Revista Facultad de Ingeniería, Universidad de Antioquia, No.110, pp. 56-64, Jan-Mar 2024
Chaotic intermittency with non-differentiable
M (x) function
Intermitencia caótica con función M (x) no diferenciable
Sergio Elaskar 1,2, Ezequiel del Río 3, Mauro Grioni 1,4*
1National Scientific and Technical Research Council (CONICET), Argentina
2Departamento de Aeronáutica, Universidad Nacional de Córdoba. Ismael Bordabehere S/N. 5016. Córdoba, Argentina.
3Departamento de Física Aplicada, Universidad Politécnica de Madrid. José Gutierrez Abascal, # 2. C. P. 28006. Madrid,
España.
4Facultad de Ingeniería, Universidad Nacional de Cuyo, Centro Universitario. C. P. 5502. Mendoza, Argentina.
CITE THIS ARTICLE AS:
S. Elaskar, E. Del Rio and M.
Grioni. ”Chaotic intermittency
with non-differentiable M(x)
function”, Revista Facultad de
Ingeniería Universidad de
Antioquia, no. 110, pp. 56-64,
Jan-Mar, 2024. [Online].
Available: https:
//www.doi.org/10.17533/
udea.redin.20230110
ARTICLE INFO:
Received: October 25, 2022
Accepted: January 27, 2023
Available online: January 27,
2023
KEYWORDS:
Intermittency; reinjection;
discontinuous reinjection
probability density function
Intermitencia; reinyección;
función de densidad de
probabilidad de reinyección
discontinua
ABSTRACT: One-dimensional maps showing chaotic intermittency with discontinuous
reinjection probability density functions are studied. For these maps, the reinjection
mechanism possesses two different processes.The M function methodology is applied
to analyze the complete reinjection mechanism and to determine the discontinuous
reinjection probability density function. In these maps the function M (x) is continuous
and non-differentiable. Theoretical equations are found for the function M (x) and for
the reinjection probability density function. Finally, the theoretical results are compared
with numerical data finding high accuracy.
RESUMEN: En este trabajo se estudian mapas unidimensionales que muestran
intermitencia caótica con funciones de densidad de probabilidad de reinyección
discontinuas. Para estos mapas, el mecanismo de reinyección posee dos procesos
diferentes. Para analizar el mecanismo de reinyección completo y determinar la función
de densidad de probabilidad de reinyección discontinua, se aplica la metodología de la
función M . Dicha función es continua y no derivable. Se encuentran ecuaciones teóricas
para la función M (x) y para la función de densidad de probabilidad de reinyección.
Finalmente, los resultados teóricos se comparan con datos numéricos encontrándose
una alta precisión entre ambos.
1. Introduction
Non-linear behavior is a ubiquitous feature in natural
phenomena and human-made mechanisms. Several of
these non-linear behaviors are described by dynamical
systems displaying chaos. A route by which the solutions of
the non-linear dynamical systems can evolve from regular
to chaotic behavior is chaotic intermittency [1]. The system
solutions are composed of laminar and chaotic phases.
The laminar or regular phases are pseudo-equilibrium or
pseudo-periodic solutions, while the bursts correspond to
chaotic evolution [1–3].
The phenomenon of chaotic intermittency has been
found in different fields of science as physics, chemistry,
medicine, engineering, and economics [4–16]. Therefore,
a better description and understanding of chaotic
intermittency phenomenon would contribute to several
fields of knowledge.
56
* Corresponding author: Mauro Grioni
E-mail: mauro.grioni@ingenieria.uncuyo.edu.ar
ISSN 0120-6230
e-ISSN 2422-2844
DOI: 10.17533/udea.redin.20230110
56
Chaotic intermittency with non-differentiable
M (x) function
Intermitencia caótica con función M (x) no diferenciable
Sergio Elaskar 1,2, Ezequiel del Río 3, Mauro Grioni 1,4*
1National Scientific and Technical Research Council (CONICET), Argentina
2Departamento de Aeronáutica, Universidad Nacional de Córdoba. Ismael Bordabehere S/N. 5016. Córdoba, Argentina.
3Departamento de Física Aplicada, Universidad Politécnica de Madrid. José Gutierrez Abascal, # 2. C. P. 28006. Madrid,
España.
4Facultad de Ingeniería, Universidad Nacional de Cuyo, Centro Universitario. C. P. 5502. Mendoza, Argentina.
CITE THIS ARTICLE AS:
S. Elaskar, E. Del Rio and M.
Grioni. ”Chaotic intermittency
with non-differentiable M(x)
function”, Revista Facultad de
Ingeniería Universidad de
Antioquia, no. 110, pp. 56-64,
Jan-Mar, 2024. [Online].
Available: https:
//www.doi.org/10.17533/
udea.redin.20230110
ARTICLE INFO:
Received: October 25, 2022
Accepted: January 27, 2023
Available online: January 27,
2023
KEYWORDS:
Intermittency; reinjection;
discontinuous reinjection
probability density function
Intermitencia; reinyección;
función de densidad de
probabilidad de reinyección
discontinua
ABSTRACT: One-dimensional maps showing chaotic intermittency with discontinuous
reinjection probability density functions are studied. For these maps, the reinjection
mechanism possesses two different processes.The M function methodology is applied
to analyze the complete reinjection mechanism and to determine the discontinuous
reinjection probability density function. In these maps the function M (x) is continuous
and non-differentiable. Theoretical equations are found for the function M (x) and for
the reinjection probability density function. Finally, the theoretical results are compared
with numerical data finding high accuracy.
RESUMEN: En este trabajo se estudian mapas unidimensionales que muestran
intermitencia caótica con funciones de densidad de probabilidad de reinyección
discontinuas. Para estos mapas, el mecanismo de reinyección posee dos procesos
diferentes. Para analizar el mecanismo de reinyección completo y determinar la función
de densidad de probabilidad de reinyección discontinua, se aplica la metodología de la
función M . Dicha función es continua y no derivable. Se encuentran ecuaciones teóricas
para la función M (x) y para la función de densidad de probabilidad de reinyección.
Finalmente, los resultados teóricos se comparan con datos numéricos encontrándose
una alta precisión entre ambos.
1. Introduction
Non-linear behavior is a ubiquitous feature in natural
phenomena and human-made mechanisms. Several of
these non-linear behaviors are described by dynamical
systems displaying chaos. A route by which the solutions of
the non-linear dynamical systems can evolve from regular
to chaotic behavior is chaotic intermittency [1]. The system
solutions are composed of laminar and chaotic phases.
The laminar or regular phases are pseudo-equilibrium or
pseudo-periodic solutions, while the bursts correspond to
chaotic evolution [1–3].
The phenomenon of chaotic intermittency has been
found in different fields of science as physics, chemistry,
medicine, engineering, and economics [4–16]. Therefore,
a better description and understanding of chaotic
intermittency phenomenon would contribute to several
fields of knowledge.
56
* Corresponding author: Mauro Grioni
E-mail: mauro.grioni@ingenieria.uncuyo.edu.ar
ISSN 0120-6230
e-ISSN 2422-2844
DOI: 10.17533/udea.redin.20230110
56
S. Elaskar et al., Revista Facultad de Ingeniería, Universidad de Antioquia, No. 110, pp. 56-64, 2024
Chaotic intermittency was classified into three types
named I, II, and III, following the loss of stability for maps
[17]. Type I intermittency happens if there is a tangent
bifurcation and an eigenvalue leaves the unit circle across
+1. Type II intermittency occurs by a Hopf bifurcation, and
two complex-conjugate eigenvalues move away from the
unit circle. Finally, type III takes place during a subcritical
period-doubling bifurcation, and an eigenvalue escapes
the unit circle by -1 [1, 3]. Posterior research detected
other intermittency types, such as X, on-off, V, eyelet, and
ring [18–27].
In one-dimensional maps, intermittency is produced by
a specific local map and a reinjection mechanism [1, 2].
The local map defines the type of intermittency, and the
reinjection process returns the trajectories to the laminar
zone. The reinjection probability density function (RPD
function) determines the trajectories’ probability of being
reinjected in the laminar zone or interval [1–3].
The correct description of the RPD function is essential
for understanding of intermittency phenomenon. Several
approaches were utilized to calculate the RPD function,
being the uniform reinjection (a constant RPD) the
most implemented [2, 3, 17, 28]. Recently, a more
general methodology to evaluate the RPD function has
been introduced [1], which is called the M function
methodology. This methodology has accurately worked
for types I, II, III, and V intermittencies with and without
noise [29–35].
This paper extends the M function methodology to
evaluate discontinuous RPD functions. These RPDs are
related to two o more overlapping reinjection mechanisms
[1, 29, 36]. The paper shows that discontinuous RPD
functions correspond to non-differentiable M (x)
functions.
The paper is organized into five sections. The second
section briefly describes the M function methodology.
