IEEE Workshop on Nonlinear Circuit Networks
December 11-12, 2015

Synchronization in Small-World Chaotic Circuit
Network under Parameter Mismatch
Kenta Ago

Yoko Uwate

Yoshifumi Nishio

Dept. of Electrical and Electronic Eng., Dept. of Electrical and Electronic Eng., Dept. of Electrical and Electronic Eng.,
Tokushima University
Tokushima University
Tokushima University
2-1 Minami-Josanjima,
2-1 Minami-Josanjima,
2-1 Minami-Josanjima,
Tokushima 770–8506, Japan
Tokushima 770–8506, Japan
Tokushima 770–8506, Japan
Email: kentago@ee.tokushima-u.ac.jp
Email: uwate@ee.tokushima-u.ac.jp
Email: nishio@ee.tokushima-u.ac.jp

Abstract—In this study, we investigate the synchronization of
coupled chaotic circuits in the presence of parameter mismatch.
We consider three different network topologies obtained fromthe small-world network model and two parameter dispersionsof mismatched circuits. By means of the computer calculations,the synchronization probabilities of the entire network areinvestigated during a certain time interval. From the simulation
results, we find that the small-world network provides the best
framework to realize synchronization.

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 significant hot topic. In order to apply for practicalapplications in many disciplines, understanding the relationship between topological structure and functional behavior
on the networks can be considered as an important problem.
As the dynamics on the networks, the synchronization is oneof the fundamental phenomenon in various fields. Especially,
chaotic synchronization is very interesting phenomenon, havereceived a great deal of attention since the report by Pecora andCarrol [3]. However, there are not many studies for complex
networks of continuous-time real physical systems such as
electrical circuits. In our previous work, we have investigated
the synchronization phenomena of coupled chaotic circuits on
a complex network with local bridge [4]. We have focused on
local bridge structure observed from the small-world network.
However, the circuit parameters were fixed with same parameters for all chaotic circuits and the only one network model
was considered.
In this study, we investigate the global synchronization ofcoupled chaotic circuits in the small-world network. Wan andChen reported the synchronization in the small-world coupled
Chua’s circuits [5]. We focus on the parameter mismatches, thesynchronization of coupled chaotic circuits in three different
network topologies obtained from the small-world networkmodel is studied. From the simulation results, synchronization
probability during a certain time interval are compared among
three network topologies. Thereby, the small-world topology(a)1718(b)

1920

16

115

214451161098718(c)19201451161098181920

121431312171615214313121716153134125116107987Fig. 1. Illustration of the WS model for N = 20 and k = 4. (a) : Regularnetwork, p = 0. (b) : Small-world network, p = 0.1. (c) : Random network,p = 1.

is shown to be effective for the achievement of the synchronization in the entire network.1

II. S MALL -W ORLD N ETWORK M ODELIn 1998, Watts and Strogatz introduced very interestingsmall-world network model, called the WS model [1]. The
WS model can be generated as shown in Fig. 1. Startingfrom a ring lattice with N nodes and k edges per nodes inFig. 1(a), each edge is rewired at randomly with probability p.
By increasing the probability p, the randomness of the network
are also increased. The small-world network is known as thegraph which is characterized by highly clustering coefficient
like a regular graph and small path length like a random graph.
Topological structures in complex networks of N nodes
and E edges can be evaluated by the typical three structural
metrics (degree, clustering coefficient and path length). First,degree (k) shows the number of edges on a node. Second,clustering coefficient (C) shows the number of actual links
between neighbors of a node divided by the number of possible
links between those neighbors. This is given as follows:

C=NN
1 ∑1 ∑2En.
Cn =N n=1
N n=1 kn (kn − 1)(1)

1 We have already presented this result in Proc. of NOLTA’15, pp. 431-444,
Dec. 2015.

- 15 -Regular00.5002.895pCL

Small-world
0.10.358
2.458

N−1 ∑N∑2l(m, n).
N (N − 1) m=1 n=m+1n-g

C1L

C2Fig. 2. Chaotic circuit.

y(2)In this research, we consider coupled chaotic circuits inthree different network topologies obtained from the WS
model in Fig. 1. Each topologies is called regular, smallworld and random networks, respectively. Table 1 shows theproperties of three networks as shown in Fig. 1.

y

1.5

1.5

-1.5
-1.5

(a)

