High and Low Type One-dimensional Function with Multi-Step Uniform Invariant Density

Abstract

Nonlinear systems described by simple mathematical models often exhibit extremely complicated behavior called chaos. In this paper, chaotic behavior, exhibited by the one-dimensional difference equation, is investigated. A method for constructing nonlinear functions with the invariant density is newly demonstrated. Furthermore, the method for finding the invariant density of the nonlinear function is also presented. Nonlinear functions, treated in this study, are piecewise linear ones and the invariant density is piecewise uniform.