2017 RISP International Workshop on Nonlinear Circuits,Communications and Signal Processing (NCSP'17)
Guam, USA, February 28th to March 3rd, 2017

Investigation of Spreading Chaotic Behavior
in Coupled Chaotic Circuit Networks with Various Features
Takahiro Chikazawa, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University,
2-1 Minamijosanjima, Tokushima, 770-8506 Japan
Email: {chikazawa, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract

25 nodes and 35 edges. The other is used 50 nodes and 85
edges. In our proposed network model, one circuit is set togenerate chaotic attractor and the other circuits are set to generate three-periodic attractors. We investigate ratio of spreading chaotic behavior in each networks. We change the initial
position of the chaos attractor according to network feature.

In this study, we investigate the spread of chaotic behavior
in coupled chaotic circuits networks. We propose two types
complex networks with hub. In each network model, thestructure is composed of coupled chaotic circuits when one
circuit is set to generate chaotic attractor and the other circuits
are set to generate three-periodic attractors. By using computer simulations, when the coupling strength increase, wehave observed that the chaotic attractor spread to the other cir- 2. Circuit model
cuits. Moreover, we investigate ratio of propagation in each
proposed network by changing the initial position of chaotic
The chaotic circuit is shown in Fig. 1. This circuit consists
attractor. The initial position of chaotic attractor changes ac- of a negative resistor, two inductors, a capacitor and dualcording to feature quantities of node.
directional diodes. This chaotic circuit is called Nishio-Inaba
circuit [4].
1. Introductioni1

i2

In our society, there are many networks. These networkmodels have various types of feature quantities. Examples
L1
L2of feature quantities are path length, degree distribution, andclustering coefﬁcient. Recently, network models become
C v
i-r
more complex and larger scale. Furthermore, various types ofvd
propagation have attracted a great deal of attention from various ﬁelds [1]. For example, the trafﬁc jam of the transportation network and the viral infection are mentioned. Hence, it
is more difﬁcult to analyze the phenomena in the networks.
Figure 1: Chaotic circuit.
Therefore, we consider that we can analyze various complicated phenomena of complex networks by investigating thespread of chaotic behavior. Additionally, it is important toinvestigate propagation phenomena observed from coupled
The circuit equations of this circuit are described as folchaotic circuits for future engineering applications.
lows:⎧diIn our previous studies, the synchronization phenomena⎪⎪
L1 = v + ri⎪⎪
and the chaos propagation has been investigated by many redt⎪⎪⎪⎪searches. The synchronization phenomena in complex net⎨
diL2 = v − vd
works has been reported to various ﬁelds [2]. On the other(1)dt⎪⎪hand, the chaos propagation have been investigated in simple⎪⎪⎪
dv⎪
networks of chaotic circuits such as a ring combination [3].
⎪= −i1 − i2 ,C⎪⎩dt
In this paper, we investigate the spread of chaotic behavior in complex network with coupled chaotic circuits. We
propose two types complex networks with hub. One is used
The characteristic of nonlinear resistance is described asC- 337 -follows:

3. Network model- -
rd -V -- -V -i−i.
+−vd =222 rd - rd -(2)By changing the variables and parameters,;⎧√
CL1 C⎪⎪
i1 =V xn , i2 =V yn , v = V z n⎪⎪⎪
L1L2⎪⎪⎪;√⎨C
L1 C
L1α=r, β=, δ = rd,⎪LLL⎪
122⎪⎪;⎪⎪
⎪⎪⎩ γ = 1 L1 , t = L1 C2 τ,R CFigure 3 shows the proposed two types complex networks.
In our proposed network model, each chaotic circuit is coupled by one resistor R. We use 25 coupled chaotic circuits
and 35 edges in Fig. 3(A). The other model use 50 coupledchaotic circuits and 85 edges in Fig. 3(B). The feature quantities of the proposed each network are summarized in Table 1.
Moreover, Fig. 4 shows degree distribution of each network.
In this graph, vertical axis denotes the number of nodes andhorizontal axis denotes the value of degree.

(3)

In the proposed system, each circuit is connected to only adjacent circuits by the resistors. The normalized circuit equations of the system are given as follows:⎧dxi⎪⎪= αxi + zi⎪⎪dτ⎪⎪⎪⎪⎨ dyi = zi − f (yi )
dτ(4)
⎪dzi
⎪
⎪
−βy−γ(z−z)
=−x⎪
iiij
⎪
dτ⎪
⎪
j∈Sn
⎪
⎩
(i, j = 1, 2, · · ·, N ).

        

In Eq. (4), N is the number of coupled chaotic circuits and γ
is the coupling strength. f (yi ) is described as follows:- -1 -1 -- -1 -(5)yi + - − -yi − - .
f (yi ) =2 δδIn this chaotic circuit, we deﬁne αc to generate the chaoticattractor (see Fig. 2(a)) and αp is deﬁned to generate thethree-periodic attractors (see Fig. 2(b)). For the computer
simulations, we calculate Eq. (4) using the fourth-order
Runge-Kutta method with the step size h = 0.01. In this study,
we set the parameters of the system as αc = 0.460, αp =0.412, β = 3.0 and δ = 470.0.

        Figure 3: Network model.

zzx() ( ) Table 1: Feature quantities of proposed each network.
Feature(A)(B)The number of nodes
2550
The number of edges
3585
Avg. degree
2.8
3.4
Avg. path length
3.34
3.51
Avg. clustering coefﬁcient
0.104
0.157

x Figure 2: Attractors of chaotic circuit.

