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Research Papers

Reinjection Probability Density for Type-III Intermittency With Noise and Lower Boundary of Reinjection

[+] Author and Article Information
Sergio Elaskar

Professor
Aeronautical Department,
National University Cordoba,
IDIT-CONICET,
Cordoba CP 5000, Argentina
e-mail: selaskar@unc.edu.ar

Ezequiel del Rio

Professor
E.T.S.I. Aeronáuticos,
Polytechnic University Madrid,
Madrid CP 28040, Spain

Andrea Costa

Professor
Aeronautical Department,
National University Cordoba,
IATE-CONICET,
Cordoba CP 5000, Argentina

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received September 8, 2015; final manuscript received August 18, 2016; published online January 19, 2017. Assoc. Editor: D. Dane Quinn.

J. Comput. Nonlinear Dynam 12(3), 031020 (Jan 19, 2017) (11 pages) Paper No: CND-15-1284; doi: 10.1115/1.4034732 History: Received September 08, 2015; Revised August 18, 2016

In this paper, we extend a methodology developed recently to study type-III intermittency considering different values of the noise intensity and the lower boundary of reinjection (LBR). We obtain accurate analytic expressions for the reinjection probability density (RPD). The proposed RPD has a piecewise definition depending on the nonlinear behavior, the LBR value, and the noise intensity. The new RPD is a sum of exponential functions with exponent α + 2, where α is the exponent of the noiseless RPD. The theoretical results are verified with the numerical simulations.

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Figures

Grahic Jump Location
Fig. 1

Left: Map (1). The parameters are ε = 0.01, a = 1, 0, and b = 1.1. Right: Map (2). The parameters are ε = 0.1 and b = 0.1.

Grahic Jump Location
Fig. 2

Map (9). Left: M(x) from the numerical simulations. The darker (0.33 < x ≤ 0.6) line indicates the data used to calculate the exponent α. Right: Theoretical and numerical RPD functions. The points correspond to numerical data, and the result of Eq. (13) is plotted as a solid line. Parameters: ε = 0.0005, c = 0.6, a = 1.1, b = 1.35, and K = 10.5833.

Grahic Jump Location
Fig. 3

Map (10). Left: M(x) from the numerical simulations. The darker (0.65 < x ≤ 1) line indicates the data used to calculate the exponent α. Right: Theoretical and numerical RPD function. The points correspond to numerical data and the result of Eq. (14) is plotted as a solid line. Parameters: ε = 0.005, c = 1.0, σ = 0.3, b = 0.05, and K≅0.

Grahic Jump Location
Fig. 4

(K+1)σ<xs and 0 < K ≤ 1. Left: M(x) function; the darker (0.75 < x ≤ 1) line shows the data used to calculate the exponent α in Eq. (8). Right: NRPD function; the numerical data and theoretical result are represented by points and line, respectively. Parameters: c = 1, ε = 0.005, σ = 0.2, xs≅0.3181, and K≅0.5366.

Grahic Jump Location
Fig. 5

(K+1)σ<xs and 1 < K ≤ 2. Left: M(x) function; the darker (1.75 < x ≤ 2) line shows the dataused to calculate the exponent α. Right: NRPD function; the points represent the numerical data and the line is the analytical NRPD. Parameters: c = 2, ε = 0.005, σ = 0.2, xs≅0.8076, and K≅1.1750.

Grahic Jump Location
Fig. 6

(K+1)σ≥xs, σ<xs, and 0 < K ≤ 1. Left: M(x) function; the darker (0.65 < x ≤ 1) line indicates the data used to calculate the exponent α. Center: NRPD function for αr =−0.656. Right: Comparison of the NRPD functions for αr = −0.656 and α = −0.61. Parameters: c = 1, ε = 0.005, σ = 0.18, xs≅0.194, and K≅0.3459.

Grahic Jump Location
Fig. 7

(K+1)σ≥xs, σ<xs, and 0 < K ≤ 1. Left: M(x) function. Right: NRPD function. Parameters: c = 1, ε = 0.005, σ = 0.18, xs≅0.194, and K≅0.3459.

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