2008 IEEE Workshop on
Nonlinear Circuit Networks, Tokushima, Dec 5-6, 2008

Quasi Synchronization Phenomena in Coupled
Chaotic Systems as a Ladder
Tomoko Yamada

Yasuteru Hosokawa

Yoshifumi Nishio

Dept. Information Science,
Shikoku University
Email: s1839074@keiei.shikoku-u.ac.jp

Dept. Information Science,
Shikoku University
Email: hosokawa@keiei.shikoku-u.ac.jp

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

Abstract— Some chaotic systems have two coexisting attractors
depending on initial values in same parameters. In these systems,changing parameters causes switching phenomena of coexisting
attractors. In this study, we investigate switching phenomena incoupled chaotic systems as a ladder. We deﬁne a quasi synchronization and conﬁrmed the quasi synchronization phenomena inproposed systems.

A. Coupled map lattice using a cubic function
The coupled element is an one-dimensional map shown
Fig. 2. The function is described as Eq. 1. Switching phenomena (red and blue) are observed in this map. The coupledsystem consists of this map. Namely, the system is a coupled
map lattice using a cubic function.
f (x) = −ax3 + ax.

I. INTRODUCTION
There are many studies about coupled chaotic systems.
Especially, many kinds of systems were proposed in a area ofelectrical engineering [1]-[3]. In some of these studies, many
interesting phenomena which is concerned in a synchronization were investigated in detail.
On the other hand, some chaotic systems have two coexisting attractors depending on initial values in same parameters [4]-[6]. In these systems, changing parameters causes
switching phenomena of coexisting attractors. We consider
that matching these two attractors in coupled systems means
one of quasi synchronization phenomena. By the conﬁrmationof this phenomena, it is expected that new interesting phenomena are observed in coupled systems.
In this study, three coupled chaotic systems are investigatedin order to conﬁrm the quasi synchronization phenomena.
Three systems are two coupled map lattices and one coupledchaotic circuits. Coupled map lattice [7] is a discrete timesystem and coupled chaotic circuits is a continuous timesystem. This differences is also important point in this study.
II. COUPLED CHAOTIC SYSTEMS

Chaoticelement

Chaoticelement

Fig. 1.

(1)

Fig. 2.
The one-dimensional maps described by Eq. (1). a = 2.8 and
x0 = 0.4.

The system is written asε
xn+1 (i) = (1 − ε)f (xn (i)) + [f (xn (i − 1)) + f (xn (i + 1))].
2(2)where n is a discrete time step and i is a lattice point. The
coupling strength is determined by parameter ε.
B. Coupled map lattice using a piece-wise linear function

Chaoticelement

System Model.

Coupled chaotic system as a ladder shown in Fig. 1 isinvestigated in this study. Three kinds of coupled elements
are applied in the system. Each system is shown as following
subsections.

The coupled element is an one-dimensional map shown as
Fig. 3. A function of the map in described as Eq. 3. Switching
phenomena (red and blue) are observed in this map. The
coupled system is a CML using a piece-wise linear function.
11g(x) = −ax + a|x + | − a|x − |.
(3)22
The system is written asεxn+1 (i) = (1 − ε)g(xn (i)) + [g(xn (i − 1)) + g(xn (i + 1))].
2(4)where n is a discrete time step and i is a lattice point. The
coupling strength is determined by parameter ε.

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The coupled system consists of chaotic circuits and resistors
as coupling elements. The circuit model is shown in Fig. 6.
By using these model, the equation of the coupled system isdescribed as follows.

Fig. 3.
The one-dimensional maps described by Eq. (3). a = 2.8 and
x0 = 0.4.

C. Coupled chaotic circuits
The coupled element is shown in Fig. 4. The circuit equation
is described as follows.

Fig. 4.

Shinriki-Mori chaotic circuit.

di
L
dt


dv1Cdtdv
 C0 2dt=v1 ,=−i − id ,=gv2 + id ,(5)where,

a(v1 − v2 − Vth ), v1 − v2 > Vth ,0,−Vth ≤ v1 − v2 ≤ Vth ,id = a(v − v + V ), v − v < −V .
12th
12th(6)The diode is modeled as a piece-wise linear function.
Figure 5 shows a computer simulation result. Switching phenomena (red and blue) are observed.

Fig. 5.
Computer simulation results of the chaotic circuit. α = 10.0,
β = 0.56 and γ = 2.605.