Section 3 extends the M function methodology to evaluate
discontinuous RPD functions. In Section 4, numerical tests
are presented. Finally, in Section 5 there are the main
conclusions.
2. The M (x) function
The RPD function determines the statistical distribution
of trajectories leaving the chaotic region and going back
into the laminar region. The RPD is the more significant
function in describing chaotic intermittency behavior.
Once the RPD function is known, the other statistical
properties can be determined [1, 2].
To evaluate the RPD function, here also called ϕ(x), we
introduce the following function (see Equation (1)) [1]: ˆx
is the lower boundary of reinjection, and ˆx ≤ x ≤ x0 + c.
Where x0 is the vanished or unstable fixed point and the
laminar interval is L = [ˆx, x0 + c]. In addition, ϕ(x) is
a C1 function in L, where a Cq is a q times continuously
differentiable function.
The M (x) function has been extensively used to determine
the statistical properties in chaotic intermittency (see
[1][29–35] and references indicated in these manuscripts).
In several maps showing intermittency, the M (x) function
showed in Equation (2) is a linear function [1] where
m ∈ (0, 1) is a free parameter. Note, the M (x) function
is defined in the laminar interval [ˆx, x0 + c].
Theorem. Let M (x) be a function defined by Equation (1)
and ϕ(x) is a C1 function. If M (x) is the linear function
given by Equation (2). Then, the reinjection probability
density function results in Equation (3).
Proof. For ∫ x
ˆx ϕ(y)dy̸ = 0 the Equation (1) can be written
as expressed in Equation (4). If we differentiate this
equation twice with x, we get the Equation (5). Assuming
that M (x) is given by Equation (2), we obtain d2M (x)
dx2 = 0,
and dM (x)
dx = m, hence the Equation (5) results in Equation
(6). The RPD function (see Equation (7)) is calculated by
integration of the last equation where b is the integration
constant.
Therefore, the RPD function can be written as expressed in
Equation (8), b(α) is a normalization parameter selected
to verify ∫
L ϕ(x) dx = 1, where L is the laminar interval.
To approximate numerically the M (x) function, we notice
that it is an average over reinjection points in the interval
[ˆx, x]. Then, we can write the Equation (9) where the data
set (N reinjection points) {xj }N
j=1 has been previously
ordered, i.e., xj ≤ xj+1. Therefore, ˆx can be approximated
by ˆx ≈ inf{xj }.
We highlight the M (x) function is a useful mathematical
tool to calculate ˆx and α determining the RPD function
and other statistical properties in chaotic intermittency.
Several papers have verified that M (x) is a linear function,
and the RPD function is given by Equation (8) (see [1, 29, 31,
32] and references indicated in these manuscripts).
3. Non-differentiable M (x) function
In this section, we analyze the reinjection mechanisms that
display non-differentiable functions M (x). These cases
happen when there are two o more reinjection processes
57
Chaotic intermittency was classified into three types
named I, II, and III, following the loss of stability for maps
[17]. Type I intermittency happens if there is a tangent
bifurcation and an eigenvalue leaves the unit circle across
+1. Type II intermittency occurs by a Hopf bifurcation, and
two complex-conjugate eigenvalues move away from the
unit circle. Finally, type III takes place during a subcritical
period-doubling bifurcation, and an eigenvalue escapes
the unit circle by -1 [1, 3]. Posterior research detected
other intermittency types, such as X, on-off, V, eyelet, and
ring [18–27].
In one-dimensional maps, intermittency is produced by
a specific local map and a reinjection mechanism [1, 2].
The local map defines the type of intermittency, and the
reinjection process returns the trajectories to the laminar
zone. The reinjection probability density function (RPD
function) determines the trajectories’ probability of being
reinjected in the laminar zone or interval [1–3].
The correct description of the RPD function is essential
for understanding of intermittency phenomenon. Several
approaches were utilized to calculate the RPD function,
being the uniform reinjection (a constant RPD) the
most implemented [2, 3, 17, 28]. Recently, a more
general methodology to evaluate the RPD function has
been introduced [1], which is called the M function
methodology. This methodology has accurately worked
for types I, II, III, and V intermittencies with and without
noise [29–35].
This paper extends the M function methodology to
evaluate discontinuous RPD functions. These RPDs are
related to two o more overlapping reinjection mechanisms
[1, 29, 36]. The paper shows that discontinuous RPD
functions correspond to non-differentiable M (x)
functions.
The paper is organized into five sections. The second
section briefly describes the M function methodology.
Section 3 extends the M function methodology to evaluate
discontinuous RPD functions. In Section 4, numerical tests
are presented. Finally, in Section 5 there are the main
conclusions.
2. The M (x) function
The RPD function determines the statistical distribution
of trajectories leaving the chaotic region and going back
into the laminar region. The RPD is the more significant
function in describing chaotic intermittency behavior.
Once the RPD function is known, the other statistical
properties can be determined [1, 2].
To evaluate the RPD function, here also called ϕ(x), we
introduce the following function (see Equation (1)) [1]: ˆx
is the lower boundary of reinjection, and ˆx ≤ x ≤ x0 + c.
Where x0 is the vanished or unstable fixed point and the
laminar interval is L = [ˆx, x0 + c]. In addition, ϕ(x) is
a C1 function in L, where a Cq is a q times continuously
differentiable function.
The M (x) function has been extensively used to determine
the statistical properties in chaotic intermittency (see
[1][29–35] and references indicated in these manuscripts).
In several maps showing intermittency, the M (x) function
showed in Equation (2) is a linear function [1] where
m ∈ (0, 1) is a free parameter. Note, the M (x) function
is defined in the laminar interval [ˆx, x0 + c].
Theorem. Let M (x) be a function defined by Equation (1)
and ϕ(x) is a C1 function. If M (x) is the linear function
given by Equation (2). Then, the reinjection probability
density function results in Equation (3).
Proof. For ∫ x
ˆx ϕ(y)dy̸ = 0 the Equation (1) can be written
as expressed in Equation (4). If we differentiate this
equation twice with x, we get the Equation (5). Assuming
that M (x) is given by Equation (2), we obtain d2M (x)
dx2 = 0,
and dM (x)
dx = m, hence the Equation (5) results in Equation
(6). The RPD function (see Equation (7)) is calculated by
integration of the last equation where b is the integration
constant.
Therefore, the RPD function can be written as expressed in
Equation (8), b(α) is a normalization parameter selected
to verify ∫
L ϕ(x) dx = 1, where L is the laminar interval.
To approximate numerically the M (x) function, we notice
that it is an average over reinjection points in the interval
[ˆx, x]. Then, we can write the Equation (9) where the data
set (N reinjection points) {xj }N
j=1 has been previously
ordered, i.e., xj ≤ xj+1. Therefore, ˆx can be approximated
by ˆx ≈ inf{xj }.
We highlight the M (x) function is a useful mathematical
tool to calculate ˆx and α determining the RPD function
and other statistical properties in chaotic intermittency.
Several papers have verified that M (x) is a linear function,
and the RPD function is given by Equation (8) (see [1, 29, 31,
32] and references indicated in these manuscripts).
3. Non-differentiable M (x) function
In this section, we analyze the reinjection mechanisms that
display non-differentiable functions M (x). These cases
happen when there are two o more reinjection processes
57
S. Elaskar et al., Revista Facultad de Ingeniería, Universidad de Antioquia, No. 110, pp. 56-64, 2024
M (x) =
{ ∫ x
ˆx y ϕ(y) dy
∫ x
ˆx ϕ(y) dy if ∫ x
ˆx ϕ(y)dy̸ = 0
0 otherwise (1)
M (x) =
{ m(x − ˆx) + ˆx if x ≥ ˆx
0 otherwise (2)
ϕ(x) = b (x − ˆx) 1−2m
m−1 (3)
M (x)
∫ x
ˆx
ϕ(y) dy =
∫ x
ˆx
y ϕ(y) dy (4)
d2M (x)
dx2
∫ x
ˆx
ϕ(y) dy + 2
( dM (x)
dx − 1
)
ϕ(x) + (M (x) − x) dϕ(x)
dx = 0 (5)
dϕ(x)
dx (x − ˆx) (m − 1) + ϕ(x)(2m − 1) = 0 (6)
ϕ(x) = b (x − ˆx) 1−2m
m−1 (7)
ϕ(x) = b(α) (x − ˆx)α where α = 1 − 2m
m − 1 (8)
M (x) = Mj ≡ 1
j
j∑
k=1
xk, xj−1 < x ≤ xj (9)
generating discontinuous RPD functions. Accordingly,
ϕ(x) is not C1 as in the previous section.
Here we introduce a general methodology to describe the
M (x) function and to obtain the RPD function when there
are two different reinjection processes.
