2015 International Symposium on Nonlinear Theory and its Applications
NOLTA2015, Kowloon, Hong Kong, China, December 1-4, 2015

Synchronization of Coupled Chaotic Circuits with Parameter Dispersion
in Small-World Network
Kenta Ago, Yoko Uwate and Yoshifumi Nishio
Department 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—This paper considers the synchronization of
coupled chaotic circuits with the parameter dispersion inthree topologies obtained from the small-world networkmodel. In particular, we focus on the parameter dispersion pattern based on the number of parameter mismatchedcircuits. By means of the computer calculations, the synchronization probabilities of each parameter dispersion pattern in three network topologies are investigated. From thesimulation results, the small-world topology is shown to beeﬀective for the synchronization in the entire network.
1. Introduction(a)1718(b)

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17166107987Figure 1: Illustration of the WS model for N = 20 andk = 4. (a) : Regular network, p = 0. (b) : Small-worldnetwork, p = 0.1. (c) : Random network, p = 1.

2. Small-World Network Model

Complex networks have attracted a great deal of attention from various fields since the discovery of “smallworld” network [1] and “scale-free” network [2]. In particular, how network topological structure influences its
dynamical behaviors, is a hot topic for understanding thestructural function on the networks and suitable for practical application in many disciplines. As the dynamics
on the networks, the synchronization is one of the typical phenomena. Especially, the synchronization phenomena of coupled chaotic systems are very interesting [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 smallworld network. However, the circuit parameters were fixedwith same parameters for all chaotic circuits and the only
one network model were considered.
In this study, we investigate the global synchronizationof coupled chaotic circuits in the small-world network.
Wan and Chen reported the synchronization in the smallworld coupled Chua’s circuits [5]. We focus on “dispersion” of the parameter mismatched chaotic circuits, thesynchronization of coupled chaotic circuits in three network topologies are studied. From the simulation results,
the synchronization probability of each parameter dispersion pattern in three network topologies are investigated.
Thereby, the small-world topology is shown to be eﬀective for the synchronization in the entire network comparedwith the regular network and the random network.

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. Startingfrom a ring lattice with N nodes and k edges per nodes inFig. 1(a), each edge is rewired at randomly with probability
p. The small-world network is known as the graph which ischaracterized by highly clustering coeﬃcient like a regulargraph 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 numberof possible links between those neighbors. This is given as
follows:
C=NN1 ∑1 ∑ 2En.
Cn =N n=1
N n=1 kn (kn − 1)(1)

Third, path length (L) shows the shortest path in the network between two nodes. This is given as follows:N−1 ∑N∑2
L=l(m, n).
N(N − 1) m=1 n=m+1(2)

In this research, we consider coupled chaotic circuitsin three network topologies obtained from WS model inFig. 1. Each topologies is called regular, small-world andrandom, respectively. Table 1 shows the properties of three
networks as shown in Fig. 1.

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v2nv1n
Table 1: Properties of three networks as shown in Fig. 1.
Regular
Small-worldRandomp00.1
1C0.500
0.358
0.216
L
2.895
2.458
2.221

nFigure 2 shows the chaotic circuit which is threedimensional autonomous circuit proposed by Shinriki etal. [6][7]. This circuit is composed by an inductor, a negative resistor, two capacitors, and dual-directional diodes. In
this study, we propose 20 coupled chaotic circuits in threenetwork topologies as shown in Fig. 1. In these networkmodels, chaotic circuits are applied to each node of the network and each edge corresponds to a coupling resistor R.
First, the circuit equations are given as follows:=v2n=

gv1n − idn −1 ∑(v1n − v1k )R k∈S(3)n=-gC1C2L

Figure 2: Chaotic circuit.

3. Coupled Chaotic Circuitdin

L


dt



dv1n

C1


dt





dv

 C2 2ndt

inidn−in + idn ,where n = 1, 2, 3, ..., 20 and S n is the set of nodes which
are directly connected to the node n. We approximate thei − v characteristics of the nonlinear resistor consisting ofthe diodes by the following three-segment piecewise-linearfunction:
Gd (v1n − v2n − V) (v1n − v2n > V)0(|v1n − v2n | ≤ V)
(4)
idn = 
 G (v − v + V) (v − v < −V).
d 1n2n1n2n
By using the parameters and the variables:
√

C2


i=V xn , v1n = Vyn , v2n = Vzn
n

L




√

dC2

t = LC2 τ, “ · ” =, α=dτC1√√√LL1LGd , γ =g, δ =,
 β=C2C2R C2(5)

y1.5

-1.5

1.5
-1.5zFigure 3: Chaotic attractor of the circuit as shown in Fig. 2.
α = 0.5, β = 20 and γ = 0.5.