1.5

-1.5z

y

1.5

-1.5

1.5

1.5

-1.5z

(b)

-1.5z

(c)

Fig. 3. Chaotic attractor of the circuit as shown in Fig. 2. β = 20, γ = 0.5.
(a) : α = 0.46, (b) : α = 0.50, (c) : α = 0.54,

III. C OUPLED C HAOTIC C IRCUIT
Figure 2 shows the chaotic circuit which is threedimensional autonomous circuit proposed by Shinriki etal. [6][7]. This circuit is composed by an inductor, a negativeresistor, two capacitors, and dual-directional diodes. In thisstudy, we propose 20 coupled chaotic circuits in three network
topologies as shown in Fig. 1. In these network models, chaotic
circuits are applied to each node of the networks and each edge
corresponds to a coupling resistor R.
First, the circuit equations are given as follows:din= v2nL

dt



dv1n
1 ∑(3)(v1n − v1k )C1= gv1n − idn −
dt
R
k∈Sn




 C2 dv2n = −in + idn ,dt
where n = 1, 2, 3, ..., 20 and Sn is the set of nodes which aredirectly 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 ) (4)
idn =


Gd (v1n − v2n + V ) (v1n − v2n < −V ).
By using the parameters and the variables:√
C2V xn , v1n = V yn , v2n = V zn
in =
L

√C2dt = LC2 τ, “ · ” =, α=dτC1√√√
1LLL β=Gd , γ =g, δ =,C2C2R C2

inidn

Random
10.216
2.221

Third, path length (L) shows the shortest path in the network
between two nodes. This is given as follows:
L=v2nv1n

TABLE IP ROPERTIES OF THREE NETWORKS AS SHOWN IN F IG . 1.

(5)

the normalized circuit equations are given as follows:x˙n = zn∑(yn − yk )y˙n = αγyn − αf (yn − zn ) − αδk∈Snz˙n = f (yn − zn ) − xn .

(6)

The nonlinear function f (yn − zn ) corresponds to the i − vcharacteristics of the nonlinear resistor consisting of the diodes
and are described as follows:β(yn − zn − 1) (yn − zn > 1)0(|yn − zn | ≤ 1) (7)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 and v2of the circuit in Fig. 2, respectively. By increasing parameter
α, the range of chaotic trajectory becomes widely formed.
IV. PARAMETER D ISPERSIONIn this research, we fix the circuit parameters as α = 0.50,
β = 20, γ = 0.50 and δ = 0.70 for all chaotic circuits.
Additionally, the parameter mismatches ∆α are added for eachcircuit parameter α which relate to the chaos degree. Namely,
the parameter α of each circuit is shown as α = 0.5 + ∆α,respectively. We propose the parameter dispersion as shownin Fig. 4. Each pattern is different in the number of theparameter mismatched circuits and the range of the parameter mismatches. By using the proposed parameter dispersionpatterns, we add the parameter mismatches for the circuits inthe computer simulations.

- 16 -

are synchronized, the value of the graph should be almost
zero. In order to analyze the synchronization state, we define
the synchronization by the following equation:|yi − yj | < 0.01(b) Pattern 2.
Fig. 4. Proposed two patterns of the parameter dispersion. Vertical axis: thenumber of circuits. Horizontal axis: parameter mismatches ∆α.

y1 - y20.01
0-0.01-0.100.10

(a) Nodes 1 and 2.

y1 - y70.01
0-0.01-0.10(8)By means of the above definition of the synchronization,
we define that the nodes 1 and 2 in Fig. 5(a) are synchronizedperfectly. However, the nodes 1 and 7 in Fig. 5(b) are almost
evaluated as the asynchronization in this definition. Thus,
we propose and investigate the synchronization probability
denoted the synchronization rate during a certain time interval. In this research, we fix a certain time interval as(τ = 1, 000, 000 and step = 0.01τ ) and statistically investigate
the synchronization probability in the entire network of 20
coupled chaotic circuits.

(a) Pattern 1.

0.10(i ̸= j).

(b) Nodes 1 and 7.
Fig. 5. The example of the voltage difference between two circuits when theparameter dispersion pattern 2 in the small-world network and the definitionof the synchronization. (a) : Synchronization. (b) : Asynchronization.

