Multifractal Measures Characterized by the Iterative Map With Two Control Parameters

Abstract

A one-dimensional iterative map with two control parameters — the Kim–Kong map — is proposed. Our purpose is to investigate the characteristic properties of this map, and to discuss numerically the multifractal behavior of the normalized first passage time. Based especially on the Monte Carlo simulation, the normalized first passage time to arrive at the absorbing barrier after starting from an arbitrary site is mainly obtained in the presence of both absorption and reflection on a two-dimensional Sierpinski gasket. We also discuss the multifractal spectra of the normalized first passage time, and the numerical result of the Kim–Kong model presented will be compared with that of the Sinai and logistic models.

Multifractal Measures on Small-World Networks

“…The Sinai model and other previous works [19] were studied for the mean and mean square displacements dependent anomalously on time. Many stochastic processes on one-dimensional lattice have argued with the mean first passage time, for the specific cases of the transition probability such as chaotic orbits of the logistic and Kim-Kong maps [20]. Recently, Kim and Kong [21] have used the box-counting method to estimate the generalized dimension and the scaling exponent for the mountain height and the sea-bottom depth.…”

Multifractal Measures in Fractional Iterative Maps

“…In this paper we present the results of our measurements of these and related quantities, multifractal spectrum, for a one-dimensional map with two control parameters -the Kim-Kong map [6]. The motivation for this work is twofold; firstly explore multifractal analysis for discrete dynamical systems and secondly to investigate the possibly competing effects of two control parameters in discrete dynamical systems.…”

“…An example of the period doubling cascade is shown in in Fig. 2 of [6] for β = 0.2 and 0 < γ < 3.78. Denoting γ k as the parameter at which the period 2 k−1 cycle becomes unstable we compute the sequence…”