dinL
dt

dvn1C
dt


dv
 C0 n2dt= vn1 ,
= −in − idn + G(v(n−1)1 − 2vn1 + v(n+1)1 )= gvn2 + idn ,(7)where,
n = 1, 2, · · · Na(vn1 − vn2 − Vth ), vn1 − vn2 > Vth ,0,−Vth ≤ vn1 − vn2 ≤ Vth ,id = a(vn1 − vn2 + Vth ), vn1 − vn2 < −Vth .
(8)
By changing parameters and variables,√1
L11xn =· i, yn =· vn1 , zn =· vn2Vth CVthVth√√√
LCLCL
α=a, β=g, γ=, δ=G,
CC0 C
C0C
1dτ=√t,=”·”dτLC(9)Normalized circuit equations are described as follows.
ẋn = yn ,
ẏn = −xn − αf (yn − zn ) + δ(yn−1 − 2yn + yn+1 ),żn = βzn + αγf (yn − zn ),(10)where,n = 1, 2, · · · N
yn − zn − 1, yn − zn > 1,
0,−1 ≤ yn − zn ≤ 1, (11)f (yn − zn ) =
 yn − zn + 1, yn − zn < −1.
III. COMPUTER SIMULATIONS
We carried out computer simulations of three models.
Figure 8 shows one of the computer simulation results in thecase of a coupled map lattice using a cubic function. Time
series of each coupled element are shown in Fig. 8 (a). Redand blue lines show differences of two attractors. Differences
between two elements are shown in Fig. 8 (b). Green lines
show differences of two attractors. Switching phenomena areobserved.
Figure 9 shows one of the computers simulation result in thecase of coupled map lattice using a piece-wise linear function.
Time series and differences of each element are shown in
Fig. 9 (a) and (b), respectively. Switching phenomena are- 69 -

Fig. 6.

Circuit Model.

also observed. Additionally, synchronization phenomena areobserved. When switching state of two elements is the same,
synchronization states (purple circle) and asynchronization
states (light blue circle) are conﬁrmed like as shown in
Fig. 7. We deﬁne a quasi synchronization as the state thatthe switching state is the same and two elements do not besynchronized.

systems. It means that this phenomena can be observed manykinds of coupled chaotic systems.
IV. CONCLUSIONSThe quasi synchronization phenomena observed in coupledchaotic systems were investigated. Three kind of systems
were investigated and quasi synchronization phenomena wereconﬁrmed in two systems. By conﬁrmation of phenomena, we
can investigate coupled systems from a new point of view.
Our future works are investigating the relationship between
parameters and quasi synchronization phenomena and investigating other coupled systems.
R EFERENCES

Fig. 7. Magniﬁcation of Fig.9 (b). A quasi synchronization is shown in alight blue circle. A synchronization is shown in a purple circle.

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on Four Simple Chaotic Oscillators Full-Coupled by Capacitors,” IEICE
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[2] M. Miyamura, Y. Nishio and A. Ushida, “Clustering in Globally Coupled
System of Chaotic Circuits,” IEEE Proc. ISCAS’02, vol. 3, pp. 57–60,
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[3] Y. Hosokawa and Y. Nishio, “Asymmetrical Chaotic Coupled System
Using Two Different Parameter Sets,” Proc. NOLTA’06, pp. 543–546,
Sep. 2006.
[4] M. Shinriki, M. Yamamoto and S. Mori, “Multi-mode Oscillations ina Modiﬁed van der Pol Oscillator Containing a Positive NonlinearConductance,” Proc. IEEE, vol. 69, pp. 394-395, 1981.
[5] N. Inaba, T. Saito and S. Mori, “Chaotic Phenomena in a Circuit witha Negative Resistance and an Ideal Switch of Diodes,” Tran. IEICE,vol. E70, no. 8, 1987.
[6] Y. Nishio, N. Inaba, S. Mori and T. Saito, “Rigorous Analysis of Windows
in a Symmetric Circuit,” IEEE Tran. Circuits Syst., vol. 37, no. 4, 1990.
[7] K. Kaneko, “Spatiotemporal Intermittency in Coupled Map Lattices,”
Prog. Theor. Phys. vol. 74, no. 5, pp. 1033-1044, 1985.
[8] M. Wada, K. Kitatsuji and Y. Nishio, “Spatio-Temporal Phase Patterns
in Coupled Chaotic Maps with Parameter Deviations,” Proc. NOLTA’05,pp. 178-181, Oct. 2005.

Figures 10 show one of the computers simulation result in
the case of coupled chaotic circuits. Time series and Differences are shown in Fig. 10 (a) and (b), respectively. Verticalaxes is zn and horizontal axes is time. Quasi synchronizing
phenomena are also observed in this case.
We have ever observed this quasi synchronization in thecase of coupled map lattice using a cubic function. In thiscase, synchronization phenomena can be observed by changing
parameters. However, quasi synchronization phenomena canbe observed. Now, we continue seeking the phenomena. If
we can obtain the phenomena, we will show the result in ourpresentation.
We could observed the quasi synchronization phenomena in
two cases. These two cases are discrete and continuous time

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(a) Time series of each coupled elements.

(b) Differences between two elements.
Fig. 8.

Computer simulation result of coupled map lattice using a cubic function. a = 2.80 and ε = 0.07.

(a) Time series of each coupled elements.

(b) Differences between two elements.
Fig. 9.

Computer simulation result of coupled map lattice using a piece-wise linear function. a = 2.48 and ε = 0.40.

(a) Time series of each coupled elements.

(b) Differences between two elements.
Fig. 10.

Computer simulation result of coupled chaotic circuits. α = 10.0, β = 0.55, γ = 2.605 and δ = 0.1.

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