Let us consider a generic one-dimensional map
xn+1 = F (xn), which has two reinjection processes,
called here a and b, respectively. Each reinjection process
verifies a linear M (x) function (see Equation (2)). We call
Ma(x) and Mb(x) these functions for each reinjection
process, and they verify the Equation (10), where ˆxa and
ˆxb are the lower boundary of reinjection for the reinjection
processes a and b respectively.
From Equation (10) and using the theory described in
Section 2, we get two independent RPD functions as
expressed in Equation (11). We analyze two general cases.
The first one assumes that the overlap of the reinjection
processes occurs in the lower part of the laminar interval.
On the other hand, the second case considers that the
reinjection processes superposition happens in the upper
part of the laminar interval.
3.1 Case 1. Reinjection overlapping in the
lower part of the laminar interval
First, we study the case with reinjection overlap or
superposition in the lower part of the laminar interval.
Then, the complete RPD function results in Equation (12),
where xs is the non-differentiable point for the global
M (x) function.
Since ba(αa) and bb(αb) are real numbers, we can write
ba(αa) = b, and bb(αb) = k b. Where k is a real number,
and b is evaluated from the normalization condition, which
is (see [1, 2]) If we introduce Equations (11) and (12) in
Equation (13), we get Equation (14) where x0 + c is the
upper limit of the laminar interval, L = [ˆxb, x0 + c],
and we have assumed ˆxb = ˆxa (see Equation (12)).
We emphasize, without loss of generality, that ˆxb could
coincide with x0 − c. Note that k and b, are always real
numbers (see Equation (14)) because of Equation (15).
In Equation (12), we have considered that ϕ1(x) is
determined by two reinjection mechanisms represented
by ϕa(x) and ϕb(x), while ϕ2(x) is calculated by only one
reinjection mechanism given by ϕb(x).
58
M (x) =
{ ∫ x
ˆx y ϕ(y) dy
∫ x
ˆx ϕ(y) dy if ∫ x
ˆx ϕ(y)dy̸ = 0
0 otherwise (1)
M (x) =
{ m(x − ˆx) + ˆx if x ≥ ˆx
0 otherwise (2)
ϕ(x) = b (x − ˆx) 1−2m
m−1 (3)
M (x)
∫ x
ˆx
ϕ(y) dy =
∫ x
ˆx
y ϕ(y) dy (4)
d2M (x)
dx2
∫ x
ˆx
ϕ(y) dy + 2
( dM (x)
dx − 1
)
ϕ(x) + (M (x) − x) dϕ(x)
dx = 0 (5)
dϕ(x)
dx (x − ˆx) (m − 1) + ϕ(x)(2m − 1) = 0 (6)
ϕ(x) = b (x − ˆx) 1−2m
m−1 (7)
ϕ(x) = b(α) (x − ˆx)α where α = 1 − 2m
m − 1 (8)
M (x) = Mj ≡ 1
j
j∑
k=1
xk, xj−1 < x ≤ xj (9)
generating discontinuous RPD functions. Accordingly,
ϕ(x) is not C1 as in the previous section.
Here we introduce a general methodology to describe the
M (x) function and to obtain the RPD function when there
are two different reinjection processes.
Let us consider a generic one-dimensional map
xn+1 = F (xn), which has two reinjection processes,
called here a and b, respectively. Each reinjection process
verifies a linear M (x) function (see Equation (2)). We call
Ma(x) and Mb(x) these functions for each reinjection
process, and they verify the Equation (10), where ˆxa and
ˆxb are the lower boundary of reinjection for the reinjection
processes a and b respectively.
From Equation (10) and using the theory described in
Section 2, we get two independent RPD functions as
expressed in Equation (11). We analyze two general cases.
The first one assumes that the overlap of the reinjection
processes occurs in the lower part of the laminar interval.
On the other hand, the second case considers that the
reinjection processes superposition happens in the upper
part of the laminar interval.
3.1 Case 1. Reinjection overlapping in the
lower part of the laminar interval
First, we study the case with reinjection overlap or
superposition in the lower part of the laminar interval.
Then, the complete RPD function results in Equation (12),
where xs is the non-differentiable point for the global
M (x) function.
Since ba(αa) and bb(αb) are real numbers, we can write
ba(αa) = b, and bb(αb) = k b. Where k is a real number,
and b is evaluated from the normalization condition, which
is (see [1, 2]) If we introduce Equations (11) and (12) in
Equation (13), we get Equation (14) where x0 + c is the
upper limit of the laminar interval, L = [ˆxb, x0 + c],
and we have assumed ˆxb = ˆxa (see Equation (12)).
We emphasize, without loss of generality, that ˆxb could
coincide with x0 − c. Note that k and b, are always real
numbers (see Equation (14)) because of Equation (15).
In Equation (12), we have considered that ϕ1(x) is
determined by two reinjection mechanisms represented
by ϕa(x) and ϕb(x), while ϕ2(x) is calculated by only one
reinjection mechanism given by ϕb(x).
58
S. Elaskar et al., Revista Facultad de Ingeniería, Universidad de Antioquia, No. 110, pp. 56-64, 2024
Ma(x) = ma(x − ˆxa) + ˆxa
Mb(x) = mb(x − ˆxb) + ˆxb
(10)
ϕa(x) = ba(αa) (x − ˆxa)αa , αa = 1−2ma
ma−1
ϕb(x) = bb(αb) (x − ˆxb)αb , αb = 1−2mb
mb−1
(11)
ϕ(x) =
ϕ1(x) = ϕa(x) + ϕb(x), ˆxb ≤ x ≤ xs
ϕ2(x) = ϕb(x), xs < x ≤ x0 + c
(12)
∫ x0+c
ˆxb
ϕ(x) dx = 1 (13)
k b
∫ x0+c
ˆxb
(x − ˆxb)αb dx + b
∫ xs
ˆxa
(x − ˆxa)αa dx = 1 (14)
(x − ˆxb)αb ≥ 0 if ˆxb ≤ x ≤ x0 + c
(x − ˆxa)αa ≥ 0 if ˆxa ≤ x ≤ xs
(15)
Note ϕb(x) is defined in the complete laminar interval L.
However, ϕa(x) acts only in the sub-interval [ˆxb, xs].
To calculate the global M (x) function, we implement
the M function methodology introduced in the previous
section. For x ≤ xs, the function M (x) result in Equation
(16) and, for x > xs we get the Equation (17). Therefore,
the complete M (x) function is given for these equations.
We notice the complete M (x) function has a
non-differentiable point at x = xs. Also, note the
M (x) function does not depend on the normalization
factor b. However, it depends on k. The parameter k is
calculated from Equation (17), and it is given by Equation
(18). In Equation (17), M (x) and x are considered for all
reinjected points verifying x > xs, and k is calculated as
the average of them.
3.2 Case 2. Reinjection overlapping in the
upper part of the laminar interval
Now, we study the second case, where the reinjection
processes overlapping happens in the upper part of the
laminar interval. Therefore, the RPD function can be
written as expressed in Equation (19), where ϕa(x) and
ϕb(x) are obtained as shown in Equation (20), where x0
is the fixed point of the map, and the laminar interval is
L = [x0 − c, x0 + c]. Note that ˆxa = x0 − c is the lower
limit of the laminar interval, L, and ˆxb = xs.
Then, the M functions are obtained in Equation (21), where
Ma(x) is defined inside the complete laminar interval
L = [x0 −c, x0 +c]. However, Mb(x) is given in [xs, x0 +c].
We calculate the parameters ma and mb, for each
reinjection process individually. We apply the M function
methodology previously explained. Then, αa and αb are
obtained in Equation (22). The factor b verifies the Equation
(23).
Therefore, the global M function is given by Equation (24)
for x < xs and by Equation (25) for x ≥ xs. From Equation
(25), we obtain k which is given by Equation (26).
In Equation (26), M (x) and x are considered for all
reinjected points verifying xs ≤ x ≤ x0 − c. We calculate
k as an average of Equation (26) evaluated for all these
values.
4. Applications. Numerical results
In this section, we present two numerical examples of
the theoretical cases described in the previous section.
To verify the generality of the methodology, the first one
uses type V intermittency, and the second analyzes type I
59
Ma(x) = ma(x − ˆxa) + ˆxa
Mb(x) = mb(x − ˆxb) + ˆxb
(10)
ϕa(x) = ba(αa) (x − ˆxa)αa , αa = 1−2ma
ma−1
ϕb(x) = bb(αb) (x − ˆxb)αb , αb = 1−2mb
mb−1
(11)
ϕ(x) =
ϕ1(x) = ϕa(x) + ϕb(x), ˆxb ≤ x ≤ xs
ϕ2(x) = ϕb(x), xs < x ≤ x0 + c
(12)
∫ x0+c
ˆxb
ϕ(x) dx = 1 (13)
k b
∫ x0+c
ˆxb
(x − ˆxb)αb dx + b
∫ xs
ˆxa
(x − ˆxa)αa dx = 1 (14)
(x − ˆxb)αb ≥ 0 if ˆxb ≤ x ≤ x0 + c
(x − ˆxa)αa ≥ 0 if ˆxa ≤ x ≤ xs
(15)
Note ϕb(x) is defined in the complete laminar interval L.
