SINCRONIZACIÓN DE MAPAS CAÓTICOS DISCRETOS

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Congreso Internacional de Investigación Tijuana.
Revista Aristas: Investigación Básica y Aplicada.
ISSN 2007-9478, Vol. 4, Núm. 7. Año 2015.
SINCRONIZACIÓN DE MAPAS CAÓTICOS DISCRETOS UNIDIMENSIONALES EN
SISTEMAS NO LINEALES ACOPLADOS
Synchronization of One-Dimensional Discrete Chaotic Maps in Coupled Nonlinear Systems
Abstract—Conditions and characteristics of chaos synchronization by
the method of coupled systems of one-dimensional chaotic maps are
presented in this paper; specifically the following maps were studied:
Bernoulli, Chebyshev, Congruent, Cosine, Exponent, Hopping,
Logistic and Tent. The speed of synchronization was determined
statistically; based on the performance of ten thousand experiments by
parameter and chaotic map. As a result, based on the promptness of
synchronization, the chaotic maps under study were classified
according with the number of iteration performed to get it. Similarly
the existence of stability in the phenomenon under study was
determined, expanding the study to different map parameters under the
chaotic behavior.
GARCÉS GUZMÁN HÉCTOR
Ingeniero en Comunicaciones y Electrónica,
Ph. D.
Profesor Investigador con perfil PROMEP
Universidad Autónoma de Ciudad Juárez
[email protected]
FIGEROA MARTELL NOEMÍ ARACELI
Ingeniero
en
Sistemas
Digitales
y
Comunicaciones
Universidad Autónoma de Ciudad Juárez
[email protected]
Keywords— Chaos, Coupled-System, Synchronization.
1. INTRODUCTION
In past decades, the advancement in information
technology and telecommunications is undoubted;
consequently, there is a great demand to explore new
techniques and tools, one of which is chaotic signals
because they feature wide bandwidth and pseudo random
behavior. Although chaos is usually associated with lack
of order, the unpredictable or confusion; it should be
noted that chaotic signals present these conditions due to
the extreme sensitivity to initial conditions.
Furthermore synchronization of chaotic oscillators is a
phenomenon that has been discussed in great detail both
experimentally and theoretically. For example, Carroll
and Pecora reported that when the Lyapunov exponent of
a chaotic signal is negative, it is possible to synchronize
it [1]. Moreover, other studies [2] led to the evaluation of
the threshold values which ensure synchronization. So
far, two methods for synchronization have been
developed for one-dimensional discrete chaotic
oscillators; they are: chaos plus noise and coupled
systems.
This article focuses on the latter, which has the advantage
of being applied to must know oscillators. The chaos
plus noise method, however, can only be used to some
chaotic oscillators. Also, in [3 - 7] some limited
experiments were presented. In this article, more results
will be presented. The study will increase from three to
eight maps and will also include a large number of map
parameters As a consequence, it was determined that
statistically, the phenomenon has the same behavior
when the parameters are changed into the chaotic regime.
2. TWO PAIRS
CHAOTIC MAPS
OF
ONE-DIMENSIONAL
An analysis of synchronization of One-Dimensional
discrete time chaotic signals is considered here. Maps
considered for this study are shown in Table 1.
Specifically, for the analysis of nonlinear functions F: 
→ , the iterated map function may be written as.
 k 1  F  k 
(1)
Table 1. One-Dimensional chaotic maps.
Map
Definition
Bernoulli
𝐹1 (𝜙(𝑘 + 1)) = 𝐴𝜙(𝑘)𝑚𝑜𝑑1
𝐹2 (𝜙(𝑘 + 1)) = 1 − 𝐵𝜙(𝑘)𝑚𝑜𝑑1
Chebyshev
𝐹(𝜙(𝑘 + 1)) = 𝑐𝑜𝑠 (𝐴𝑎𝑐𝑜𝑠(𝜙(𝑘)))
Congruent
𝐹(𝜙(𝑘 + 1)) = {
𝐵𝜙(𝑘) − 2𝐴
𝐵𝜙(𝑘)
𝐵𝜙(𝑘) + 2𝐴
𝜙(𝑘) > 𝐴
|𝜙(𝑘)| ≤ 𝐴
𝜙(𝑘) < −𝐴
Cosine
𝐹(𝜙(𝑘 + 1)) = 𝐴𝑐𝑜𝑠(𝜙(𝑘) + 𝐵)
Exponent
𝐹(𝜙(𝑘 + 1)) = 𝜙(𝑘)𝑒𝑥𝑝 (𝐵(𝐴 − 𝜙(𝑘)))
Hopping
𝐹(𝜙(𝑘 + 1)) = {
𝐷(𝜙(𝑘) − 𝐴) + 𝐶
𝐵𝜙(𝑘)
𝐷(𝜙(𝑘) + 𝐴) + 𝐶
𝜙(𝑘) > 𝐴
|𝜙(𝑘)| ≤ 𝐴
𝜙(𝑘) < −𝐴
Logistic
𝐹(𝜙(𝑘 + 1)) = 𝐴𝜙(𝑘)(1 − 𝜙(𝑘)2 )
Tent
𝐹(𝜙(𝑘 + 1)) = 𝐴 − 𝐵|𝜙(𝑘)|
Source: Own elaboration from Multiplexing Chaotic Signals
Using Synchronization [2].
