discretizations} in which the finite machine arithmetic space replaces the continuum state space of the original system. Any trajectory of a spatial discretization of a dynamical system is thus eventually periodic, so the dynamical behaviour of such computations are essentially determined by the cycles of the discretized map. Such dynamical behaviour depends seemingly randomly on the fineness of the discretization mesh. In this paper statistical properties of the maximal cycles of spatial discretizations are investigated for some systems such as the tent map, rotations on a circle and toral endomorphisms.