Parameter renormalization of maps based on potential function

Abstract

A systematic way for deriving the parameter renormalization group equation for one-dimensional maps is presented and the critical behavior of periodic doubling is investigated. Introducing a formal potential function in one-parameter cases, it is shown that accumulation points correspond to local potential maxima and universal constants are easily determined. The estimates of accumulation points and universal constants match the known values asymptotically when the order of potential grows large. The potential function shows scaling in the parameter space with the universal convergent rate at the accumulation point similar to the Feigenbaum universal function. For two-parameter cases, a parameter reduction transformation is found to be useful to determine some important fixed points. A locally defined potential function is introduced and its scaling property is discussed. (c) 1997 American Institute of Physics.

Parameter Renormalization for Asymmetrically Coupled Maps

“…An important issue to address at this point is that c and d become 0 when c and d are 0, respectively, which makes possible to apply the parameter reduction technique allowing one to simplify the analysis as proposed in [Matsuba, 1997b]. …”

“…Its associated set of eigenvalues is Λ(P 1+ ) = (5.73205, 1, 0, −2.73205). For the oneparameter uncoupled map, the critical behavior is fully investigated by means of the potential function [Matsuba, 1997b].…”

“…In this section, the critical lines are investigated using the parameter reduction technique proposed in the previous paper [Matsuba, 1997b]. The goal of this method is to seek a set of parameters which gives simpler maps having an equivalent critical behavior.…”

“…In the previous papers, the parameter RG method has been proposed to investigate critical behaviors of periodic-doubling [Matsuba, 1997a] by extending the idea in [Helleman, 1980], and the parameter scaling function has been derived [Matsuba, 1997b]. The renormalized map takes the same form as the original one, which makes the present treatment easy to generalize to higher-dimensional maps.…”

“…Furthermore, the critical behavior of two coupled one-dimensional maps has also been investigated in great detail using the Feigenbaum's RG method in [Kuznetsov et al, 1992a[Kuznetsov et al, , 1992b[Kuznetsov et al, , 1997. For the coupled maps having four parameters considered in the present paper, the sections of the four-dimensional parameter space leading to the parameter planes on which we study the critical behavior using the parameter RG method developed in [Matsuba, 1997b] are considered as a foliated structure as introduced in [Kuznetsov et al, 1992b]. Then one has multistability with fold cusp bifurcation points, crossroads area hierarchy leading to infinitely many tricritical situations.…”