RISP

NCSP’16

International Workshop on
Nonlinear Circuits, Communications and Signal Processing

Honolulu, Hawaii, USA, March 6-9, 2016

Synchronization of Small-World Coupled Chaotic Circuits with or without Parameter Mismatch
Kenta Ago, Yoko Uwate and Yoshifumi Nishio
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(a)Abstract

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This paper considers the synchronization of 100 coupledchaotic circuits in five different topologies obtained from thesmall-world network model. By using the computer simulations, we calculate the synchronization probability of theentire network during a certain time interval when the coupling strength is changed. Moreover, the case under parameFigure 1: Illustration of the WS model for N = 20 and k = 4.
ter mismatch is also investigated. From the simulation results,(a) : Regular network, p = 0. (b) : Small-world network,
the small-world topology is shown to be effective for the synp = 0.1. (c) : Random network, p = 1.
chronization compared with regular and random networks.
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1. Introduction
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, understanding the relation between topological structure and functional behavior is considered a significant hot topic for applying to practical applications in many disciplines [3]. As the
dynamical behaviors on the networks, the synchronization is
one of the typical phenomena. Therefore, many researchers
have studied the synchronization on complex networks andreported that the small-world topology enhances the synchronization [4]-[6].
On the other hand, the synchronization of coupled chaotic
systems [7]-[10] is very interesting and good models to describe various higher-dimensional nonlinear phenomena in
the field of natural science. However, there are not manystudies for complex networks of continuous-time real physical systems such as electrical circuits. Wan and Chen reported the synchronization in the small-world coupled Chua’s
circuits [5]. In our previous work, we have studied the synchronization of 20 coupled chaotic circuits in the presence ofparameter mismatch [11]. We considered three different network topologies, namely regular, small-world, and random
networks, we found that the small-world network provides
the best framework to realize synchronization. Synchronization under parameter mismatch is an important issue to be
more clearly the structural function on the networks, but notfrequently studied.

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In this study, we investigate the synchronization of 100
coupled chaotic circuits as more larger-scale networks. We
consider five different network topologies including regular,
small-world, and random networks. We define the synchronization state in a quantitative way, and the synchronization
probability of the entire network during a certain time interval
is calculated in five networks. First, we investigate the relation between the synchronization probability and the couplingstrength which corresponds to the value of the coupling resistors for chaotic circuits. Next, the maximum cluster sizes ofsmall-world and random networks are investigated, it shows
the difference of how to achieve the synchronization between
two networks. Furthermore, the case under parameter mismatch is also considered. Thereby, the small-world topology
is shown to be effective for the synchronization of the entire
network compared with regular and random networks.
2. Small-World Network Model
In 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. Starting froma ring lattice with N nodes and k edges per nodes in Fig. 1(a),each edge is rewired at randomly with probability p. By increasing probability p, the randomness of the network is also
increased. The small-world network is known as the graphwhich is characterized by highly clustering coefficient like aregular graph and small path length like a random graph.

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v2n

v1n

in

idn

n

-g

C1

L

C2

R

Figure 3: Chaotic circuit and coupling resistor.

yFigure 2: Characteristic clustering coefficient C(p) and path
length L(p) for randomly rewired probability p. N = 100
and k = 6.

1.5

1.5

-1.5
-1.5

z

Figure 4: Chaotic attractor of the circuit as shown in Fig. 3.
α = 0.5, β = 20, and γ = 0.5.

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, 3. Coupled Chaotic Circuits
clustering coefficient (C) shows the number of actual links
Figure 3 shows the chaotic circuit which is threebetween neighbors of a node divided by the number of possidimensional autonomous circuit proposed by Shinriki etble links between those neighbors. It is given as follows:al. [12][13]. In this study, we propose 100 coupled chaotic
circuits in five network topologies. In these network models,
N
N
1 ∑
1 ∑
2En
C=
Cn =
.
(1) chaotic circuits are applied to each node of the network and
N n=1
N n=1 kn (kn − 1)each edge corresponds to a coupling resistor R (see Fig. 3).
The normalized circuit equations are given as follows:
Third, path length (L) shows the shortest path in the network x˙n = zn
between two nodes. It is given as follows:


