IEEE Workshop on Nonlinear Circuit Networks
December 13-14, 2013

Synchronization Phenomena of Coupled ChaoticCircuit Network with Bridge
Kenta Ago, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Eng.,
Tokushima University,
2-1 Minami–Josanjima, Tokushima,
770-8506 JAPAN
Email: {kentago, uwate, nishio}@ee.tokushima-u.ac.jp

v2n

v1n

Abstract— In this study, we investigate synchronization phenomena of coupled chaotic circuit network with the bridge. Bycomputer simulations, synchronization of the bridge on thenetwork is easy to break down and clustering of circuits isoccurred from the bridge. Moreover, we statistically investigate
synchronous rate.

inidn

CCn-g

c1

Fig. 1.

Chaotic circuit.

c2

LR

CC1I. I NTRODUCTION
Recently, engineering applications of chaos have received
a lot of attention by many researchers. For example, communication systems of chaos, control of chaos, synchronizationof chaos and so on. Especially, synchronization of chaos isvery interesting phenomena that chaotic elements synchronizein spite of different initial values[1]. Additionally, coupledsystems of chaotic elements generate many kinds of complexity phenomena such as spatiotemporal intermittency[2],clustering[3], and so on. Coupled Map Lattice (CML) andGlobally Coupled Map (GCM) proposed by Kaneko arevery simple and carried out for discrete–time mathematicalmodel. However, many of nonlinear phenomena generated innature would be not so simple. Therefore, it is important toinvestigate the complex phenomena observed in continuous–time real physical system such as electrical circuits[4]–[6].
In this study, we investigate synchronization phenomena
in coupled chaotic circuit network with the bridge. The
Bridge is the edge which provide the only route between twonodes. In order to analyze complex phenomena of the bridgeon the network, each node is applied chaotic circuits andconnected via registers. By computer simulations, we observe
synchronization phenomena between circuits. Moreover, we
statistically investigate synchronous rate.

CC7
R

CC6

R

R

CC4
R

R

CC3

↑
bridgeRR

CC2

CC5Fig. 2.

Model of coupled chaotic circuits network with bridge.

First, the circuit equations are given as follows:


din= v2ndt
dv1n1 ∑
C1= gv1n − idn −(v1n − v1k )dtRk∈Cndv C2 2n = −in + idn ,dt(n = 1, 2, 3, ..., 7),

II. N ETWORK MODEL
Figure 1 shows chaotic circuit which is three-dimensional
autonomous circuit proposed by Mori et al.[7][8]. A Proposed
network model is shown in Fig. 2. In this study, chaotic circuits
are applied to nodes of the network and are used resistors as
coupling elements. In this model, the bridge is the resistor
between CC3 and CC4.

RL

(1)

where Cn is set of nodes which are connected to CCn. We
approximate the i − v characteristics of the nonlinear resistors
consisting of the diodes by the following 3-segment piecewiselinear function,- 27 - Gd (v1n − v2n − a)0idn =Gd (v1n − v2n + a)

(v1n − v2n > a)(|v1n − v2n | ≤ a)(v1n − v2n < −a).

(2)

By using the parameters and variables as follows:√C2axn , v1n = ayn , v2n = azn
L√dC2t = LC2 τ, “ · ” =, α=dτC1√√√
LLL1β=Gd , γ =g, δ =,
C2
C2
R C2
in =(3)the normalized circuit equations are given as follows:x˙n = zn∑y˙n = αγyn − αβf (yn − zn ) − αδ(yn − yk )
k∈Cn


z˙n = βf (yn − zn ) − xn ,(4)

where the nonlinear function corresponding to the characteristics of the nonlinear resistor of the diodes and are describedas follows: yn − zn − 1
0f (yn − zn ) =yn − zn + 1(yn − zn > 1)(|yn − zn | ≤ 1)(yn − zn < −1).

(5)of synchronous rate. Synchronous rate of Fig. 5 corresponds
to Fig. 4. Also, iteration time is fixed with 10,000,000. InFig. 5, we confirmed that CC1 and CC2 synchronized almost
completely. On the other hands, the other combinations show
low synchronous rate. In addition, the bridge (CC3 and CC4)
is lowest synchronous rate of all combinations. This result
depending on the initial values can be observed since eachcircuit generates asymmetric attractor.

y1-y2
y1-y3
y2-y3
y3-y4
(bridge)y4-y5
y4-y6y4-y7
y5-y6y6-y7This circuit generates chaotic attractor as shown in Fig. 3.

→τy1

Fig. 4.

Computer simulation result. δ = 0.25.

Z1Fig. 3.

Example of chaotic attractor. α = 0.4, β = 0.5 and γ = 20.

III. S IMULATION R ESULTIn this study, we fix the parameters as α = 0.4, β = 0.5,
γ = 20 and δ on all circuits. We focus on the couplingstrength δ. Each circuit is given different initial values eachother. Figure 4 shows the computer simulation result in thecase of δ = 0.25. Also, the vertical axes are the differences
between the voltage of the two chaotic circuits. Namely, if thetwo chaotic circuits synchronize, the value of the graph shouldbe almost zero like y1 − y2 . In Fig. 4, synchronization of thebridge (y3 − y4 ) is easy to break down compared with othercircuit combinations and clustering of circuits is occurred fromthe bridge.
Moreover, in order to analyze synchronization, we define
the synchronization as following equation,|yn − yk | < 0.01,(6)where n is the number of circuits and k is circuits which connected circuit n. Figure 5 shows statistically investigated result

Fig. 5.

Synchronous rate. δ = 0.25.

IV. C ONCLUSION
In this study, we have investigated influence of the bridgeon the network via synchronization phenomena of coupledchaotic circuits. By computer simulations, synchronization of
the bridge is easy to break down and clustering of circuits isoccurred from the bridge. In our future works, we investigate
the other networks with the bridge and carry out circuit
experiment.

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