However, ϕa(x) acts only in the sub-interval [ˆxb, xs].
To calculate the global M (x) function, we implement
the M function methodology introduced in the previous
section. For x ≤ xs, the function M (x) result in Equation
(16) and, for x > xs we get the Equation (17). Therefore,
the complete M (x) function is given for these equations.
We notice the complete M (x) function has a
non-differentiable point at x = xs. Also, note the
M (x) function does not depend on the normalization
factor b. However, it depends on k. The parameter k is
calculated from Equation (17), and it is given by Equation
(18). In Equation (17), M (x) and x are considered for all
reinjected points verifying x > xs, and k is calculated as
the average of them.
3.2 Case 2. Reinjection overlapping in the
upper part of the laminar interval
Now, we study the second case, where the reinjection
processes overlapping happens in the upper part of the
laminar interval. Therefore, the RPD function can be
written as expressed in Equation (19), where ϕa(x) and
ϕb(x) are obtained as shown in Equation (20), where x0
is the fixed point of the map, and the laminar interval is
L = [x0 − c, x0 + c]. Note that ˆxa = x0 − c is the lower
limit of the laminar interval, L, and ˆxb = xs.
Then, the M functions are obtained in Equation (21), where
Ma(x) is defined inside the complete laminar interval
L = [x0 −c, x0 +c]. However, Mb(x) is given in [xs, x0 +c].
We calculate the parameters ma and mb, for each
reinjection process individually. We apply the M function
methodology previously explained. Then, αa and αb are
obtained in Equation (22). The factor b verifies the Equation
(23).
Therefore, the global M function is given by Equation (24)
for x < xs and by Equation (25) for x ≥ xs. From Equation
(25), we obtain k which is given by Equation (26).
In Equation (26), M (x) and x are considered for all
reinjected points verifying xs ≤ x ≤ x0 − c. We calculate
k as an average of Equation (26) evaluated for all these
values.
4. Applications. Numerical results
In this section, we present two numerical examples of
the theoretical cases described in the previous section.
To verify the generality of the methodology, the first one
uses type V intermittency, and the second analyzes type I
59
S. Elaskar et al., Revista Facultad de Ingeniería, Universidad de Antioquia, No. 110, pp. 56-64, 2024
M (x) =
∫ x
ˆxa y ϕa(y) dy+∫ x
ˆxb y ϕb(y) dy
∫ x
ˆxa ϕa(y) dy+∫ x
ˆxb ϕb(y) dy =
∫ x
ˆxa y b (y−ˆxa)αa dy+∫ x
ˆxb y b k (y−ˆxb)αb dy
∫ x
ˆxa b (y−ˆxa)αa dy+∫ x
ˆxb b k (y−ˆxb)αb dy
= (−ˆxa+x)1+αa (ˆxa+x(1+αa)) (1+αb)(2+αb)+k(x−ˆxb)1+αb (ˆxb+x(1+αb)) (1+αa)(2+αa)
(2+αa)(2+αb) ((−ˆxa+x)1+αa (1+αb)+k(x−ˆxb)1+αb (1+αa))
(16)
M (x) =
∫ xs
ˆxa y ϕa(y) dy+∫ x
ˆxb y ϕb(y) dy
∫ xs
ˆxa ϕa(y) dy+∫ x
ˆxb ϕb(y) dy =
∫ xs
ˆxa y b (y−ˆxa)αa dy+∫ x
ˆxb y b k (y−ˆxb)αb dy
∫ xs
ˆxa b (y−ˆxa)αa dy+∫ x
ˆxb b k (y−ˆxb)αb dy
= (xs−ˆxa)1+αa (ˆxa+xs(1+αa)) (1+αb)(2+αb)+k(x−ˆxb)1+αb (ˆxb+x(1+αb)) (1+αa)(2+αa)
(2+αa)(2+αb) ((xs−ˆxa)1+αa (1+αb)+k(x−ˆxb)1+αb (1+αa))
(17)
k = M (x) (2+αa)(2+αb) ((xs−ˆxa)1+αa (1+αb)−(xs−ˆxa)1+αa (ˆxa+xs(1+αa)) (1+αb)(2+αb)
(x−ˆxb)1+αb (ˆxb+x(1+αb)) (1+αa)(2+αa)−M (x) (2+αa)(2+αb)(1+αa)(x−ˆxb)1+αb (18)
ϕ(x) =
ϕ1(x) = ϕa(x), x0 − c ≤ x < xs
ϕ2(x) = ϕa(x) + ϕb(x), xs ≤ x ≤ x0 + c
(19)
ϕa(x) = b (x − ˆxa)αa = b (x − x0 + c)αa
ϕb(x) = b k (x − ˆxb)αb = b k (x − xs)αb
(20)
Ma(x) = ma (x − x0 + c) + x0 − c
Mb(x) = mb(x − xs) + xs
(21)
αa = 2 ma − 1
1 − ma
, αb = 2 mb − 1
1 − mb
(22)
∫ x0+c
x0−c
b (x − x0 + c)αa dx +
∫ x0+c
ˆxs
b k (x − xs)αb dx = 1 (23)
M (x) =
∫ x
x0−c y ϕa(y) dy
∫ x
x0−c ϕa(y) dy = x(1 + αa) + x0 − c
2 + αa
(24)
M (x) =
∫ x
x0−c y ϕa(y) dy+∫ x
xs y ϕb(y) dy
∫ x
x0−c ϕa(y) dy+∫ x
xs ϕb(y) dy
= (x−x0+c)1+αa (x0−c+x(1+αa)) (1+αb)(2+αb)+k (x−xs)1+αb (xs+x(1+αb)) (1+αa)(2+αa)
(2+αb)(2+αa) ((x−x0+c)1+αa (1+αb)+k (x−xs)1+αa (1+αa))
(25)
k = (x−x0+c)1+αa (x0−c+x(1+αa)) (1+αb) (2+αb)−M (x) (2+αb) (2+αa) (x−x0+c)1+αa (1+αb)
M (x) (2+αb) (2+αa) (1+αa) (x−xs)1+αa −(x−xs)1+αb (xs+x(1+αb)) (1+αa) (2+αa)
(26)
intermittency.
To accomplish the numerical tests, we make an iterative
process using the corresponding map. Later, we divide
the laminar interval L into Ns sub-intervals, and finally,
we estimate the reinjection’s histogram and the numerical
RPD function. To do it, we utilize N reinjected points
inside the laminar interval. Typically, millions of iterations
are required. This procedure has been applied previously
[1, 29, 37].
60
M (x) =
∫ x
ˆxa y ϕa(y) dy+∫ x
ˆxb y ϕb(y) dy
∫ x
ˆxa ϕa(y) dy+∫ x
ˆxb ϕb(y) dy =
∫ x
ˆxa y b (y−ˆxa)αa dy+∫ x
ˆxb y b k (y−ˆxb)αb dy
∫ x
ˆxa b (y−ˆxa)αa dy+∫ x
ˆxb b k (y−ˆxb)αb dy
= (−ˆxa+x)1+αa (ˆxa+x(1+αa)) (1+αb)(2+αb)+k(x−ˆxb)1+αb (ˆxb+x(1+αb)) (1+αa)(2+αa)
(2+αa)(2+αb) ((−ˆxa+x)1+αa (1+αb)+k(x−ˆxb)1+αb (1+αa))
(16)
M (x) =
∫ xs
ˆxa y ϕa(y) dy+∫ x
ˆxb y ϕb(y) dy
∫ xs
ˆxa ϕa(y) dy+∫ x
ˆxb ϕb(y) dy =
∫ xs
ˆxa y b (y−ˆxa)αa dy+∫ x
ˆxb y b k (y−ˆxb)αb dy
∫ xs
ˆxa b (y−ˆxa)αa dy+∫ x
ˆxb b k (y−ˆxb)αb dy
= (xs−ˆxa)1+αa (ˆxa+xs(1+αa)) (1+αb)(2+αb)+k(x−ˆxb)1+αb (ˆxb+x(1+αb)) (1+αa)(2+αa)
(2+αa)(2+αb) ((xs−ˆxa)1+αa (1+αb)+k(x−ˆxb)1+αb (1+αa))
(17)
k = M (x) (2+αa)(2+αb) ((xs−ˆxa)1+αa (1+αb)−(xs−ˆxa)1+αa (ˆxa+xs(1+αa)) (1+αb)(2+αb)
(x−ˆxb)1+αb (ˆxb+x(1+αb)) (1+αa)(2+αa)−M (x) (2+αa)(2+αb)(1+αa)(x−ˆxb)1+αb (18)
ϕ(x) =
ϕ1(x) = ϕa(x), x0 − c ≤ x < xs
ϕ2(x) = ϕa(x) + ϕb(x), xs ≤ x ≤ x0 + c
(19)
ϕa(x) = b (x − ˆxa)αa = b (x − x0 + c)αa
ϕb(x) = b k (x − ˆxb)αb = b k (x − xs)αb
(20)
Ma(x) = ma (x − x0 + c) + x0 − c
Mb(x) = mb(x − xs) + xs
(21)
αa = 2 ma − 1
1 − ma
, αb = 2 mb − 1
1 − mb
(22)
∫ x0+c
x0−c
b (x − x0 + c)αa dx +
∫ x0+c
ˆxs
b k (x − xs)αb dx = 1 (23)
M (x) =
∫ x
x0−c y ϕa(y) dy
∫ x
x0−c ϕa(y) dy = x(1 + αa) + x0 − c
2 + αa
(24)
M (x) =
∫ x
x0−c y ϕa(y) dy+∫ x
xs y ϕb(y) dy
∫ x
x0−c ϕa(y) dy+∫ x
xs ϕb(y) dy
= (x−x0+c)1+αa (x0−c+x(1+αa)) (1+αb)(2+αb)+k (x−xs)1+αb (xs+x(1+αb)) (1+αa)(2+αa)
(2+αb)(2+αa) ((x−x0+c)1+αa (1+αb)+k (x−xs)1+αa (1+αa))
(25)
k = (x−x0+c)1+αa (x0−c+x(1+αa)) (1+αb) (2+αb)−M (x) (2+αb) (2+αa) (x−x0+c)1+αa (1+αb)
M (x) (2+αb) (2+αa) (1+αa) (x−xs)1+αa −(x−xs)1+αb (xs+x(1+αb)) (1+αa) (2+αa)
(26)
intermittency.
