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Research Papers

# Discontinuous Reinjection Probability Density functions in Type V Intermittency

[+] Author and Article Information
Sergio Elaskar

Professor
Instituto de Estudios Avanzados
en Ingeniería y Tecnología,
IDIT Departamento de Aeronáutica,
FCEFyN Universidad Nacional de Córdoba
and CONICET,
Córdoba 5000, Argentina
e-mail: selaskar@unc.edu.ar

Ezequiel del Río

Professor
E.T.S.I. Aeronáutica y Espacio,
Universidad Politécnica de Madrid,
Madrid 28040, Spain

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.

###### FIGURES IN THIS ARTICLE
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Copyright © 2018 by ASME
Topics: Density , Probability
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## References

Schuster, H. , and Just, W. , 2005, Deterministic Chaos, Wiley VCH, Mörlenbach, Germany.
Nayfeh, A. , and Balachandran, B. , 1995, Applied Nonlinear Dynamics, Wiley, New York.
Marek, M. , and Schreiber, I. , 1995, Chaotic Behaviour of Deterministic Dissipative Systems, Cambridge University Press, Cambridge, UK.
Elaskar, S. , and del Rio, E. , 2017, New Advances on Chaotic Intermittency and Its Applications, Springer, New York.
Kaplan, H. , 1992, “ Return to Type-I Intermittency,” Phys. Rev. Lett., 68(5), pp. 553–557. [PubMed]
Price, T. , and Mullin, P. , 1991, “ An Experimental Observation of a New Type of Intermittency,” Phys. D, 48(1), pp. 29–52.
Platt, N. , Spiegel, E. , and Tresser, C. , 1993, “ On-Off Intermittency: A Mechanism for Bursting,” Phys. Rev. Lett., 70(3), pp. 279–282. [PubMed]
Pikovsky, A. , Osipov, G. , Rosenblum, M. , and Zaks, M. J. K. , 1997, “ Attractor-Repeller Collision and Eyelet Intermittency at the Transition to Phase Synchronization,” Phys. Rev. Lett., 79(1), pp. 47–50.
Lee, K. , Kwak, Y. , and Lim, T. , 1998, “ Phase Jumps Near a Phase Synchronization Transition in Systems of Two Coupled Chaotic Oscillators,” Phys. Rev. Lett., 81(2), pp. 321–324.
Hramov, A. , Koronovskii, A. , Kurovskaya, M. , and Boccaletti, S. , 2006, “ Ring Intermittency in Coupled Chaotic Oscillators at the Boundary of Phase Synchronization,” Phys. Rev. Lett., 97, p. 114101. [PubMed]
Dubois, M. , Rubio, M. , and Berge, P. , 1983, “ Experimental Evidence of Intermittencies Associated With a Subharmonic Bifurcation,” Phys. Rev. Lett., 51, p. 1446.
Malasoma, J. , Werny, P. , and Boiron, M. , 2004, “ Multichannel Type-I Intermittency in Two Models of Rayleigh-Benard Convection,” Phys. Rev. Lett., 51(3), pp. 487–500.
Stavrinides, S. , Miliou, A. , Laopoulos, T. , A. , and Anagnostopoulos, A. , 2008, “ The Intermittency Route to Chaos of an Electronic Digital Oscillator,” Int. J. Bifurcation Chaos, 18(5), pp. 1561–1566.
Sanmartin, J. , Lopez-Rebollal, O. , del Rio, E. , and Elaskar, S. , 2004, “ Hard Transition to Chaotic Dynamics in Alfven Wave-Fronts,” Phys. Plasmas, 11(5), pp. 2026–2035.
Sanchez-Arriaga, G. , Sanmartin, J. , and Elaskar, S. , 2007, “ Damping Models in the Truncated Derivative Nonlinear Schrödinger Equation,” Phys. Plasmas, 14(8), p. 082108.
Pizza, G. , Frouzakis, G. , and Mantzaras, J. , 2012, “ Chaotic Dynamics in Premixed Hydrogen/Air Channel Flow Combustion,” Combust. Theor. Model, 16(2), pp. 275–299.
Nishiura, Y. , Ueyama, D. , and Yanagita, T. , 2005, “ Chaotic Pulses for Discrete Reaction Diffusion Systems,” SIAM J. Appl. Dyn. Syst., 4(3), pp. 723–754.
de Anna, P. , Borgne, T. L. , Dentz, M. , Tartakovsky, A. , Bolster, D. , and Davy, P. , 2013, “ Flow Intermittency, Dispersion and Correlated Continuous Time Random Walks in Porous Media,” Phys. Rev. Lett., 110, p. 184502. [PubMed]
Stan, C. , Cristescu, C. , and Dimitriu, D. , 2010, “ Analysis of the Intermittency Behavior in a Low-Temperature Discharge Plasma by Recurrence Plot Quantification,” Phys. Plasmas, 17(4), p. 042115.
Chian, A. , 2007, Complex System Approach to Economic Dynamics (Lecture Notes in Economics and Mathematical Systems, Vol. 592), Springer, Berlin.
Zebrowski, J. , and Baranowski, R. , 2004, “ Type-I Intermittency in Nonstationary Systems: Models and Human Heart-Rate Variability,” Phys. A, 336(1-2), pp. 74–86.