diodes and are described as follows:
β(yn − zn − 1) (yn − zn > 1)
0(|yn − zn | ≤ 1) (7)f (yn − zn ) =  β(y − z + 1) (y − z < −1).
nnnn
This circuit generates asymmetric chaotic attractor asshown in Fig. 3. The values y and z in Fig. 3 correspond tov1 and v2 of the circuit in Fig. 2, respectively.
4. Parameter Dispersion
In this research, we fix the circuit parameters as α = 0.5,
β = 20, γ = 0.5 and δ = 0.7 for all chaotic circuits.
Additionally, the parameter mismatches ∆α are added foreach circuit parameter α which is relating to the chaos degree. Namely, the parameter α of each circuit is shown as
α = 0.5 + ∆α, respectively. We propose four patterns of
the parameter dispersion as shown in Fig. 4. Each patternis diﬀerent in the number of the parameter mismatched circuits and the range of the parameter mismatches. By using
the proposed four patterns of the parameter dispersion, weadd the parameter mismatches for the circuits in the computer simulations.

the normalized circuit equations are given as follows:
x˙n







y˙n






z˙n

=zn

=αγyn − α f (yn − zn ) − αδ=f (yn − zn ) − xn ,5. Synchronization∑(yn − yk )5.1. Definition of Synchronization
(6)

k∈S n

The nonlinear function f (yn − zn ) corresponds to the i −v characteristics of the nonlinear resistor consisting of theFigure 5 shows the example of the computer simulation results when the parameter dispersion pattern 4 (seeFig. 4(d)) in the small-world network. The vertical axes are
the diﬀerence between the voltages (corresponding to v1 ofthe circuit in Fig. 2) of the nodes 1 and 2 or 7. Namely,- 432 -

5.2. Synchronization Probability(a) Pattern 1.

(b) Pattern 2.

(c) Pattern 3.

(d) Pattern 4.

Figure 4: Proposed four pattern of the parameter dispersion. Horizontal axis: the number of circuits. Verticalaxis: parameter mismatches ∆α.
0.10

y1 - y20.01
0-0.01-0.10(a) Nodes 1 and 2.
0.10

y1 - y70.01
0-0.01-0.10(b) Nodes 1 and 7.
Figure 5: The example of the voltage diﬀerence between
two circuits when the parameter dispersion pattern 4 in thesmall-world network and the definition of the synchronization. (a) : Synchronization. (b) : Asynchronization.

Figure 6 shows the investigation results of the synchronization probability and the combinational sample of eachparameter dispersion pattern in three network topologies.
Each pattern corresponds to the four pattern in Fig. 4. The
vertical axes denote the synchronization probability in theentire network during the certain time interval. If the synchronization probability equals 100%, we can consider thatthe entire network is synchronized perfectly. The horizontal axes denote the combinational samples consideredfrom each parameter dispersion pattern. The combinationalof the parameter dispersion is considered a large number.
Therefore, we choose the 10 samples randomly as the combinational of the parameter dispersion in each pattern ofFig. 4. From Fig. 6, we confirm that the networks become
to be diﬃcult for the global synchronization by increasing
the number of the parameter mismatched circuits. On theother hand, the synchronization probability in each patterndepends on the combinational sample. Especially, when
the parameter mismatched circuits are the small number,the global synchronization is strongly influenced from thenetwork topologies and the combinational of the parameter dispersion. Figure 7 shows the average synchronization
probability in each network of Fig. 6. From this result, weconsider that the small-world topology is eﬀective for thesynchronization in the entire network.
In addition, the complex relation between the parametermismatched circuit and the network topological structure
can be confirmed in this research. More detailed these relation considering more large number of samples should beinvestigated for our future works.
6. Conclusion
This paper considered the synchronization of coupledchaotic circuits with the parameter dispersion in three network topologies obtained from the WS model. In particular, we focused on the parameter dispersion patternbased on the number of parameter mismatched circuits. Byif the two nodes are synchronized, the value of the graphshould be almost zero. In order to analyze the synchronization state, we define the synchronization by the following
equation:|yi − y j | < 0.01 (i , j).

(8)

By means of the above definition of the synchronization,
we define that the nodes 1 and 2 in Fig. 5(a) are synchronized perfectly. 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 timeinterval as (τ= 1, 000, 000 and step = 0.01τ) and statistically investigate the synchronization probability in the entire network of 20 coupled chaotic circuits.

Figure 7: Average synchronization probability of the combinational sample of each parameter dispersion pattern inthree network topologies.

- 433 -(a) Pattern 1.

(b) Pattern 2.

(c) Pattern 3.

(d) Pattern 4.

Figure 6: Relationship between the synchronization probability and the combinational sample of each parameter dispersion pattern in three network topologies.
means of the computer calculations, the synchronization
probabilities of each parameter dispersion pattern in three
network topologies were investigated. From the simulation
results, we consider that the small-world topology is eﬀective for the synchronization in the entire network. More detailed investigation considering more large-scale networks
should be carried out in our future works.
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