To accomplish the numerical tests, we make an iterative
process using the corresponding map. Later, we divide
the laminar interval L into Ns sub-intervals, and finally,
we estimate the reinjection’s histogram and the numerical
RPD function. To do it, we utilize N reinjected points
inside the laminar interval. Typically, millions of iterations
are required. This procedure has been applied previously
[1, 29, 37].
60
S. Elaskar et al., Revista Facultad de Ingeniería, Universidad de Antioquia, No. 110, pp. 56-64, 2024
4.1 Case 1. Type V intermittency with
overlapping in the lower part of the
laminar interval
Let us introduce the following map [38] in Equation (27)
(see below), where F (xm) = ym = 1, ˆx is the lower
boundary of reinjection, and ε is the control parameter.
This map has a fixed point x = 0 for ε = 0. If 0 < ε ≪ 1,
type V intermittency occurs. Figure 1 shows the map for
a1 = 0.9, a2 = 2, ε = 0.001, and ˆx = −0.2.0 0.5 1
-0.2
0
0.2
0.4
0.6
0.8
Figure 1 Map given by Equation (27) for a1 = 0.9, a2 = 2,
ε = 0.001, and ˆx = −0.2. Black line: map. Blue dashed line:
bisector.-0.1 -0.05 0 0.05 0.1
-0.11
-0.1
-0.09
-0.08
-0.07
-0.06
Figure 2 The complete M function given by equations (16)-(17)
for a1 = 0.9, a2 = 2, ε = 0.001, and ˆx = −0.2. Blue line:
numerical M (x) function. Black line: theoretical M (x)
function.
To evaluate the equations described in Section 3, we
analyze the following test: a1 = 0.9, a2 = 2, and
ε = 0.001, ˆx = −0.2, c = 0.114, and N = 100000,
where N is the number of reinjected points. We obtain
k = 0.1868, b = 7.25689, ma = 0.4077 (αa =
−0.3117), mb = 0.4518 (αb = −0.1757). Figures 2
and 3 show the complete M (x) function and the RPD
function, respectively. The M (x) function is given by
equations (16)-(17), and the RPD function is determined by
Equation (12). From the figures, we observe that the theory
here described works accurately regarding the numerical
results. Note that the M function is non-differentiable at
xs = −0.1016, and the RPD function is discontinued at
that point.
4.2 Case 2. Type I intermittency with
overlapping in the upper part of the
laminar interval
In a recent paper, the reinjection mechanism for type I
intermittency in the logistic map was studied [36]. Here, we
use the results of this paper to verify the theory developed
in the previous section.
Because the logistic map displays type I intermittency
close a period-3 window, the third iterate of this map is
studied in Equation (28), where the Equation (29) is the
logistic map.
We consider the fixed point x0 = 0.5143552770619905.
We shift the third iteration of the logistic map, I3
μ(x), so
that the fixed point x0 matches the origin of the coordinate
system. Then, the map results in Equation (30).
For μc = 1 + √8, a period-3 cycle is a solution of
the logistic map. However, for μ < μc but near it,
there is a tangent bifurcation, and type I intermittency
occurs. Figure 4 shows the map given by Equation (30) for
μ = 3.8278 < μc.
For the map given by Equation (30), the reinjection process
around the fixed point depends on the relation between
cl, cd and the laminar interval semi-amplitude, c (L =
[−c, c]). The parameter cl verifies F (−cl) = cl, and it is
the laminar interval limit where there are pre-reinjection
points close to −c. For c ≤ cl there is reinjection from
neighboring points to points to −c. The parameter cd,
satisfies d F (x)
d x
∣
∣
∣x=cd
= 0 and F (cd) ∈ L.
For the sub-interval |cd| ≤ c ≤ cl the map shows
two reinjection mechanisms, then the RPD function
possesses two components. The first one is produced
by pre-reinjection points away from the laminar interval,
and the trajectories going by for these pre-reinjection
points are reinjected in the complete laminar interval,
L = [−c, c]. The second component is given by
pre-reinjection points neighboring to the laminar interval
lower limit, and they satisfy F (xn−1) ∈ (F (−c), c], where
xn−1 are the pre-reinjection points [36].
To study the reinjection processes, we consider the
following test: μ = 3.8278 and |cd| = 0.014355277062 <
61
4.1 Case 1. Type V intermittency with
overlapping in the lower part of the
laminar interval
Let us introduce the following map [38] in Equation (27)
(see below), where F (xm) = ym = 1, ˆx is the lower
boundary of reinjection, and ε is the control parameter.
This map has a fixed point x = 0 for ε = 0. If 0 < ε ≪ 1,
type V intermittency occurs. Figure 1 shows the map for
a1 = 0.9, a2 = 2, ε = 0.001, and ˆx = −0.2.0 0.5 1
-0.2
0
0.2
0.4
0.6
0.8
Figure 1 Map given by Equation (27) for a1 = 0.9, a2 = 2,
ε = 0.001, and ˆx = −0.2. Black line: map. Blue dashed line:
bisector.-0.1 -0.05 0 0.05 0.1
-0.11
-0.1
-0.09
-0.08
-0.07
-0.06
Figure 2 The complete M function given by equations (16)-(17)
for a1 = 0.9, a2 = 2, ε = 0.001, and ˆx = −0.2. Blue line:
numerical M (x) function. Black line: theoretical M (x)
function.
To evaluate the equations described in Section 3, we
analyze the following test: a1 = 0.9, a2 = 2, and
ε = 0.001, ˆx = −0.2, c = 0.114, and N = 100000,
where N is the number of reinjected points. We obtain
k = 0.1868, b = 7.25689, ma = 0.4077 (αa =
−0.3117), mb = 0.4518 (αb = −0.1757). Figures 2
and 3 show the complete M (x) function and the RPD
function, respectively. The M (x) function is given by
equations (16)-(17), and the RPD function is determined by
Equation (12). From the figures, we observe that the theory
here described works accurately regarding the numerical
results. Note that the M function is non-differentiable at
xs = −0.1016, and the RPD function is discontinued at
that point.
4.2 Case 2. Type I intermittency with
overlapping in the upper part of the
laminar interval
In a recent paper, the reinjection mechanism for type I
intermittency in the logistic map was studied [36]. Here, we
use the results of this paper to verify the theory developed
in the previous section.
Because the logistic map displays type I intermittency
close a period-3 window, the third iterate of this map is
studied in Equation (28), where the Equation (29) is the
logistic map.
We consider the fixed point x0 = 0.5143552770619905.
We shift the third iteration of the logistic map, I3
μ(x), so
that the fixed point x0 matches the origin of the coordinate
system. Then, the map results in Equation (30).
For μc = 1 + √8, a period-3 cycle is a solution of
the logistic map. However, for μ < μc but near it,
there is a tangent bifurcation, and type I intermittency
occurs. Figure 4 shows the map given by Equation (30) for
μ = 3.8278 < μc.