In this study, we deﬁne hub as the node which is the highest
value of degree. In Model-A, we deﬁne hub as 1st node. InModel-B, we deﬁne hub as 31st node.

- 338 -Figure 4: Degree distribution of each network.
Figure 5: Ratio of propagation according to the shortest path length
between the hub node and the initial chaos position in model-A.

4. Simulation results
In this section, we investigate the spread of chaotic behavior by increasing the coupling strength and changing the initial chaos position. In the previous studies, chaos propagation have been investigated simple network system of coupled chaotic circuits [5]. Moreover, the three-periodic attractors are affected from the chaotic attractors when the couplingstrength are increasing.
In addition, we use two type network in this study. The
number of node and edge in each network is different. However, one circuit is set to generate chaotic attractor and theother circuits are set to generate three-periodic attractors ineach network.

4.2 Model-B (50 coupled chaotic circuits)

Next, we construct a larger scale complex network and weinvestigate the ratio of propagation when we change the shortest path length between the hub(31st) node and the initialchaos position. The simulation results of ratio of propagation according to the shortest path length are shown in Fig. 6.
For example, in model-B, when the shortest path length between the hub(1st) node and the initial chaos position is 2,
we set the chaotic attractor in 6th, 7th, 10th, 11th, 12th, 16th,19th, 26th, 28th, 34th, 36th, 38th, 39th, 42nd, 44th or 50th
node. Furthermore, we average the ratio of propagation ineach node under the same condition. In addition, we inves4.1 Model-A (25 coupled chaotic circuits)tigate the ratio of propagation in the static state when we ﬁx
We investigate ratio of propagation when we change the coupling strength as γ = 0.001, 0.005. Here, we deﬁne theshortest path length between the hub(1st) node and the initial ratio of propagation as number of chaotic circuits of wholechaos position. The simulation results of ratio of propaga- network at steady state.
tion according to the shortest path length are shown in Fig. 5.
For example, in model-A, when the shortest path length between the hub(1st) node and the initial chaos position is 1,
we set the chaotic attractor in 2nd, 3rd, 4th, 5th, 6th, 7th or
8th node. Also, when the shortest path length is 4, we setthe chaotic attractor in 18th, 21st or 22nd node. Furthermore,
we average the ratio of propagation in each node under thesame condition. In addition, we investigate the ratio of propagation in the static state when we ﬁx coupling strength asγ = 0.001, 0.005. Here, we deﬁne the ratio of propagation as
number of chaotic circuits of whole network at steady state.
From the result, ratio of propagation in large couplingstrength is higher than small coupling strength. When we
increase the path length between the hub(1st) node and theinitial chaos position, each three-periodic attractors are easily
affected from the chaotic attractors. Moreover, when we set Figure 6: Ratio of propagation according to the shortest path length
the initial chaos position in hub(1st) node, ratio of propaga- between the hub node and the initial chaos position in model-B.
tion is the lowest than other node.

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From the result, the highest ratio of propagation is 88.67%
when the coupling strength γ = 0.005 and the shortest path
length is 4. As with the model-A, when we increase the pathlength between the hub(31st) node and the initial chaos position, each three-periodic attractors are easily affected fromthe chaotic attractors. As the path length in the initial chaos
position increases, the spread of chaotic behavior become
more easy.
Next, we investigate the spread of chaotic behavior in detailed condition. When we ﬁx the shortest path length as 2,
the ratio of propagation according to the value of degree ineach node are summarized in Fig. 7. For example, in thiscondition, when the value of degree in the initial chaos position is 2, we set the chaotic attractor in 6th or 19th node.
Moreover, when we ﬁx the shortest path length as 3, the ratio of propagation according to the value of degree in eachnode are summarized in Fig. 8. For example, in this condition, when the value of degree in the initial chaos position is2, we set the chaotic attractor in 4th, 17th, 23rd, 24th, 35th or
48th node. In addition, we ﬁx coupling strength as γ = 0.001
in each result.

From each result, when we increase the value of degree ininitial chaos position, ratio of propagation become low. In
other words, the spread of chaotic behavior become more difﬁcult by increasing the value of degree in the initial chaos
position. Consequently, three-periodic attractors are easily
affected from the chaotic attractors when we set the initialchaos position in faraway nod from hub and smaller value ofdegree.
5. Conclusion
In this study, we have investigated spreading chaotic behavior in complex networks. By the computer simulations,
we investigated ratio of propagation by changing the initialchaos position. We conﬁrmed that chaos propagation is more
difﬁcult when we increase the and the value of degree in theinitial chaos position. We consider that the spread of chaotic
behavior affect the low degree node. Furthermore, the low
degree node has important role in complex network.
Acknowledgment
This work was partly supported by JSPS Grant-in-Aid for
Scientiﬁc Research 16K06357.
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Figure 7: Ratio of propagation according to degree distribution
when we ﬁx the shortest(31st) node as 2.

[3] Y. Uwate and Y. Nishio, “Chaos Propagation in a Ringof Coupled Circuits Generating Chaotic and ThreePeriodic Attractors,” Proceedings of IEEE Asia Paciﬁc
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Figure 8: Ratio of propagation according to degree distribution
when we ﬁx the shortest(31st) node as 3.

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