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Research Papers

# Discontinuous Reinjection Probability Density functions in Type V Intermittency

[+] Author and Article Information

Professor
en Ingeniería y Tecnología,
IDIT Departamento de Aeronáutica,
and CONICET,
Córdoba 5000, Argentina

Ezequiel del Río

Professor
E.T.S.I. Aeronáutica y Espacio,

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received March 30, 2018; final manuscript received September 14, 2018; published online October 15, 2018. Assoc. Editor: Mohammad Younis.

J. Comput. Nonlinear Dynam 13(12), 121001 (Oct 15, 2018) (10 pages) Paper No: CND-18-1137; doi: 10.1115/1.4041577 History: Received March 30, 2018; Revised September 14, 2018

## Abstract

This paper reports theoretical and numerical results about the reinjection process in type V intermittency. The M function methodology is applied to a simple mathematical model to evaluate the reinjection process through the reinjection probability density function (RPD), the probability density of laminar lengths, and the characteristic relation. We have found that the RPD can be a discontinuous function and it is a sum of exponential functions. The RPD shows two reinjection behaviors. Also, the probability density of laminar lengths has two different behaviors following the RPD function. The dependence of the RPD function and the probability density of laminar lengths with the reinjection mechanisms and the lower boundary of return are considered. On the other hand, we have obtained, for the analyzed map, that the characteristic relation verifies $l¯≈ε−0.5$. Finally, we highlight that the M function methodology is a suitable tool to analyze type V intermittency and there is a very high accuracy between the new theoretical equations and the numerical data.

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Topics: Density , Probability

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## Figures

Fig. 1

Map described by Eq. (7). The parameters are γ = 2, ε = 0.001, a1 = 0.5, a2 = 1. x0 is the vanished fixed point and x̃ is the lower boundary of return. A trajectory moving through x̃ is also indicated.

Fig. 9

Function M(x) for map (7). The parameters are: γ = 0.5,ε = 0.001, a1 = 0.9, a2 = 1, Nj = 30,000, c = 0.1128 and x̃=−1.00006. Clearer line is obtained from numerical data, and the darker one is calculated using Eqs. (15)(17).

Fig. 10

RPD for map (7). The parameters are: γ = 0.5, ε = 0.001, a1 = 0.9, a2 = 1, Nj = 30,000, c = 0.1128 and x̃=−1.00006. Points represent the numerical data, and the continuous line corresponds to the theoretical RPD calculated using Eqs. (9)(11). (b) is an enlarged image of the (a) inside the Δ2 interval.

Fig. 13

Probability density of the laminar length, ψ(l), for map (7). The parameters are: γ = 0.5, ε = 0.001, a1 = 0.9, a2 = 1, Nj = 30,000, c = 0.1128 and x̃=−1.00006. Points represent the numerical data, and the line the theoretical results calculated using Eqs. (9), (10), (18) and (19).

Fig. 2

Bifurcation diagram for map (7) with γ = 1, a1 = 0.9, a2 = 1 and x̃=−1.00006. Clearer points correspond with the first numerical test.

Fig. 3

M(x) function for map (7) with γ = 1, ε = 0.001, a1 = 0.9, a2 = 1, Nj = 30,000, c = 0.1128 and x̃=−1.00006. Clearer line is thenumerical data and the continuous line represents the theoretical M(x) function.

Fig. 4

Numerical Mi(x) function Δ1 = [x0 − c, xs] for map (7) calculated using only reinjected points coming from x < x0 − c. Parameters: γ = 1, ε = 0.001, a1 = 0.9, a2 = 1, Nj = 30,000, c = 0.1128 and x̃=−1.00006. From this figure we obtain: mi≅0.5006 and αi≅−0.00249.

Fig. 5

Numerical Ms(x) function inside Δ = [x0 − c, x0 + c] for map (7) obtained for reinjected points coming from x > xm. Parameters: γ = 1, ε = 0.001, a1 = 0.9, a2 = 1, Nj = 30,000, c = 0.1128 and x̃=−1.00006. From this figure we calculate: ms≅0.5089 and αs≅0.03623.

Fig. 6

RPD for map (7) with γ = 1, ε = 0.001, a1 = 0.9, a2 = 1, Nj = 30,000, c = 0.1128 and x̃=−1.00006. Points are numerical results and the continuous line represents the theoretical RPD calculated using Eqs. (9)(11).

Fig. 7

Probability density of the laminar length for γ = 1, ε = 0.001, a1 = 0.9, a2 = 1, Nj = 30,000, c = 0.1128 and x̃=−1.00006. Points represent the numerical data, and the line the theoretical results calculated using Eqs. (9), (10), (18) and (19). (b) is an enlargement of (a) for reinjected points inside of Δ2 interval.

Fig. 8

Iterative evolution of the map (7) with γ = 1, a1 = 0.9, a2 = 1 and x̃=−1.00006. (a) uses ε = −0.001 < 0 and x0 = x1 is a stable fixed point. (b) considers ε = 0.001 > 0, the iterative process shows intermittency.

Fig. 11

Numerical Mi(x) function inside Δ1 = [x0 − c, xs) for map(7) obtained using only reinjected points coming from x < x0 − c. The parameters are: γ = 0.5, ε = 0.001, a1 = 0.9, a2 = 1, Nj = 30,000, c = 0.1128 and x̃=−1.00006. Mi(x) is a linear function with slope mi≅0.5095.

Fig. 12

Numerical Ms(x) function inside Δ2 = [xs, x0 + c] for map (7) obtained for reinjected points coming from x > xm. The parameters are: γ = 0.5, ε = 0.001, a1 = 0.9, a2 = 1, Nj = 30,000, c = 0.1128 and x̃=−1.00006. Ms(x) is a linear function with slope ms≅0.5212.

Fig. 15

Characteristic relation for γ = 1.5, ε = 0.001–0.0000001, a1 = 0.9, a2 = 1. The number of iterations from x̃ to x0 − c are 1 and 20 for the lower and upper lines respectively. Points: numerical data. Line: theoretical approach.

Fig. 14

Characteristic relation for γ = 1, ε = 0.001–0.0000001, a1 = 0.9, a2 = 1, the number of iterations from x̃ to x0 − c is 20. Points: numerical data. Line: theoretical approach given by Eqs. (23)(27).

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