For the map given by Equation (30), the reinjection process
around the fixed point depends on the relation between
cl, cd and the laminar interval semi-amplitude, c (L =
[−c, c]). The parameter cl verifies F (−cl) = cl, and it is
the laminar interval limit where there are pre-reinjection
points close to −c. For c ≤ cl there is reinjection from
neighboring points to points to −c. The parameter cd,
satisfies d F (x)
d x
∣
∣
∣x=cd
= 0 and F (cd) ∈ L.
For the sub-interval |cd| ≤ c ≤ cl the map shows
two reinjection mechanisms, then the RPD function
possesses two components. The first one is produced
by pre-reinjection points away from the laminar interval,
and the trajectories going by for these pre-reinjection
points are reinjected in the complete laminar interval,
L = [−c, c]. The second component is given by
pre-reinjection points neighboring to the laminar interval
lower limit, and they satisfy F (xn−1) ∈ (F (−c), c], where
xn−1 are the pre-reinjection points [36].
To study the reinjection processes, we consider the
following test: μ = 3.8278 and |cd| = 0.014355277062 <
61
S. Elaskar et al., Revista Facultad de Ingeniería, Universidad de Antioquia, No. 110, pp. 56-64, 2024
F (x) =
F1(x) = a1 x + ε , ˆx ≤ x < 0
F2(x) = a2 x + ε , 0 ≤ x < xm
F3(x) = ˆx + (ym − ˆx)
( ym−x
ym−xm
)γ
, xm ≤ x ≤ ym
(27)
xn+1 = I3
μ(xn) (28)
Iμ(x) = μ x (1 − x) (29)
F (x) = −x0 + μ3 (1 − (x + x0)) (x + x0) (1 + μ (−1 + (x + x0)) (x + x0))
×(1 + μ2 (−1 + (x + x0)) (x + x0) (1 + μ (−1 + (x + x0)) (x + x0)))
(30)-0.1 -0.05 0 0.05 0.1
0
20
40
60
80
Figure 3 RPD function given by map given by Equation (12) for
a1 = 0.9, a2 = 2, ε = 0.001, and ˆx = −0.2. Black points:
theoretical RPD function. Blue points: numerical RPD function.-0.5 0 0.5
-0.5
0
0.5
Figure 4 Map given by Equation (30) for μ = 3.8278. Black
points: map. Blue dashed line: bisector.
c = 0.04 < cl = 0.058036869.
The RPD function, ϕ(x), is calculated by adding two
reinjection mechanisms described by ϕa(x) and ϕb(x).-0.02 0 0.02 0.04
5
10
15
20
25
Figure 5 RPD functions for μ = 3.8278 and c = 0.04. Black
line: the RPD obtained using the theoretical development of the
previous section. Blue points: the numerical RPD function.
αa = −0.0168018, αb = −0.0981602.-0.04 -0.02 0 0.02 0.04
-0.04
-0.03
-0.02
-0.01
0
0.01
F(-c)
Figure 6 M (x) function for μ = 3.8278 and c = 0.04. The
green line corresponds to xs = F (−c) = 0.0167937.
To get the slopes ma and mb, we study each reinjection
mechanism individually. We order the numerical data and
apply the M function methodology described in Section 2.
62
F (x) =
F1(x) = a1 x + ε , ˆx ≤ x < 0
F2(x) = a2 x + ε , 0 ≤ x < xm
F3(x) = ˆx + (ym − ˆx)
( ym−x
ym−xm
)γ
, xm ≤ x ≤ ym
(27)
xn+1 = I3
μ(xn) (28)
Iμ(x) = μ x (1 − x) (29)
F (x) = −x0 + μ3 (1 − (x + x0)) (x + x0) (1 + μ (−1 + (x + x0)) (x + x0))
×(1 + μ2 (−1 + (x + x0)) (x + x0) (1 + μ (−1 + (x + x0)) (x + x0)))
(30)-0.1 -0.05 0 0.05 0.1
0
20
40
60
80
Figure 3 RPD function given by map given by Equation (12) for
a1 = 0.9, a2 = 2, ε = 0.001, and ˆx = −0.2. Black points:
theoretical RPD function. Blue points: numerical RPD function.-0.5 0 0.5
-0.5
0
0.5
Figure 4 Map given by Equation (30) for μ = 3.8278. Black
points: map. Blue dashed line: bisector.
c = 0.04 < cl = 0.058036869.
The RPD function, ϕ(x), is calculated by adding two
reinjection mechanisms described by ϕa(x) and ϕb(x).-0.02 0 0.02 0.04
5
10
15
20
25
Figure 5 RPD functions for μ = 3.8278 and c = 0.04. Black
line: the RPD obtained using the theoretical development of the
previous section. Blue points: the numerical RPD function.
αa = −0.0168018, αb = −0.0981602.-0.04 -0.02 0 0.02 0.04
-0.04
-0.03
-0.02
-0.01
0
0.01
F(-c)
Figure 6 M (x) function for μ = 3.8278 and c = 0.04. The
green line corresponds to xs = F (−c) = 0.0167937.
To get the slopes ma and mb, we study each reinjection
mechanism individually. We order the numerical data and
apply the M function methodology described in Section 2.
62
S. Elaskar et al., Revista Facultad de Ingeniería, Universidad de Antioquia, No. 110, pp. 56-64, 2024
The RPD function is shown in Figure 5. The blue points are
the numerical data, and the black line is the theoretical
RPD. The RPD function shows two distinct behaviors, one
for x < xs, and another for x ≥ xs. Where xs = F (−c).
For the same test, the M (x) function is displayed in Figure
6. We notice the M (x) function has a non-differentiable
point at xs = F (−c), where the RPD function is
discontinuous.
5. Conclusions
In this paper, we presented a systematic methodology to
obtain the reinjection probability density function in chaotic
intermittency when there are two overlapping reinjection
processes. We extended the M function methodology
developed and verified in previous studies [1, 29–32, 34].
Two different cases were studied and described. The first
one considers that the overlapping occurs in the lower
part of the laminar interval. The second case assumes an
overlapping in the upper part of the laminar interval. For
both cases, we developed theoretical equations for M (x)
and RPD functions.
To satisfy the normalization condition we introduced a new
real parameter, called here k. This parameter takes into
account the different number of reinjected points in both
sub-intervals that make up the complete laminar interval.
We have verified the accurate behavior of the theoretical
background here introduced by comparison with numerical
data. We carried out two numerical tests. The first
one studied the overlapping close to the lower part of
the laminar interval, and the second test analyzed the
reinjection mechanism when there is overlapping in the
upper part of the laminar interval. One test considered type
V intermittency, and another one used type I intermittency.
In both tests, the theoretical results are very accurate
regarding the numerical data.
6. Declaration of competing interest
We declare that we have no significant competing interests,
including financial or non-financial, professional, or
personal interests interfering with the full and objective
presentation of the work described in this manuscript.
7. Acknowledgements
The authors are grateful to Universidad Nacional de
Córdoba and Universidad Politécnica de Madrid.
8. Funding
This work was supported by SECyT of Universidad
Nacional de Córdoba, and Ministerio de Ciencia,
Innovación y Universidades of Spain under grand No
RTI2018-094409-B-I00.
9. Author contributions
S.E. Conceptualization, methodology, software, validation,
formal analysis, investigation, resources, writing the
paper, visualization and project administration. E.dR.
Methodology, validation, formal analysis, resources and
writing the paper. M.G. Numerical results and Validation.
All authors have read and agreed to the published version
of the manuscript.
10. Data availability statement
The authors confirm that the data supporting the findings
of this study are available within the article [and/or] its
supplementary materials.
References
[1] S. Elaskar and E. Del Río, New advances on chaotic intermittency and
its applications. Springer, 2017.
[2] H. G. Schuster and W. Just, Deterministic chaos. John Wiley & Sons,
2005.
[3] A. H. Nayfeh and B. Balachandran, Applied nonlinear dynamics. John
Wiley & Sons, 1995.
[4] G. Pizza, C. E. Frouzakis, and J. Mantzaras, “Chaotic dynamics
in premixed hydrogen/air channel flow combustion,” Combustion
Theory and Modelling, vol. 16, no. 2, pp. 275–299, 2012.
[5] P. De Anna, T. Le Borgne, M. Dentz, A. M. Tartakovsky, D. Bolster, and
P. Davy, “Flow intermittency, dispersion, and correlated continuous
time random walks in porous media,” Physical review letters, vol. 110,
no. 18, p. 184502, 2013.
[6] C. Stan, C. Cristescu, and D. Dimitriu, “Analysis of the intermittent
behavior in a low-temperature discharge plasma by recurrence plot
quantification,” Physics of Plasmas, vol. 17, no. 4, p. 042115, 2010.
[7] A. Chian, Complex systems approach to economic dynamics. Springer
Science & Business Media, 2007.
