This letter studies a digital return map that is a mapping from a set of lattice points to itself. The digital map can exhibit various periodic orbits. As a typical example, we present the digital logistic map based on the logistic map. Two fundamental results are shown. When the logistic map has a unique periodic orbit, the digital map can have plural periodic orbits. When the logistic map has an unstable period-3 orbit that causes chaos, the digital map can have a stable period-3 orbit with various domain of attractions.

Because of its simple structure, many reports on the logistic map have been presented. To implement this map on computers, finite precision is usually used, and therefore rounding is required. There are five major methods to implement rounding, but, to date, no study of rounding applied to the logistic map has been reported. In the present paper, we present experimental results showing that the properties of sequences generated by the logistic map are heavily dependent on the rounding method used and give a theoretical analysis of each method. Then, we describe why using the map with a floor function for rounding generates long aperiodic subsequences.

The logistic map is a chaotic mapping. Although several studies have examined logistic maps over real domains with infinite/finite precisions, there has been little analysis of the logistic map over integers. Focusing on differences between the logistic map over the real domain with infinite precision and the logistic map over integers with finite precision, we herein show the characteristic properties of the logistic map over integers and discuss the sequences generated by the map.

A moment matrix analysis (MMA) method can derive macroscopic statistical properties such as moments, response time, and power spectra of non-linear equations without solving the equations. MMA expands a non-linear equation into simultaneous linear equations of moments, and reduces it to a linear equation of their coefficient matrix and a moment vector. We can analyze the statistical properties from the eigenvalues and eigenvectors of the coefficient matrix. This paper presents (1) a systematic procedure to linearize non-linear equations and (2) an expansion of the previous work of MMA to derive the statistical properties of various non-linear equations. The statistical properties of the logistic map were evaluated by using MMA and computer simulation, and it is shown that the proposed systematic procedure was effective and that MMA could accurately approximate the statistical properties of the logistic map even though such a map had strong non-linearity.

Nonlinear dynamics of xn+1=λ {4xn (1-xn)}q is studied in this paper. Different from the logistic map (q=1), in the case of q

It is shown that a self–recurrent fuzzy inference can cause chaotic responses at least three membership functions, if the inference rules are set to represent nonlinear relations such as pie–kneading transformation. This system has single input and single output both with crisp values, in which membership functions is taken to be triangular. Extensions to infinite memberships are proposed, so as to reproduce the continuum case of one–dimensional logistic map f(x)=Ax(1–x). And bifurcation diagrams are calculated for number N of memberships of 3, 5, 9 and 17. It is found from bifurcation diagrams that different periodic states coexist at the same bifurcation parameter for N9. This indicates multistability necessarily accompanied with hysteresis effects. Therefore, it is concluded that the final states are not uniquely determined by fuzzy inferences with sufficiently large number of memberships.