2014 RISP International Workshop on Nonlinear Circuits,Communications and Signal Processing
NCSP'14, Honolulu, Hawaii, USA. February 28 - March 3, 2014

Influence of Bridge on Coupled Chaotic Circuit Network
Kenta Ago, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University,
2-1 Minami-Josanjima, Tokushima, 770-8506, Japan
Phone: +81-88-656-7470, Fax: +81-88-656-7471
E-mail: {kentago,uwate,nishio}@ee.tokushima-u.ac.jp

Abstract
How network topological structure influences its dynamical behaviors, is currently becoming a topic of great important. In this study, we investigate synchronization phenomena of coupled chaotic circuit network with the bridge. Bycomputer simulations, synchronization of the bridge is easy
to break down and clustering of circuits is occurred from thebridge. Moreover, we statistically investigate the relation between synchronous rate and coupling strength.
1. Introduction
Recently, complex networks have attracted increasing attention from various field of science and engineering[1]–[4]. Additionally, how network topological structure influences its dynamical behaviors, is currently becoming a topicof great important. Many of complex networks can be described by node and edge, by its topological structure generates various synchronization phenomena. On the otherhand, chaos synchronization is very interesting phenomenathat two chaotic circuits are synchronized regardless of differences circuit parameters[5]. Also, as coupled systems ofchaotic elements, Coupled Map Lattice (CML) and Globally Coupled Map (GCM) were proposed by Kaneko. Thesemodels generate many kinds of complexity phenomena
such as spatiotemporal intermittency[6], clustering[7] and soon. However these models are very simple and discrete–time
mathematical model, many of nonlinear phenomena generated in nature would be not so simple. Therefore, it is important to investigate the complex phenomena observed incontinuous–time real physical system such as electrical circuits.
In this study, we focus on the relation between chaos
synchronization and network topological structure. Therefore, we focus on the influence of the bridge on the network. The bridge is the edge which provide the only route
between two nodes. Example of the bridge is shown in Fig. 1.

Figure 1: Example of the bridge.
In the Fig.1, the bridge between 3 and 4 provides the only
route which can flow information or influence from any contact of 3 and 4. In order to analyze complex phenomena of
the bridge on the network, chaotic circuits are applied to eachnode of the network and coupling elements between each circuit are used resistors. By computer simulations, we investigate the synchronization phenomena between each circuiton the coupled chaotic circuit network with bridge. Moreover, we statistically investigate the relation between synchronous rate and coupling strength. Thereby, we reveal theinfluence of the bridge on the network.
2. Network model
Figure 2 shows chaotic circuit which is three-dimensional
autonomous circuit proposed by Shinriki et al.[8]. A proposed network model is shown in Fig. 3. In this study, chaotic
circuits are applied to each node of the network and coupling elements between each circuit are used resistors. In thismodel, the bridge is the resistor between CC3 and CC4.

v2nv1nCCn-gc1c2

Figure 2: Chaotic circuit.

- 417 -inidnLthe normalized circuit equations are given as follows:

CC7RCC6RRRRRCC4R↑bridgex˙n = zn∑ y˙n = αγyn − αβ f (yn − zn ) − αδ(yn − yk )k∈Cn z˙n = β f (yn − zn ) − xn ,

CC1

CC3

RR

CC2

where the nonlinear function corresponding to the characteristics of the nonlinear resistor of the diodes and are describedas follows:

CC5
Figure 3: Model of coupled chaotic circuit network withbridge.

First, the circuit equations are given as follows:din
L= v2ndt∑ C1 dv1n = gv1n − idn − 1(v1n − v1k )


dt
R k∈C
n



dv2n



= −in + idn , C2dt(n = 1, 2, 3, ..., 7),(4)y − zn − 1 (yn − zn > 1)
 n0(|yn − zn | ≤ 1)f (yn − zn ) =  y − z + 1 (y − z < −1).
nnnn

(5)

This circuits generates chaotic attractor as shown in Fig. 5.

y1(1)

Z1
where Cn is set of nodes which are connected to CCn. We
approximate the i − v characteristics of the nonlinear resisFigure 5: Example of chaotic attractors. α = 0.4, β = 0.5 andtors consisting of the diodes by the following 3–segment
γ = 20.
piecewise–linear function as shown in Fig.4.

idn-a

3. Simulation Result

Gda (vn1-vn2)

Figure 4: Approximation of the nonlinear resistor.
G (v − v2n − a) (v1n − v2n > a)
 d 1n0(|v1n − v2n | ≤ a)idn =  G (v − v + a) (v − v < −a).
d 1n2n1n2n
By using the parameters and variables as follows:√C2in =axn , v1n = ayn , v2n = azn
L√dC2t = LC2 τ, “ · ” =, α=dτ
C1
√√
√
1LL
LGd , γ =g, δ =,β=C2
C2
R C2(2)In this study, we fix the same 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 6 shows the computer simulation results. In theFig.6, the vertical axes are the diﬀerences between the voltageof the two chaotic circuits. Namely, if the two chaotic circuits
synchronize, the value of the graph should be almost zero likey1 − y2 of Fig. 6(b). In the case of Fig. 6(a), all circuits behave
asynchronously. In the case of Fig. 6(b), y1 − y2 shows fullsynchronization. In the case of Fig. 6(c), additionally y5 − y6and y6 − y7 show full synchronization. In the Fig. 6(b)(c), thebridge (y3 − y4 ) is bursting waveform compared with others.
In other words, the bridge is easy to break down.
In order to analyze synchronization, we define the synchronization as following equation,|yn − yk | < 0.01,

(3)

(6)

where n is number of circuits and k is circuits which connected circuit n. Figure 7 shows statistically investigated result of synchronous rate. The iteration time τ is fixed with- 418 -y1-y2
y1-y3
y2-y3
y3-y4
(bridge)
y4-y5
y4-y6
y4-y7
y5-y6
y6-y7

→τ
(a) δ = 0.10.

(a) δ = 0.10.

y1-y2
y1-y3
y2-y3
y3-y4(bridge)
y4-y5
y4-y6
y4-y7
y5-y6
y6-y7→τ(b) δ = 0.25.

(b) δ = 0.25.

y1-y2
y1-y3
y2-y3
y3-y4
(bridge)
y4-y5
y4-y6
y4-y7
y5-y6
y6-y7→τ(c) δ = 0.40.

(c) δ = 0.40.

Figure 6: Computer simulation result.

Figure7: Synchronous rate.

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route〜Figure 8: The relation between synchronous rate and coupling strength.

10,000,000. Synchronous rate of the Fig. 7 corresponds to theFig. 6. In the Fig.7(c), full synchronization is shown as 100% numerically like routes 1–2, 5–6 and 6–7. On the otherhands, other routes show low synchronous rate. Additionally, bridge (3-4) is lowest synchronous rate of all routes.
Moreover, we statistically investigate the relation between
synchronous rate and the coupling strength δ. Figure 8
shows the relation between synchronous rate and the couplingstrength δ. By increasing the coupling strength δ, some combination of chaotic circuits become synchronous state. CC1
and CC2 are synchronized completely (around δ=0.3). CC5
and CC6, CC6 and CC7 are also synchronized completelyafter CC1 and CC2. The other combinations also become
synchronous state, however the other combinations are notsynchronized completely. Even if the diﬀerences between theinitial value of each circuit are set very small, we can not beobserved full synchronization of all circuits. Thereby, clustering of circuits is occurred from the bridge. Some results
depending on the initial values can be observed since eachcircuit generate asymmetric attractor.
4. Conclusion
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 work, we investigate
the other network topological structure with the bridge andcarry out circuit experiment.

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