[8] J. Żebrowski and R. Baranowski, “Type i intermittency in
nonstationary systems—models and human heart rate variability,”
Physica A: Statistical Mechanics and Its Applications, vol. 336, no. 1-2,
pp. 74–83, 2004.
[9] P. Paradisi, P. Allegrini, A. Gemignani, M. Laurino, D. Menicucci,
and A. Piarulli, “Scaling and intermittency of brain events as a
manifestation of consciousness,” in AIP Conference Proceedings, vol.
1510, no. 1. American Institute of Physics, 2013, pp. 151–161.
[10] J. Bragard, J. Vélez, J. Riquelme, L. Pérez, R. Hernández-García,
R. Barrientos, and D. Laroze, “Study of type-iii intermittency in the
landau–lifshitz-gilbert equation,” Physica Scripta, vol. 96, no. 12, p.
124045, 2021.
[11] P. Ge and H. Cao, “Intermittent evolution routes to the periodic or
the chaotic orbits in rulkov map,” Chaos: An Interdisciplinary Journal
of Nonlinear Science, vol. 31, no. 9, p. 093119, 2021.
[12] I. Belyaev, D. Biryukov, D. N. Gerasimov, and E. Yurin, “On-off
intermittency and hard turbulence in the flow of fluid in the magnetic
63
The RPD function is shown in Figure 5. The blue points are
the numerical data, and the black line is the theoretical
RPD. The RPD function shows two distinct behaviors, one
for x < xs, and another for x ≥ xs. Where xs = F (−c).
For the same test, the M (x) function is displayed in Figure
6. We notice the M (x) function has a non-differentiable
point at xs = F (−c), where the RPD function is
discontinuous.
5. Conclusions
In this paper, we presented a systematic methodology to
obtain the reinjection probability density function in chaotic
intermittency when there are two overlapping reinjection
processes. We extended the M function methodology
developed and verified in previous studies [1, 29–32, 34].
Two different cases were studied and described. The first
one considers that the overlapping occurs in the lower
part of the laminar interval. The second case assumes an
overlapping in the upper part of the laminar interval. For
both cases, we developed theoretical equations for M (x)
and RPD functions.
To satisfy the normalization condition we introduced a new
real parameter, called here k. This parameter takes into
account the different number of reinjected points in both
sub-intervals that make up the complete laminar interval.
We have verified the accurate behavior of the theoretical
background here introduced by comparison with numerical
data. We carried out two numerical tests. The first
one studied the overlapping close to the lower part of
the laminar interval, and the second test analyzed the
reinjection mechanism when there is overlapping in the
upper part of the laminar interval. One test considered type
V intermittency, and another one used type I intermittency.
In both tests, the theoretical results are very accurate
regarding the numerical data.
6. Declaration of competing interest
We declare that we have no significant competing interests,
including financial or non-financial, professional, or
personal interests interfering with the full and objective
presentation of the work described in this manuscript.
7. Acknowledgements
The authors are grateful to Universidad Nacional de
Córdoba and Universidad Politécnica de Madrid.
8. Funding
This work was supported by SECyT of Universidad
Nacional de Córdoba, and Ministerio de Ciencia,
Innovación y Universidades of Spain under grand No
RTI2018-094409-B-I00.
9. Author contributions
S.E. Conceptualization, methodology, software, validation,
formal analysis, investigation, resources, writing the
paper, visualization and project administration. E.dR.
Methodology, validation, formal analysis, resources and
writing the paper. M.G. Numerical results and Validation.
All authors have read and agreed to the published version
of the manuscript.
10. Data availability statement
The authors confirm that the data supporting the findings
of this study are available within the article [and/or] its
supplementary materials.
References
[1] S. Elaskar and E. Del Río, New advances on chaotic intermittency and
its applications. Springer, 2017.
[2] H. G. Schuster and W. Just, Deterministic chaos. John Wiley & Sons,
2005.
[3] A. H. Nayfeh and B. Balachandran, Applied nonlinear dynamics. John
Wiley & Sons, 1995.
[4] G. Pizza, C. E. Frouzakis, and J. Mantzaras, “Chaotic dynamics
in premixed hydrogen/air channel flow combustion,” Combustion
Theory and Modelling, vol. 16, no. 2, pp. 275–299, 2012.
[5] P. De Anna, T. Le Borgne, M. Dentz, A. M. Tartakovsky, D. Bolster, and
P. Davy, “Flow intermittency, dispersion, and correlated continuous
time random walks in porous media,” Physical review letters, vol. 110,
no. 18, p. 184502, 2013.
[6] C. Stan, C. Cristescu, and D. Dimitriu, “Analysis of the intermittent
behavior in a low-temperature discharge plasma by recurrence plot
quantification,” Physics of Plasmas, vol. 17, no. 4, p. 042115, 2010.
[7] A. Chian, Complex systems approach to economic dynamics. Springer
Science & Business Media, 2007.
[8] J. Żebrowski and R. Baranowski, “Type i intermittency in
nonstationary systems—models and human heart rate variability,”
Physica A: Statistical Mechanics and Its Applications, vol. 336, no. 1-2,
pp. 74–83, 2004.
[9] P. Paradisi, P. Allegrini, A. Gemignani, M. Laurino, D. Menicucci,
and A. Piarulli, “Scaling and intermittency of brain events as a
manifestation of consciousness,” in AIP Conference Proceedings, vol.
1510, no. 1. American Institute of Physics, 2013, pp. 151–161.
[10] J. Bragard, J. Vélez, J. Riquelme, L. Pérez, R. Hernández-García,
R. Barrientos, and D. Laroze, “Study of type-iii intermittency in the
landau–lifshitz-gilbert equation,” Physica Scripta, vol. 96, no. 12, p.
124045, 2021.
[11] P. Ge and H. Cao, “Intermittent evolution routes to the periodic or
the chaotic orbits in rulkov map,” Chaos: An Interdisciplinary Journal
of Nonlinear Science, vol. 31, no. 9, p. 093119, 2021.
[12] I. Belyaev, D. Biryukov, D. N. Gerasimov, and E. Yurin, “On-off
intermittency and hard turbulence in the flow of fluid in the magnetic
63
S. Elaskar et al., Revista Facultad de Ingeniería, Universidad de Antioquia, No. 110, pp. 56-64, 2024
field,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 29,
no. 8, p. 083119, 2019.
[13] P. Bordbar and S. Ahadpour, “Type-i intermittency from markov
binary block visibility graph perspective,” Journal of Applied
Statistics, vol. 48, no. 7, pp. 1303–1318, 2021.
[14] L.-W. Kong, H. Fan, C. Grebogi, and Y.-C. Lai, “Emergence of
transient chaos and intermittency in machine learning,” Journal of
Physics: Complexity, vol. 2, no. 3, p. 035014, 2021.
[15] S. G. Stavrinides, A. N. Miliou, T. Laopoulos, and A. Anagnostopoulos,
“The intermittency route to chaos of an electronic digital oscillator,”
International Journal of Bifurcation and Chaos, vol. 18, no. 05, pp.
1561–1566, 2008.
[16] S. Zambrano, I. P. Mariño, and M. A. Sanjuán, “Controlling
crisis-induced intermittency using its relation with a boundary
crisis,” New Journal of Physics, vol. 11, no. 2, p. 023025, 2009.
[17] P. Manneville and Y. Pomeau, “Intermittency and the lorenz model,”
Physics Letters A, vol. 75, no. 1-2, pp. 1–2, 1979.
[18] M. Bauer, S. Habip, D.-R. He, and W. Martienssen, “New type of
intermittency in discontinuous maps,” Physical review letters, vol. 68,
no. 11, p. 1625, 1992.
[19] D.-R. He, M. Bauer, S. Habip, U. Krueger, W. Martienssen,
B. Christiansen, and B.-H. Wang, “Type v intermittency,” Physics
Letters A, vol. 171, no. 1-2, pp. 61–65, 1992.
[20] J. Fan, F. Ji, S. Guan, B.-H. Wang, and D.-R. He, “The distribution of
laminar lengths in type v intermittency,” Physics Letters A, vol. 182,
no. 2-3, pp. 232–237, 1993.
[21] T. Price and T. Mullin, “An experimental observation of a new type
of intermittency,” Physica D: Nonlinear Phenomena, vol. 48, no. 1, pp.
29–52, 1991.
[22] N. Platt, E. Spiegel, and C. Tresser, “On-off intermittency: A
mechanism for bursting,” Physical Review Letters, vol. 70, no. 3, p.
279, 1993.
[23] J. Heagy, N. Platt, and S. Hammel, “Characterization of on-off
intermittency,” Physical Review E, vol. 49, no. 2, p. 1140, 1994.
[24] A. Pikovsky, G. Osipov, M. Rosenblum, M. Zaks, and J. Kurths,
“Attractor-repeller collision and eyelet intermittency at the
transition to phase synchronization,” Physical review letters,
vol. 79, no. 1, p. 47, 1997.