V. S YNCHRONIZATION
A. Definition of Synchronization
Figure 5 shows the example of the computer simulation
results when the parameter dispersion pattern 2 (see Fig. 4(b))
in the small-world network. The vertical axes are the difference
between the voltages (corresponding to v1 of the circuit inFig. 2) of the nodes 1 and 2 or 7. Namely, if the two nodes

B. Synchronization Probability
Figure 6 shows the investigation results of the relationshipbetween the synchronization probability and the combinational
samples of two parameter dispersion patterns in three network
topologies. Each pattern corresponds to the two patterns inFig. 4. The vertical axes denote the synchronization probability
in the entire network during the time interval. The horizontal
axes denote the combinational samples considered from eachparameter dispersion pattern. The combinational of the parameter dispersion is considered a large number. Therefore,
we choose the 10 samples randomly as the combinationalof the parameter dispersion in each pattern of Fig. 4. FromFig. 6, we confirm that the networks become to be difficult
for the global synchronization by increasing the number ofthe parameter mismatched circuits. On the other hand, thesynchronization probability in each pattern depends on thecombinational sample. This reason is that N = 20 would be
too small. However, we found that the synchronization probability tends to be higher in the small-world network. Therefore,
we consider that the small-world topology is effective for thesynchronization in the entire network. In addition, the complex
relation between global synchronization of the location of
mismatched circuit can be confirmed in this research. Moredetailed these relation considering more large number ofsamples should be investigated for our future works.
VI. C ONCLUSION
This paper considered the synchronization of coupledchaotic circuits with the parameter mismatches in three network topologies obtained from the WS model. In particular,
we focused on the parameter distribution based on the numberof parameter mismatched circuits. By means of the computer
calculations, the synchronization probabilities of the entire
network were investigated during a certain time interval. Fromthe simulation results, we found that the synchronization probability tends to be higher in the small-world network. We consider that the small-world network enhances synchronization.
However, the fluctuation is large among the combinationalsamples. This reason is that N = 20 would be too small.
Therefore, more detailed investigation considering more largescale networks should be carried out in our future works.

- 17 -

[13] J. Lu, X. Yu, G. Chen and D. Chen, “Characterizing the synchronizability of small-world dynamical networks”, IEEE Trans., vol. 51, pp. 787796, 2004.
[14] K. Ago, Y. Uwate and Y. Nishio, “Synchronization Phenomena of
Coupled Chaotic Circuit Network with Bridge”, Proc. NCN’13, pp. 27-29,
2013.
[15] K. Ago, Y. Uwate and Y. Nishio, “Investigation of Partial Synchronization in Coupled Chaotic Circuit Network with Local Bridge”, Proc.
NCN’14, pp. 63-66, 2014.

(a) Pattern 1.

(b) Pattern 2.
Fig. 6. Relationship between the synchronization probability and the combinational sample in three network topologies.

R EFERENCES[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] L. M. Pecora and T. L. Carrol, “Synchronization in Chaotic Systems”,
Physical Review Letters, vol. 64, pp. 821-824, 1990.
[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] X. F Wang and G. Chen, “Synchronization in small-world dynamicalnetworks”, Bifurcation and Chaos, vol. 12, no. 1, pp. 187-192, 2002.
[6] 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.
[7] 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. 744-754, 1987.
[8] H. Sekiya, S. Moro, S. Mori and I. Sasase, “Synchronization Phenomena
on Four Simple Chaotic Oscillators Full-Coupled by Capacitors”, IEICE
Trans. Fundamentals, vol. J82-A, no. 3, pp. 375-385, 1999.
[9] M. Miyamura, Y. Nishio and A. Ushida, “Clustering in Globally Coupled
System of Chaotic Circuits”, Proc. ISCAS’02, vol. 3, pp. 57-60, 2002.
[10] Y. Hosokawa, R. Tsujioka and Y. Nishio, “Relation between Synchronous Rate and Small Variations on an Asymmetrical Coupled ChaoticSystem”, Proc. NOLTA’05, pp. 166-169, 2005.
[11] Y. Hosokawa and Y. Nishio, “Asymmetrical Chaotic Coupled System
Using Two Different Parameter Sets”, Proc. NOLTA’06, pp. 543-546,2006.
[12] M. Barahona and L. M. Pecora, “Synchronization in small-world systems”, Physical Review Letters, vol. 89, no. 5, 054101, 2002.

- 18 -