On Mechanism Yielding Fractal Invariant Domains of One-Dimensional Discrete Nonlinear Systems

Abstract

The purpose of this paper is to investigate the invariant domain with fractal structures of a class of one-dimensional nonlinear systems described by difference equations. The invariant domain treated here is the set of initial conditions satisfying a condition that the state variable is included in the normalized interval] 0, 1 [for any time. Under the condition that the nonlinearity of the system is described by the unimodal nonlinear function, the mechanism yielding fractal invariant domains and conditions for its existence are clarified by using the notation of the symbolic dynamics. First, it is shown that the fractal structure of the invariant domain is generated by the inverse mapping of the nonlinear function. Furthermore, the mechanism yielding the fractal invariant domain is classified to the five different mechanisms by using the notation of the symbolic dynamics. One of them is similar to that yielding the Cantor centor 1/3 set. The other four types of mechanisms are newly demonstrated by theorems of this paper.