[25] A. Pikovsky, M. Rosenblum, and J. Kurths, “Synchronization: a
universal concept in nonlinear science,” 2002.
[26] M. Kurovskaya, “Distribution of laminar phases at eyelet-type
intermittency,” Technical Physics Letters, vol. 34, no. 12, pp.
1063–1065, 2008.
[27] A. E. Hramov, A. A. Koronovskii, M. K. Kurovskaya, and S. Boccaletti,
“Ring intermittency in coupled chaotic oscillators at the boundary
of phase synchronization,” Physical review letters, vol. 97, no. 11, p.
114101, 2006.
[28] J. Hirsch, B. Huberman, and D. Scalapino, “Theory of intermittency,”
Physical Review A, vol. 25, no. 1, p. 519, 1982.
[29] S. Elaskar, E. Del Rio, and J. M. Donoso, “Reinjection probability
density in type-iii intermittency,” Physica A: Statistical Mechanics and
its Applications, vol. 390, no. 15, pp. 2759–2768, 2011.
[30] E. del Rio, S. Elaskar, and V. A. Makarov, “Theory of intermittency
applied to classical pathological cases,” Chaos: An Interdisciplinary
Journal of Nonlinear Science, vol. 23, no. 3, p. 033112, 2013.
[31] E. del Rio, S. Elaskar, and J. M. Donoso, “Laminar length and
characteristic relation in type-i intermittency,” Communications in
Nonlinear Science and Numerical Simulation, vol. 19, no. 4, pp.
967–976, 2014.
[32] G. Krause, S. Elaskar, and E. del Río, “Noise effect on statistical
properties of type-i intermittency,” Physica A: Statistical Mechanics
and its Applications, vol. 402, pp. 318–329, 2014.
[33] S. Elaskar, E. del Rio, G. Krause, and A. Costa, “Effect of the lower
boundary of reinjection and noise in type-ii intermittency,” Nonlinear
Dynamics, vol. 79, no. 2, pp. 1411–1424, 2015.
[34] S. Elaskar and E. Del Río, “Discontinuous reinjection probability
density functions in type v intermittency,” Journal of Computational
and Nonlinear Dynamics, vol. 13, no. 12, p. 121001, 2018.
[35] S. Elaskar, E. Del Rio, and A. Costa, “Reinjection probability
density for type-iii intermittency with noise and lower boundary
of reinjection,” Journal of Computational and Nonlinear Dynamics,
vol. 12, no. 3, p. 031020, 2017.
[36] S. Elaskar, E. del Río, and S. Elaskar, “Intermittency reinjection in
the logistic map,” Symmetry, vol. 14, no. 3, p. 481, 2022.
[37] S. Elaskar, E. del Río, and W. Schulz, “Analysis of the type
v intermittency using the perron-frobenius operator,” Symmetry,
vol. 14, p. 2519, 2022.
[38] S. Elaskar, E. del Río, and L. Gutierrez Marcantoni, “Nonuniform
reinjection probability density function in type v intermittency,”
Nonlinear Dynamics, vol. 92, p. 683697, 2018.
64
field,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 29,
no. 8, p. 083119, 2019.
[13] P. Bordbar and S. Ahadpour, “Type-i intermittency from markov
binary block visibility graph perspective,” Journal of Applied
Statistics, vol. 48, no. 7, pp. 1303–1318, 2021.
[14] L.-W. Kong, H. Fan, C. Grebogi, and Y.-C. Lai, “Emergence of
transient chaos and intermittency in machine learning,” Journal of
Physics: Complexity, vol. 2, no. 3, p. 035014, 2021.
[15] S. G. Stavrinides, A. N. Miliou, T. Laopoulos, and A. Anagnostopoulos,
“The intermittency route to chaos of an electronic digital oscillator,”
International Journal of Bifurcation and Chaos, vol. 18, no. 05, pp.
1561–1566, 2008.
[16] S. Zambrano, I. P. Mariño, and M. A. Sanjuán, “Controlling
crisis-induced intermittency using its relation with a boundary
crisis,” New Journal of Physics, vol. 11, no. 2, p. 023025, 2009.
[17] P. Manneville and Y. Pomeau, “Intermittency and the lorenz model,”
Physics Letters A, vol. 75, no. 1-2, pp. 1–2, 1979.
[18] M. Bauer, S. Habip, D.-R. He, and W. Martienssen, “New type of
intermittency in discontinuous maps,” Physical review letters, vol. 68,
no. 11, p. 1625, 1992.
[19] D.-R. He, M. Bauer, S. Habip, U. Krueger, W. Martienssen,
B. Christiansen, and B.-H. Wang, “Type v intermittency,” Physics
Letters A, vol. 171, no. 1-2, pp. 61–65, 1992.
[20] J. Fan, F. Ji, S. Guan, B.-H. Wang, and D.-R. He, “The distribution of
laminar lengths in type v intermittency,” Physics Letters A, vol. 182,
no. 2-3, pp. 232–237, 1993.
[21] T. Price and T. Mullin, “An experimental observation of a new type
of intermittency,” Physica D: Nonlinear Phenomena, vol. 48, no. 1, pp.
29–52, 1991.
[22] N. Platt, E. Spiegel, and C. Tresser, “On-off intermittency: A
mechanism for bursting,” Physical Review Letters, vol. 70, no. 3, p.
279, 1993.
[23] J. Heagy, N. Platt, and S. Hammel, “Characterization of on-off
intermittency,” Physical Review E, vol. 49, no. 2, p. 1140, 1994.
[24] A. Pikovsky, G. Osipov, M. Rosenblum, M. Zaks, and J. Kurths,
“Attractor-repeller collision and eyelet intermittency at the
transition to phase synchronization,” Physical review letters,
vol. 79, no. 1, p. 47, 1997.
[25] A. Pikovsky, M. Rosenblum, and J. Kurths, “Synchronization: a
universal concept in nonlinear science,” 2002.
[26] M. Kurovskaya, “Distribution of laminar phases at eyelet-type
intermittency,” Technical Physics Letters, vol. 34, no. 12, pp.
1063–1065, 2008.
[27] A. E. Hramov, A. A. Koronovskii, M. K. Kurovskaya, and S. Boccaletti,
“Ring intermittency in coupled chaotic oscillators at the boundary
of phase synchronization,” Physical review letters, vol. 97, no. 11, p.
114101, 2006.
[28] J. Hirsch, B. Huberman, and D. Scalapino, “Theory of intermittency,”
Physical Review A, vol. 25, no. 1, p. 519, 1982.
[29] S. Elaskar, E. Del Rio, and J. M. Donoso, “Reinjection probability
density in type-iii intermittency,” Physica A: Statistical Mechanics and
its Applications, vol. 390, no. 15, pp. 2759–2768, 2011.
[30] E. del Rio, S. Elaskar, and V. A. Makarov, “Theory of intermittency
applied to classical pathological cases,” Chaos: An Interdisciplinary
Journal of Nonlinear Science, vol. 23, no. 3, p. 033112, 2013.
[31] E. del Rio, S. Elaskar, and J. M. Donoso, “Laminar length and
characteristic relation in type-i intermittency,” Communications in
Nonlinear Science and Numerical Simulation, vol. 19, no. 4, pp.
967–976, 2014.
[32] G. Krause, S. Elaskar, and E. del Río, “Noise effect on statistical
properties of type-i intermittency,” Physica A: Statistical Mechanics
and its Applications, vol. 402, pp. 318–329, 2014.
[33] S. Elaskar, E. del Rio, G. Krause, and A. Costa, “Effect of the lower
boundary of reinjection and noise in type-ii intermittency,” Nonlinear
Dynamics, vol. 79, no. 2, pp. 1411–1424, 2015.
[34] S. Elaskar and E. Del Río, “Discontinuous reinjection probability
density functions in type v intermittency,” Journal of Computational
and Nonlinear Dynamics, vol. 13, no. 12, p. 121001, 2018.
[35] S. Elaskar, E. Del Rio, and A. Costa, “Reinjection probability
density for type-iii intermittency with noise and lower boundary
of reinjection,” Journal of Computational and Nonlinear Dynamics,
vol. 12, no. 3, p. 031020, 2017.
[36] S. Elaskar, E. del Río, and S. Elaskar, “Intermittency reinjection in
the logistic map,” Symmetry, vol. 14, no. 3, p. 481, 2022.
[37] S. Elaskar, E. del Río, and W. Schulz, “Analysis of the type
v intermittency using the perron-frobenius operator,” Symmetry,
vol. 14, p. 2519, 2022.
[38] S. Elaskar, E. del Río, and L. Gutierrez Marcantoni, “Nonuniform
reinjection probability density function in type v intermittency,”
Nonlinear Dynamics, vol. 92, p. 683697, 2018.
64