IEEE Workshop on Nonlinear Circuit Networks
December 12-13, 2014

Investigation of Partial Synchronization in Coupled
Chaotic Circuit Network with Local Bridge
Kenta Ago, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: {kentago,uwate,nishio}@ee.tokushima-u.ac.jp

Abstract—In our previous work, the influence of local bridge
on a complex network of 25 coupled chaotic circuits has been
investigated. In this study, we focus on partial synchronizationin the case of 5 clusters that are divided from local bridges.
By means of computer simulations, the states of significant
node is located in between two local bridges in the networkand partial synchronization probability of each cluster are
investigated. Moreover, we statistically analyze the sojourn timeof synchronization in various sizes of the clusters.

I. I NTRODUCTION
Complex networks have attracted a great deal of attention
from various fields since the discovery of “small-world”
network [1] and “scale-free” network [2]. In particular, how
network topological structure influences its dynamical behaviors, is a hot topic for understanding the structural function
on the networks and suitable for practical application in
many disciplines such as biology, sociology and economics.
Most of complex networks can be represented as a graphby nodes (vertices) and edges (links). On the other hand,synchronization is one of the typical phenomena in the field of
natural science. In particular, synchronization phenomena on
the networks of coupled chaotic systems are very interesting.
However, there are not many studies for complex networks
of continuous-time real physical systems such as electricalcircuits.
In sociology, there is a famous theory called “The strengthof weak ties” by Granovetter [3]. “Local bridge” is the edge
which predominantly shorter than other routes between two
nodes, and is essential for information propagation in complex
networks. In our previous work, we have investigated theinfluence of local bridge on a complex network by means ofsynchronization phenomena of 25 coupled chaotic circuits [4].
Local bridges almost behave asynchronously, partial synchronization in the network is almost occurred from local bridge.
In this study, partial synchronization in a complex networkwith local bridge of 25 coupled chaotic circuits is investigated.
We focus on partial synchronization in the case of 5 clusters
that are divided from local bridges. For the investigations,
we define the synchronization and a time interval. By meansof computer simulations, the states of significant node islocated in between two local bridges in the network and partial
synchronization probability of each cluster are investigated.
When partial synchronization probabilities of each cluster areevaluated, we consider the typical three structural metrics ofeach cluster. Moreover, we statistically analyze the sojourntime of synchronization in various sizes of the clusters.
II. N ETWORK MODEL
Figure 1 shows the chaotic circuit which is threedimensional autonomous circuit proposed by Shinriki etal. [5][6]. This circuit is composed by an inductor, a negative
resistor, two capacitors and dual-directional diodes. A proposed network model of 25 coupled chaotic circuits with local
bridge is shown in Fig. 2. In this model, chaotic circuits areapplied to each node of the network and each edge corresponds
to a coupling resistor R. In this model, local bridges are
considered to be the edges 1–25, 8–9, 14–15, 15–16 and 22–23.

- 63 -v2nv1ninidn-gCCn

C1L

C2Fig. 1. Chaotic circuit.

CC17
CC12R

R

R

R

R

R CC19

CC16

R

RR
CC20

CC8

CC7R

R

CC22

R
CC2

CC5R

CC21

R

R

R
R

R
CC6R

CC4R

CC23

R

R

R

R

R
CC1

R
R
CC3Fig. 2. Network model.

CC25

R

R

R

CC13

R

R
R

CC15

R

CC11
CC9

R
CC14

R

CC18

R

R

R
CC10R

CC24

First, the circuit equations are given as follows:din
= v2nLdt

dv1n1 ∑(1)
C1= gv1n − idn −(v1n − v1k )
dt
Rk∈S
n


 C2 dv2n = −in + idn ,dt
where n = 1, 2, 3, ..., 25 and Sn is the set of nodes which are
directly connected to the node n. We approximate the i − vcharacteristics of the nonlinear resistor consisting of the diodes
by the following three-segment piecewise-linear function:
G (v − v2n − V ) (v1n − v2n > V ) d 1n0(|v1n − v2n | ≤ V ) (2)idn =
Gd (v1n − v2n + V ) (v1n − v2n < −V ).
By using the parameters and the variables:√
C2V xn , v1n = V yn , v2n = V zni=
 n

