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¯ε0.5. 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.

References

References
1.
Schuster
,
H.
, and
Just
,
W.
,
2005
,
Deterministic Chaos
,
Wiley VCH
,
Mörlenbach, Germany
.
2.
Nayfeh
,
A.
, and
Balachandran
,
B.
,
1995
,
Applied Nonlinear Dynamics
,
Wiley
,
New York
.
3.
Marek
,
M.
, and
Schreiber
,
I.
,
1995
,
Chaotic Behaviour of Deterministic Dissipative Systems
,
Cambridge University Press
,
Cambridge, UK
.
4.
Elaskar
,
S.
, and
del Rio
,
E.
,
2017
,
New Advances on Chaotic Intermittency and Its Applications
,
Springer
,
New York
.
5.
Kaplan
,
H.
,
1992
, “
Return to Type-I Intermittency
,”
Phys. Rev. Lett.
,
68
(
5
), pp.
553
557
.
6.
Price
,
T.
, and
Mullin
,
P.
,
1991
, “
An Experimental Observation of a New Type of Intermittency
,”
Phys. D
,
48
(
1
), pp.
29
52
.
7.
Platt
,
N.
,
Spiegel
,
E.
, and
Tresser
,
C.
,
1993
, “
On-Off Intermittency: A Mechanism for Bursting
,”
Phys. Rev. Lett.
,
70
(
3
), pp.
279
282
.
8.
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
.
9.
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
.
10.
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
.
11.
Dubois
,
M.
,
Rubio
,
M.
, and
Berge
,
P.
,
1983
, “
Experimental Evidence of Intermittencies Associated With a Subharmonic Bifurcation
,”
Phys. Rev. Lett.
,
51
, p. 1446.
12.
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
.
13.
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
.
14.
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
.
15.
Sanchez-Arriaga
,
G.
,
Sanmartin
,
J.
, and
Elaskar
,
S.
,
2007
, “
Damping Models in the Truncated Derivative Nonlinear Schrödinger Equation
,”
Phys. Plasmas
,
14
(
8
), p.
082108
.
16.
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
.
17.
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
.
18.
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
.
19.
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
.
20.
Chian
,
A.
,
2007
,
Complex System Approach to Economic Dynamics
(Lecture Notes in Economics and Mathematical Systems, Vol. 592), Springer, Berlin.
21.
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
.
22.
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.
23.
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
.
24.
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.
25.
del Rio
,
E.
, and
Elaskar
,
S.
,
2010
, “
New Characteristic Relation in Type-II Intermittency
,”
Int. J. Bifurcation Chaos
,
20
(
4
), pp.
1185
1191
.
26.
Elaskar
,
S.
,
del Rio
,
E.
, and
Donoso
,
J.
,
2011
, “
Reinjection Probability Density in Type-III Intermittency
,”
Phys. A
,
390
(
15
), pp.
2759
2768
.
27.
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
.
28.
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
.
29.
del Rio
,
E.
,
Elaskar
,
S.
, and
Makarov
,
V.
,
2013
, “
Theory of Intermittency Applied to Classical Pathological Cases
,”
Chaos
,
23
(
3
), p.
033112
.
30.
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
.
31.
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
.
32.
Krause
,
G.
,
Elaskar
,
S.
, and
del Rio
,
E.
,
2014
, “
Noise Effect on Statistical Properties of Type-I Intermittency
,”
Phys. A
,
402
, pp.
318
329
.
33.
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
.
34.
del Rio
,
E.
, and
Elaskar
,
S.
,
2016
, “
On the Intermittency Theory in 1D Maps
,”
Int. J. Bifurcation Chaos
,
26
(14), p.
1620228
.
35.
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
.
36.
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
.
37.
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
.
38.
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
.
39.
Fan
,
J.
,
Ji
,
F.
,
Guan
,
S.
,
Wang
,
B.
, and
He
,
D.
,
1993
, “
Type V Intermittency
,”
Phys. Lett. A
,
182
(
2–3
), pp.
232
237
.
40.
Wu
,
S.
, and
He
,
D.
,
2001
, “
Characteristics of Period-Doubling Bifurcation Cascades in Quasidiscontinuous Systems
,”
Commun. Theor. Phys.
,
35
(3), pp.
275
282
.
41.
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
.
42.
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
.
43.
Bai-lin
,
H.
,
1989
,
Elementary Symbolic Dynamics Chaos Dissipative Systems
,
World Scientific
,
Singapore
.
You do not currently have access to this content.