2011 International Workshop on Nonlinear Maps and their Applications
NOMA'11, Evora, Portugal, September 15-16, 2011

Switching of Synchronization States in Coupled
Maps by Intermittency Chaos
Takahiro Nagai

Yoko Uwate

Yoshifumi Nishio

Dept. Electrical and Electronic Eng.,The Univ. of Tokushima
Email: nagataka@ee.tokushima-u.ac.jp

Dept. Electrical and Electronic Eng.,The Univ. of Tokushima
Email: uwate@ee.tokushima-u.ac.jp

Dept. Electrical and Electronic Eng.,The Univ. of Tokushima
Email: nishio@ee.tokushima-u.ac.jp

Abstract— In this study, we investigate switching synchronization phenomena which are observed in coupled chaotic maps.
Two logistic maps with generating intermittency chaos near
three periodic window are coupled as a CML coupling topology.
Moreover, we compare obtained synchronization states of coupledmaps with coupled chaotic circuits.

I. Introduction
Synchronization is one of the most typical nonlinear phenomena, there have been many studies until now. In addition, coupled oscillatory system can produce interesting
synchronization phenomena. Also, the coupled chaotic circuits
systems are suitable model to express the many various
high-dimensional nonlinear phenomena. In particular, chaotic
synchronization phenomena are paid attention to many researchers in various science fields because those mechanism
are clear by piecemeal. Therefor, their investigations of chaotic
synchronization phenomena have been done in many research
fields including physics, and the biologies. In our previousstudy, we have investigated the complex behavior in coupled chaotic circuits [1]. Figure 1 shows the circuit model
which used our previous study. In this circuit, two identicalchaotic circuits are coupled by a coupling resistor R. We
could find the three diﬀerent types of synchronization states
when the two circuits generate three periodic attractors inthis coupled chaotic circuit. These three synchronization states
can be obtained by giving diﬀerent initial conditions. Next,
we vary the bifurcation parameter of each chaotic circuit togenerate intermittency chaos near three-periodic window. In
this case, we could observe a complex behavior of the threesynchronization states. Namely, intermittency bursts disturb
the synchronizations and diﬀerent synchronizations reappear
after the bursts settle down.
Does such synchronization phenomenon of chaotic circuits
observe in coupled chaotic maps or not? In order to makeclear this question, we investigate synchronization phenomena
in coupled chaotic map as a Coupled Map Lattice (CML)
topology. In particular, CML is well known to as a discretetime mathematical models and we can observe the various
dynamics in these corollary. In the investigations of systemsuch as CML, it is very important to analyze the behavior forthe application of the engineering of the nonlinear phenomena,
biological networks.

In this study, we investigate synchronization phenomenon
when two logistic maps are coupled. We observe the diﬀerenceof behavior between adjacent maps by changing the parameterof the logistic map, and make a comparison the diﬀerenceof coupled maps and coupled chaotic circuits. Also, we pay
attention to the bursting part, and investigate the behavior of
the maps before and after the burst occurs.
II. Coupled Maps Lattice
A CML is known as suitable models to describe chaos andpattern dynamics. CML is a network of grid to generate theinteraction between oneself and adjacent maps. Figure 2 shows
an example of CML when two maps are coupled. In the CML,
the each map connects to only adjacent maps. In this study,
we use the logistic maps as chaos maps. A logistic map isexpressed as following equation:- 85 -fi,n = αxi,n (1 − xi,n )Fig. 1.

Two coupled chaotic circuits model [1].

Fig. 2.

Coupled map lattice.

(1)

Fig. 3.

Logistic map.
(a)where, α is a bifurcation parameter. The time series obtained
form the logistic map for α = 4.0 is shown in Fig. 3.
A general coupled map lattice is described as followingequation.
xi,n+1 = (1 − ε) fi,n + 2ε ( fi−1,n + fi+1,n )(2)III. Simulation Result
A. Three Types of Synchronization
First, we consider simple model of CML. The parameters
are fixed as α = 3.835 and ε = 0.000015 which produce threeperiodic solution. We can obtain the three diﬀerent types ofsynchronization states by diﬀerential initial values as shownin Fig. 4.
(b)Fig. 5. Switching of three types of synchronization. (a) ε = 0.000015.
(b) ε = 0.0015.

B. Switching of Three Types of Synchronization(a)(b)(c)Fig. 4. Three synchronization states depending on initial values. (a) x1 =0.20, x2 = 0.60. (b) x1 = 0.10, x2 = 0.90. (c) x1 = 0.40, x2 = 0.70.

Next, we vary the bifurcation parameter from α = 3.835 to
3.828427. Under this parameter α, the intermittency chaos near
the three-periodic window is obtained. In Fig. 5, we show thesimulation result when the bifurcation parameters is fixed as
α = 3.828427, and the initial conditions of each logistic map
is set to x1,0 = 0.159126, x2,0 = 0.956548. In these figures,
we confirm that the three types of synchronization states
are switching via bursting part. Furthermore, the switchingfrequency becomes fast by increasing the value of the couplingstrength ε.
In order to make clear the switching mechanism, we partition the obtained synchronization state (Fig. 5(a)) into four
patterns ((a):50 ≤ t ≤ 100, (b):250 ≤ t ≤ 300, (c):1000 ≤ t ≤1050, (d):75 ≤ t ≤ 500) as shown in Fig. 6.
Figure 6 (a),(b) and (c) are corresponding the three typesof synchronization. In Fig. 6(d), we show the switching ofsynchronization part, and we can observe that after a burst ofx2 is occurred, a burst of x1 is also occurred.
Finally, as an example, the long span of switching synchronization is shown in Fig. 7. We can see that the three types- 86 -Fig. 6.

(a)

(b)

(c)

(d)

Detailed drawing of the switching synchronization. (a) 50 ≤ t ≤ 100. (b) 250 ≤ t ≤ 300. (c) 1000 ≤ t ≤ 1050. (d) 75 ≤ t ≤ 500.

Fig. 7.

Long span of switching synchronization.

of synchronization states appear and disappear via burstingbehavior.
IV. Conclusions
In this study, we have investigated synchronization phenomena observed in two coupled chaotic maps. By setting the certain parameters we could confirm the interesting phenomena
for burst. Three types of synchronization states changed bybursting behavior.
In our future works, we would like to study the synchronization phenomena when the number of the couped maps areincreased.

References[1] Y. Uwate, Y.Nishio, “Modeling Using 1-D Map of Complex Behavior inCoupled Chaotic Circuits with Intermittency” Proceedings of European
Conference on Circuit Theory and Design (ECCTD’05), vol. 1, pp. 229232, Aug. 2005.

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