Paradisi, P. , Allegrini, P. , Gemignani, A. , Laurino, M. , Menicucci, D. , and Piarulli, A. , 2012, “ Scaling and Intermittency of Brains Events as a Manifestation of Consciousness,” AIP Conference Proceedings, 1510(1), p. 151.
Kye, W. , and Kim, C. , 2000, “ Characteristic Relations of Type-I Intermittency in Presence of Noise,” Phys. Rev. E, 62 (5 Pt A), pp. 6304–6307.
Kye, W. , Rim, S. , Kim, C. , Lee, J. , Ryu, J. , Yeom, B. , and Park, Y. , 2003, “ Experimental Observation of Characteristic Relations of Type-III Intermittency in the Presence of Noise in a Simple Electronic Circuit,” Phys. Rev. E, 68(3) p. 036203.
del Rio, E. , and Elaskar, S. , 2010, “ New Characteristic Relation in Type-II Intermittency,” Int. J. Bifurcation Chaos, 20(4), pp. 1185–1191.
Elaskar, S. , del Rio, E. , and Donoso, J. , 2011, “ Reinjection Probability Density in Type-III Intermittency,” Phys. A, 390(15), pp. 2759–2768.
del Rio, E. , Sanjuan, M. , and Elaskar, S. , 2012, “ Effect of Noise on the Reinjection Probability Density in Intermittency,” Commun. Nonlinear Sci. Numer. Simul., 17(9), pp. 3587–3596.
Elaskar, S. , and del Rio, E. , 2012, “ Intermittency Reinjection Probability Function With and Without Noise Effects,” Latest Trends in Circuits, Automatics Control and Signal Processing, WSEAS, Barcelona, Spain, pp. 145–154.
del Rio, E. , Elaskar, S. , and Makarov, V. , 2013, “ Theory of Intermittency Applied to Classical Pathological Cases,” Chaos, 23(3), p. 033112. [PubMed]
del Rio, E. , Elaskar, S. , and Donoso, J. , 2014, “ Laminar Length and Characteristic Relation in Type-I Intermittency,” Commun. Nonlinear Sci. Numer. Simul., 19(4), pp. 967–976.
Krause, G. , Elaskar, S. , and del Rio, E. , 2014, “ Type-I Intermittency With Discontinuous Reinjection Probability Density in a Truncation Model of the Derivative Nonlinear Schrödinger Equation,” Nonlinear Dyn., 77(3), pp. 455–466.
Krause, G. , Elaskar, S. , and del Rio, E. , 2014, “ Noise Effect on Statistical Properties of Type-I Intermittency,” Phys. A, 402, pp. 318–329.
Elaskar, S. , del Rio, E. , Krause, G. , and Costa, A. , 2015, “ Effect of the Lower Boundary of Reinjection and Noise in Type-II Intermittency,” Nonlinear Dyn., 79(2), pp. 1411–1424.
del Rio, E. , and Elaskar, S. , 2016, “ On the Intermittency Theory in 1D Maps,” Int. J. Bifurcation Chaos, 26(14), p. 1620228.
Elaskar, S. , del Rio, E. , and Costa, A. , 2017, “ Reinjection Probability Density for Type-III Intermittency With Noise and Lower Boundary of Reinjection,” ASME J. Comput. Nonlinear Dyn., 12(3), p. 031020.
Elaskar, S. , del Rio, E. , and Marcantoni, L. G. , 2018, “ Nonuniform Reinjection Probability Density Function in Type V Intermittency,” Nonlinear Dyn., 92, pp. 683–697.
Bauer, M. , Habip, S. , He, D. , and Martiessen, W. , 1992, “ New Type of Intermittency in Discontinuous Maps,” Phys. Rev. Lett., 68(11), pp. 1625–1628. [PubMed]
He, D. , Bauer, M. , Habip, S. , Kruger, U. , Martiessen, W. , Christiansen, B. , and Wang, B. , 1992, “ New Type of Intermittency in Discontinuous Maps,” Phys. Lett. A, 171(1–2), pp. 61–65.
Fan, J. , Ji, F. , Guan, S. , Wang, B. , and He, D. , 1993, “ Type V Intermittency,” Phys. Lett. A, 182(2–3), pp. 232–237.
Wu, S. , and He, D. , 2001, “ Characteristics of Period-Doubling Bifurcation Cascades in Quasidiscontinuous Systems,” Commun. Theor. Phys., 35(3), pp. 275–282.
Wang, D. , Mo, J. , Zhao, X. , Gu, H. , Qu, S. , and Ren, W. , 2011, “ Intermittent Chaotic Neural Firing Characterized by Non-Smooth like Features,” Chin. Phys. Lett., 27(7), p. 070503.
Gu, H. , and Xiao, W. , 2014, “ Difference Between Intermittent Chaotic Bursting and Spiking of Neural Firing Patterns,” Int. J. Bifurcation Chaos, 24(6), p. 1450082.
Bai-lin, H. , 1989, Elementary Symbolic Dynamics Chaos Dissipative Systems, World Scientific, Singapore.

## 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. 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. 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. 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. 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. 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).

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.

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