L



√
dC2t = LC2 τ, “ · ” =, α=
dτ
C1

√
√
√



LL1L

 β=Gd , γ =g, δ =,C2C2
R C2
the normalized circuit equations are given as follows:x˙n = zn∑y˙n = αγyn − αf (yn − zn ) − αδ(yn − yk )k∈Sn z˙ = f (y − z ) − x .
nnnn(3)(4)

The nonlinear function f () corresponds to the i − v characteristics of the nonlinear resistances consisting of the diodesand are described as follows:β(yn − zn − 1) (yn − zn > 1)0(|yn − zn | ≤ 1) (5)f (yn − zn ) =β(yn − zn + 1) (yn − zn < −1).
This circuit generates asymmetric chaotic attractor as shownin Fig. 3. The values y and z in Fig. 3 correspond to v1 andv2 of the circuit in Fig. 1, respectively.

yIII. S YNCHRONIZATIONS STATESIn this research, we fix the circuit parameters as α = 0.4,
β = 20, γ = 0.5 and δ = 1.0 for all chaotic circuits. Each circuit is given different initial values for computer simulations.
Figure 4 shows an example of computer simulation results.
The vertical axes denote the differences between the voltages(corresponding to v1 of the circuit in Fig. 1) of the two chaoticcircuits. Namely, if the two chaotic circuits are synchronized,
the value of the graph should be almost zero like 19-20. We
can confirm that synchronizations of local bridges (8-9, 14-15,15-16, 22-23 and 1-25) are easy to break down compared withothers. In this paper, we define two synchronization states of“global synchronization” and “partial synchronization”. Figure
4 shows that the network switches to two synchronization
states. Additionally, partial synchronization is almost occurredfrom local bridges.
1-2
1-3
1-4
2-4
3-4
4-5
4-6
5-6
5-7
5-8
6-7
7-8
8-9
9-10
9-11
10-11
10-12
11-12
11-13
12-13
12-14
13-14
14-1515-16
16-17
16-18
16-19
16-20
17-18
18-19
19-20
19-21
19-22
20-21
20-22
21-22
22-23
23-24
23-25
24-25
1-25
Fig. 4. Phase difference waveform (δ = 1.0).

1.5

1.5

-1.5
-1.5

In order to analyze synchronization states, we define thesynchronization by the following equation:|yn − yk | < 0.01z

Fig. 3. Example of chaotic attractor (α = 0.4, β = 20 and γ = 0.5).

(k ∈ Sn ).

(6)

We fix a certain time interval as (τ = 10, 000 and step =0.01τ ) and statistically investigate partial synchronization ineach cluster of 25 coupled chaotic circuits by means of theabove definition of the synchronization.

- 64 -IV. PARTIAL S YNCHRONIZATIONIn this section, we consider partial synchronization in thecase of 5 clusters (nodes 1〜8, 9〜14, 15, 16〜22 and 23〜25) that are divided from local bridges. Then, the network isshown in Fig. 5.

Fig. 7. Partial synchronization probability of each cluster.

Fig. 5. A partial synchronization state from local bridges.

First, node 15 is located in between two local bridges (1415 and 15-16) in the network. Therefore the investigating ofthe states of significant node 15 in the network, is important.
Figure 6 shows the distribution of the states of node 15 during
the time interval. In this figure, the state A shows that node
15 are not synchronized both nodes 14 and 15. The state
B shows that nodes 14, 15 and 16 are all synchronized.
The state C shows that nodes 15 and 16 are synchronizedhowever nodes 14 and 15 are not synchronized. The state Dshows the opposite the state C. In four states, state A is thehighest distribution. Thus, we consider that node 15 faces withdilemma between two clusters, and hence tend to be isolated.
However, state B is the second highest distribution that isinterested.

Next, we statistically investigate partial synchronization ineach cluster of the network by means of the above definition of
the synchronization. Figure 7 shows the partial synchronization probability of each cluster during the time interval. We
consider that the partial synchronization probabilities of theseclusters depend on network topological structures. Topologicalstructures in complex networks can be evaluated by thetypical three structural metrics (degree, clustering coefficientand path length). Degree shows the number of edges on anode. Clustering coefficient shows the number of actual links
between neighbors of a node divided by the number of possible
links between those neighbors. Path length shows the shortest
path in the network between two nodes. Table I shows theproperties of each cluster. In Table I, N denotes the number ofnodes, E denotes the number of edges, k denotes the average
degree, C denotes the average of clustering coefficient and L
denotes the average of path length. We assume that the resultof Fig. 7 depends on three structural metrics in Table I. Inparticular, partial synchronization probability of cluster 16〜
22 is most highest in 4 clusters. The structural metrics k and Cof cluster 16〜22 show the largest values in 4 clusters. Also,
the path length of cluster 16〜22 is the second shortest in 4clusters. Thereby, we consider that the community strengthcharacterized by large clustering coefficient and short path
length can be evaluated by partial synchronization probabilityof each cluster.
TABLE IP ROPERTIES OF EACH CLUSTER .