∑
(3)
y˙n = αγyn − αf (yn − zn ) − αδ(yn − yk )
N
−1
N

∑ ∑

2
k∈S

n
L=
(2) 
l(m, n).
z˙n = f (yn − zn ) − xn ,N (N − 1)m=1 n=m+1

In this research, we consider 100 coupled chaotic circuits
in five network topologies obtained from the WS model. The
data shown in the Fig. 2 are the characteristic C and L for randomly rewired probability p of the rewiring process describedin Fig. 1. C(p) and L(p) have been normalized by using thevalues L(0) and C(0) for a regular network (p = 0). All
generated networks have N = 100 and the average degree ofk = 6 edges per nodes. By increasing p, L(p) drops rapidly,
corresponding to the onset of the small-world phenomenon.
On the other hand, during this drop of L(p), C(p) remains
at highly value like the regular network. Generally, the networks at such regions of p are called small-world network.
We choose five values as p = 0, 0.06, 0.1, 0.2, and 1, respectively. Each topologies is considered as regular (p = 0), random (p = 1), and small-world (p = 0.06, 0.1, and 1) networks. Namely, we obtain five network models for 100 coupled chaotic circuits and investigate the synchronization.

where n = 1, 2, 3, ..., 100 and Sn is the set of nodes which
are directly connected to the node n. The parameter δ is corresponding to coupling resister R, it determines the couplingstrength in this study. The nonlinear function f (yn − zn ) corresponds to the i − v characteristics of the nonlinear resistor
consisting of the diodes as follows:β(yn − zn − 1) (yn − zn > 1)0(|yn − zn | ≤ 1)f (yn − zn ) =(4)β(yn − zn + 1) (yn − zn < −1).
This circuit generates asymmetric chaotic attractor as
shown in Fig. 4. The values y and z in Fig. 4 correspond to v1
and v2 of the circuit in Fig. 3, respectively. In this study, we
fix the circuit parameters as α = 0.5, β = 20, and γ = 0.5 forall chaotic circuits. For the computer simulations, we calculate Eq. (3) using the fourth-order Runge-Kutta method withstep size h = 0.01.

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4. Synchronization
In this section, we show the computer simulation results
about the synchronization of 100 coupled chaotic circuits. In
particular, two cases of interest are considered: global synchronization of coupled chaotic circuits with or without parameter mismatch in five networks (p = 0, 0.06, 0.1, 0.2, and
1) described in Sec. 2.
4.1 Definition of Synchronization
Now, we define the synchronization state in a quantitative
way by the following equation:|yi − yj | < 0.01 (i ̸= j),(5)where y corresponds to v1 of the circuit in Fig. 3, it denotes
the voltage in the coupling terminal. Namely, if the two nodes(i and j) are synchronized, the value of |yi − yj | should bealmost zero. However, this chaotic circuit model is difficult toachieve complete synchronization. Therefore, we determine
the synchronization threshold as 0.01, and temporal phasesynchronization states in the networks are considered.
By means of the above definition of the synchronization,
we propose and investigate the “synchronization probability” denoted the synchronization rate during a certain timeinterval. In this research, we fix a certain time interval as(τ = 1, 00, 000 and step = 0.01τ ) and statistically investigate the synchronization probability in the entire network of
100 coupled chaotic circuits.

Figure 5: Relation between the synchronization probability
and the coupling strength δ in five network topologies.

Figure 6: Difference of the maximum cluster sizes in smallworld (p = 0.1) and random (p = 1) networks.

4.2 Synchronization Probability
Figure 5 shows the investigation results of the synchronization probability in five network topologies when the couplingstrength is changed by δ = 0.01. The vertical axis denotes thesynchronization probability in the entire network during thecertain time interval. We check every step whether all nodes
pares are satisfied Eq. (5). If the synchronization probability is 1, we can consider that the entire network is synchronized perfectly. On the other hand, if the synchronization
probability is 0, we can consider that all nodes have no time
for the synchronization during the time interval. We compare the relation between the synchronization probability andthe coupling strength δ among five network topologies. By
increasing δ, the synchronization probabilities in four networks excluding p = 0 are also increased. Namely, all nodes
become to be easily synchronized. However, the synchronization probability remains constant value at 0 in the network only p = 0 (regular network) even though the couplingstrength is increased. This result could not be obtained in thecase of N = 20 of our previous work. Therefore, we consider its reason the influence of the path length among nodes