18 al 20 de febrero 2015. Facultad de Ciencias Químicas e Ingeniería. UABC. Copyright 2015. Tijuana, Baja California, México.
74
Chaotic behavior has been observed in numerous OneDimensional discrete time dynamical systems; in fact,
Table 1 shows the definition for the maps considered for
this analysis. In addition to the mathematical relationship
Table 1 indicates the region where the map has chaotic
behavior. From this group of eight maps Bernoulli stands
out because it shows the chaos synchronization for two
shift maps [2].
In fact, functions shown in Table 1 are deterministic;
however, they have chaotic characteristics. Hence, a way
to observe their behavior is to vary the value of the
constant parameters (A, B, etc.); thus, a bifurcation
diagram is obtained. In order to guarantee chaotic
response of oscillators, Table 2 shows the specific
parameter range for all the maps considered. In addition,
the increment considered between ranges is also shown.
Particularly for those maps that have more than one
parameter, the study was performed considering the
variation in only one of them.
Φ1 (𝑘 + 1) = 𝐹1 (Φ1 (𝑘)) + 𝜀[𝐹1 (𝜙1 (𝑘)) + 𝐹2 (𝜙2 (𝑘)) −
𝐹1 (Φ1 (𝑘)) − 𝐹2 (Φ2(𝑘))] (4)
Φ2 (𝑘 + 1) = 𝐹2 (Φ2 (𝑘)) + 𝜀[𝐹1 (𝜙1 (𝑘)) + 𝐹2 (𝜙2 (𝑘)) −
𝐹1 (Φ1 (𝑘)) − 𝐹2 (Φ2(𝑘))]
(5)
Where ε is a coupling factor. Also it was determined that
synchronization is simplified if ε = ½ [2].
The purpose of synchronization is that two chaotic
systems, which initially evolved from different initial
conditions, coupled in some way to the end follow an
equal trajectory. This synchronization is obtained when
one of the two systems changes the path to a different but
also common path to both systems. Moreover, it can be
established that the system shown in Figure 1 is
synchronized when
Φ1 (k) = ϕ1 (k)
Φ2 (k) = ϕ2 (k)
Table 2. Chaotic Regime.
Map
Chaotic Regime Analyzed
Bernoulli
𝐴 ∈ [2, 3.8] Δ = 0.2
Chebyshev
𝐴 ∈ [3, 6] Δ = 0.3
Congruent
𝐴 = 0.25
𝐵 ∈ [1.91, 2] ∆ = 0.01
𝐶 = 2𝐴
Cosine
Exponent
One factor to evaluate the performance of chaos
synchronization, as shown in Figure 1, is the speed to
achieve it; that is, the number of iterations performed to
get it, namely the synchronization index (SI).
Considering the eight maps studied, the slower map is
Logistics; with an SI mean value of 1438.50. In contrast,
the faster map is Chebyshev; with an SI mean value of
only 7.66. Then the eight maps can be classified as
follows: those that required less than 100 iterations for
synchronization; they are: Chebyshev, Tent, and
Exponent. Those that demand from 100 to 1000
iterations: Hopping, Bernoulli and Cosine. Finally the
slow maps that necessitate more than a thousand of
iteration: Congruent and Logistic.
𝐴 ∈ [2.18, 2.27] ∆ = 0.01
𝐵 = 47𝜋/64
𝐴=1
𝐵 ∈ [3.8, 5.8 ] ∆ = 0.2
𝐵 ∈ [2.3, 3.1] ∆ = 0.1
Figure 1. Synchronization configuration for two pairs of
chaotic oscillators.
𝐶 = 0.5
𝐷 = −2
Logistic
𝐴 ∈ [3.68, 3.93] Δ = 0.01
Tent
𝐴 ∈ [1.5, 1.95] Δ = 0.05
1(k)
Master
Osc 1
Next, a configuration of the synchronization system with
a simple model of two system pairs of One-dimensional
chaotic map is shown in Figure 1 [2]. The system
includes four chaotic oscillators: two master oscillators
(1 and 2) and two slave oscillators (1 and 2), in the
form
𝜙1 (𝑘 + 1) = 𝐹1 (𝜙1 (𝑘))
𝜙2 (𝑘 + 1) = 𝐹2 (𝜙2 (𝑘))
(6)
(7)
3. MAIN RESULTS
𝐴 = 𝐶/𝐵
Hopping
ISSN 2007-9478, Vol. 4, Núm. 7. Año 2015.
(2)
(3)
Slave
Osc 1
+ -

Master
Osc 2
1(k+1)
+
Slave
Osc 2
2(k+1)
2(k)
-1
75
Another factor studied was observing the behavior of
each map during the synchronization by varying its
parameters, into the chaotic regime as shown in Table 2.
ISSN 2007-9478, Vol. 4, Núm. 7. Año 2015.