Cluster
1〜89〜14
16〜2223〜25Fig. 6. Distribution of the states of node 15.

N8673E129123k
3.25
3.17
3.43
2.67

C
0.60
0.50
0.66
0.56

L
1.79
1.57
1.52
1.00

Moreover, we focus on the sojourn time of partial synchronization in each cluster. Figure 8 shows the distribution of thesojourn time of synchronization in various sizes of the clusters.
The cluster sizes denote the number of node in each cluster.
The slots in the horizontal axes of the figure denote the ranges- 65 -

of the sojourn time in Table II. Figure 8(a) shows the 1 cluster
state that the entire network are synchronized. From Fig. 8(a),the ratio of slot 1 is predominantly high. Namely, synchronization of the entire network tend to disappear immediately.
From Figs. 8(b), (c) and (e), these graphs are very similar andthe ratio of slot 1 is accounted for a substantial fraction ofdistribution. However, the graph of cluster 16〜22 shows thatthe ratio of slot 3 is the highest in 6 slots from Fig. 8(d),hence the sojourn time of synchronization of cluster 16〜22
is longer than other clusters. Namely, cluster 16〜22 can beconsidered as a robust cluster.

(a) 1〜25 (Cluster size = 25).

TABLE IIR ANGES OF SLOTS IN F IG . 8.

Slot123456Sojourn time (τ )
τ <11 ≤τ < 22 ≤τ < 33 ≤τ <44 ≤τ <5τ ≥5(b) 1〜8 (Cluster size = 8).

V. C ONCLUSIONIn this study, a complex network of 25 coupled chaoticcircuits with local bridge has been analyzed based on synchronization phenomena. The nodes and edges on the network
correspond to the chaotic circuits and coupling resistors,respectively. By means of computer simulations, partial synchronization is almost occurred from local bridges. Therefore
we investigated partial synchronization in the case of 5 clusters
that are divided from local bridges. As the results, significant
node is located in between two local bridges in the networkthat tend to be isolated and partial synchronization probabilityof each cluster almost depends on three structural metrics.
From these results, we consider that local bridge is suitablefor finding cluster and significant node on complex networks
and partial synchronization probability of each cluster can beeffective for evaluating as the community strength on complex
networks. In order to understand the phenomena correctly,
more detailed investigation considering more various clusters
should be carried out in our future works.

(c) 9〜14 (Cluster size = 6).

R EFERENCES(d) 16〜22 (Cluster size = 7).

[1] D. J. Watts and S. H. Strogatz, “Collective dynamics of small-world”,
Nature, vol. 393, pp. 440-442, 1998.
[2] A. L. Barabasi and R. Albert, “Emergence of scaling in random networks”, Science, vol. 286, pp. 509-512, 1999.
[3] M. S. Granovetter, “The Strength of Weak Ties”, Amerian Journal ofSociology, vol. 78, pp. 1360-1380, 1973.
[4] K. Ago, Y. Uwate and Y. Nishio, “Influence of Local Bridge on a Complex
Network of Coupled Chaotic Circuits”, Proc. NOLTA’14, pp. 731-734,2014.
[5] M. Shinriki, M. Yamamoto and S. Mori, “Multimode Oscillations ina Modified van der Pol Oscillator Containing a Positive NonlinearConductance”, Proc. IEEE, vol. 69, pp. 394-395, 1981.
[6] N. Inaba, T. Saito and S. Mori, “Chaotic Phenomena in a Circuit witha Negative Resistance and an Ideal Switch of Diodes”, Trans. of IEICE,
vol. E70, no. 8, pp. 744754, 1987.

(e) 23〜25 (Cluster size = 3).
Fig. 8. Distribution of the sojourn time of synchronization in various sizesof the clusters.

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