according to N = 100. From Fig. 5, we can evaluate effective topologies for the synchronization among five networks
as follows; p = 0.2 > 1 > 0.1 > 0.06 > 0.
In order to understand the process of the synchronization,
we investigate the maximum cluster sizes which show the
maximum number of the synchronized nodes at all times during the time interval. Figure 6 shows the characteristics of the
maximum cluster sizes in small-world (p = 0.1) and random(p = 1) networks. Here, the synchronization probability corresponds to 1 when the maximum cluster size is 100. In thecase of the small-world network, the maximum cluster size
remains constant value at 1 even though the synchronization
probability is gradually increased (see Fig. 5). From this result, we find that the small-world network have no pairs of
nodes that are synchronized at all time. Thereby, we consider
that the small-world network behave like chaotic itinerancy
and approach to achieve the synchronization. On the other
hand, in the case of the random network, the maximum cluster size shows the large number (around 70) even when the- 39 -

work (p = 1). Correctively, we conclude that the small-world
topology provides the best framework to be effective for thesynchronization.
5. Conclusion

Figure 7: Relation between the synchronization probability
and the coupling strength δ in five network topologies underparameter mismatch.

synchronization probability is almost 0 (see Fig. 5). Whereas
the small-world network is gradually approached to achieve
the synchronization, the random network becomes to be synchronized rapidly by exceeding a certain value δ. This reason
can be considered that a few nodes tend to be isolated in the
random network by spreading the degree distribution. Thus,
we find the difference of how to achieve the synchronizationin two networks.

This paper considered the synchronization of 100 coupledchaotic circuits in five different topologies obtained from the
WS model. In particular, we focused on the global synchronization of the networks when the coupling strength waschanged. By means of the computer calculations, we proposed and investigated the synchronization probability duringa certain time interval. From the simulation results, we foundthat the regular network was difficult to achieve synchronization according to the influence of large-scale network. Furthermore, the case under parameter mismatch was also investigated. By comparing between two cases with or withoutparameter mismatch, we found that the networks with highrandomness are greatly influenced by parameter mismatch.
Hence, it can be said that small-world topology provides thebest framework to realize synchronization even if there isvariability among the nodes on the networks. More detailed
investigation considering more various networks should be
carried out in our future works.
References[1] D. J. Watts and S. H. Strogatz, “Collective dynamics of small-world”, Nature,
vol. 393, pp. 440-442, 1998.

4.3 Parameter Mismatch[2] A. L. Barabasi and R. Albert, “Emergence of scaling in random networks”, Science,
vol. 286, pp. 509-512, 1999.

Moreover, we consider the synchronization under parameter mismatch. Now, the parameter mismatches ∆α are added
for all circuits parameter α which is related to the range ofchaotic trajectory. We choose α = 0.5 as a standard parameter which is same parameter in the non-mismatch case. The
parameter mismatches ∆α are generated at randomly and therange of ∆α is set to [-0.01:0.01]. Namely, the parameter α
of all circuits are shown as α = 0.5 + ∆α, respectively.
Figure 7 shows the synchronization probability of 100 coupled chaotic circuits with the parameter mismatches. The
network condition excluding parameter mismatch is the same
condition in the case of Fig. 5. By comparing between Fig. 5
and Fig. 7, we can find the influence of parameter mismatchand more clearly the structural function of the networks.
From Fig. 7, we can evaluate effective topologies for the synchronization under parameter mismatch among five networks
as follows; p = 0.1 > 0.2 > 0.06 > 1 > 0. It is striking that the random network is greatly influenced by parameter mismatch and difficult to achieve synchronization. In
addition, the network of p = 0.2 is also difficult to achieve
synchronization perfectly although rising the fastest among
five networks. We consider that the network of p = 0.2 hashigh randomness along many rewiring edges. Therefore, thegraph of p = 0.2 in Fig. 7 is very similar to random net-

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