Figure 3, whereas in the rest of the diagram the behavior
is stable, with no significant variation.
Figure 3. Synchronization for Tent map.
To this purpose three- dimensional graphics were
generated. With this it is found a statistical stability in the
phenomenon under study for the eight chaotic maps but
with different parameter values.
Table 3. Mean of Synchronization Index.
Mean of
Map
Synchronization
Index
Chebyshev
7.66
Tent
62.88
Exponent
70.00
Hopping
296.46
Bernoulli
569.29
Cosine
684.75
Congruent
1148.59
Logistic
1438.50
Source: Own elaboration from Simulink Matlab.
Chebyshev map displays a relatively constant behavior
for the whole range of values. SI varies around 6 with
increasing trends up to 12 in the end, maintaining an
average of 7.66 which places it as one of the lowest
averages; therefore, this map is one of those having faster
synchronization in all cases.
Figure 4. Synchronization for Chebyshev map.
Source: Own elaboration from Statistical Analysis of Bernoulli,
Logistic and Tent Maps with Applications to Radar Signal
Design [4].
In addition, Logistic map shows an unstable behavior. It
can be observed on Figure 2 that if the master and slave
oscillators’ parameter value separation is large, then the
SI is lower (less than 1000). Nevertheless, if separation is
short, then SI rises up to 3500, or there is not
synchronization.
Figure 2. Synchronization for Logistic map.
4. CONCLUSIONS
In summary, with this work, a deeper understanding of
chaos synchronization of One-Dimensional discrete time
signals was achieved, particularly for coupled systems.
Also it extends the results previously reported in the
chaos synchronization literature.
5. SOURCES
Moreover, three of the eight maps show similar behavior
(Cosine, Tent, and Exponent) because there is no
synchronization when the master and slave oscillators
have the same value. This is noted on the diagonal on
[1]
Carroll T.L, Pecora L.M., “Synchronization in
Chaotic Systems”, Phys. Rev. Lett., vol. 64, pp 821-824,
1990.
[2]
Tsimring
L.S.
and
Sushchik
M.M.,
“Multiplexing Chaotic Signals Using Synchronization”,
Phys. Lett. A, vol. 213, pp 155-166, 1996
[3]
Garcés H., “Wideband Chaotic Signal Analysis
and Processing”, Ph.D. Dissertation, The University of
Texas at El Paso, 2007.
[4]
Garcés Héctor, Flores Benjamin C., “Statistical
Analysis of Bernoulli, Logistic and Tent Maps with
76
Applications to Radar Signal Design”, Proceedings of
SPIE The Int. Soc. for Opt. Eng., Defense and Security
Symposium 2006, Orlando FL., vol. 6210, 62100G, pp
512-521, ISSN 0277-786X, April 2006.
[5]
Ochoa Carrillo Rosa Isela, Garcés Guzmán
Héctor, Hinostroza Zubía Victor, Mendoza Carreón
Alejandra, “Sincronización de señales caóticas
unidimensionales por el método de sistemas acoplados”,
Proceedings ICSS, 31th International Congress of
Electronic Engineering, Chihuahua, Chih., vol. XXXI,
pp. 168 - 171, ISSN 1405-2172, Octubre 2009.
[6]
Garcés Guzmán Héctor, Ochoa Carrillo Rosa
Isela, “Estudio de la sincronización de osciladores
caóticos por el método de sistemas acoplados”,
Proceedings ICSS, 34th International Congress of
Electronic Engineering, Chihuahua, Chih., vol. XXXIV,
pp. 23 - 27, ISSN 1405-2172, Octubre 2012.
[7] Figueroa Martell Nohemí Araceli, “Estudio de la
sincronización
de
señales
caóticas
discretas
ISSN 2007-9478, Vol. 4, Núm. 7. Año 2015.
unidimensionales”, Tesis profesional,
Autónoma de Ciudad Juárez, Mayo 2013.
Universidad
Garcés Guzmán Héctor: Doctor en Filosofía (PhD) en el área de
ingeniería computacional por The University of Texas at El Paso en
2007. Maestro en Ciencias en el área de la ingeniería eléctrica por The
University of Texas at El Paso en 2002. Ingeniero en Comunicaciones y
Electrónica por el Instituto Politécnico Nacional en 1981. Además es
perito en telecomunicaciones con autorización vigente No. 401 por el
Instituto Federal de Telecomunicaciones y profesor con perfil
PROMEP. Experiencia académica de más de treinta años y profesional
de veinte años en la industria de las telecomunicaciones. Es autor y/o
coautor de más de veinte ensayos, artículos y tesis publicados en
revistas nacionales e internacionales. Interés de investigación: Señales
caóticas unidimensionales en el procesamiento de imágenes de radar de
alta resolución.
Figueroa Martell Nohemí Araceli: Ingeniera en Sistemas Digitales y
Comunicaciones por la Universidad Autónoma de Ciudad Juárez.
Experiencia académica y profesional por más de cinco años en la
industria maquilador
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SINCRONIZACIÓN DE MAPAS CAÓTICOS DISCRETOS

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