New Advances on Chaotic Intermittency and its Applications by Sergio Elaskar & Ezequiel del Río

Author:Sergio Elaskar & Ezequiel del Río
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(5.16)

where k is a normalization constant such that ∫ 0 c ϕ(τ)d τ = 1. For the expression (5.16) evaluated for the function given by Eq. (5.7) we have

(5.17)

where F 1 ′ (x) indicates the derivative of the function F 1(x). In the linear approximation of F 1(x) in the interval (x r , F 2 −1(c)), we can consider F 1 ′ (x) as a constant. Now, if the density ρ(x ′ ) is uniform, we get the following reinjection probability density

(5.18)

where b is a normalization constant given by Eq. (5.15). Note that the function PRD will strongly depend on parameter γ, that determines the curvature of the map in region marked by colored segments in Fig. 5.1. Only the points in that region will be mapped inside of the laminar region. For a numerical verification of Eq. (5.18) see [13]. It is interesting to note that a power law like (5.18) is already verified in a wide class of one-dimensional maps even in some classical “pathological” cases that deviate significantly from the classical predictions as will be explained in Chap. 7

Regarding with the classical hypothesis of uniform RPD, it holds for the map of Fig. 5.1 only in the case of γ = 1, where x r + is not an extreme point having a bounded slope. However it is false for γ ≠ 1 where the RPD is given by Eq. (5.18). We notice that the hypothesis of uniform reinjection does not work in general, but it usually works for ρ(x ′ ) when it is generated in no extreme points as it is indicated by the green arrow in Fig. 5.1.

To consider the effect of an LBR ≠ 0 we propose the following modification of the map (5.5) having type-I intermittency:

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