STATISTICAL LAWS FOR COMPUTATIONAL COLLAPSE OF DISCRETIZED CHAOTIC MAPPINGS

Abstract

When a dynamical system is realized on a computer, the computation is of a discretization, where finite machine arithmetic replaces continuum state space. For chaotic dynamical systems, the discretizations often have collapsing effects to a fixed point or to short cycles. Statistical properties of this phenomenon can be modeled by random mappings with an absorbing center. The model gives results which are very much in line with computational experiments and there appears to be a type of universality summarised by an Arcsine Law. The effects are discussed with special reference to the family of mappings f l (x)=1−|1−2x| l ,x∈[0, 1], 1<l≤2. Computer experiments show close agreement with predictions of the model.