IEEE Workshop on Nonlinear Circuit Networks
December 15-16, 2017

Spread of Chaotic Behavior
in Switching Ring Network Topology
by Coupled Chaotic Circuits
Takahiro Chikazawa, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: {chikazawa,uwate,nishio}@ee.tokushima-u.ac.jp

Abstract—In this study, we investigate the spread of chaotic
behavior in switching ring network topology. We propose ring
network model by coupled chaotic circuits. In proposed networkmodel, one circuit is set to generate chaotic attractor and theother circuits are set to generate three-periodic attractors. By
using computer simulations, we investigate the chaos propagationby changing the switching coupling patterns. From the simulation
result, the chaos propagation depends on the previous switchingpattern.

I. I NTRODUCTION
We have various types network in our society. Most ofnetworks can be represented as a graph by nodes and edges.
Moreover, the network model has various types of feature
quantities. Examples of feature quantities are path length, degree distribution and clustering coefficient. However, networkmodel become more large scale and complicated network.
These network model is called by complex networks .
Recently, complex networks have attracted a great deal ofattention from various fields since the discovery of smallworld
network [1] andscale-free
network [2]. Aswith the complex networks, various types of propagation incomplex network have attracted a great deal of attention
from various fields. For example, friendship in social network,neural activities with information processing in human brain
network and the pandemic outbreak of viral infection in
biology. Therefore, we consider that we can analyze various
complicated phenomena of complex networks by investigating
the spread of chaotic behavior. Additionally, it is important
to investigate propagation phenomena observed from coupledchaotic circuits for future engineering applications.
In our research group, we have investigated the spread ofchaotic behavior in various type networks. Various research
results of spread of chaotic behavior in static network model
have been reported by using coupled chaotic circuits [3]. In
this study, we confirmed that the three-periodic attractors areaffected from the chaotic attractors when the coupling strengthincreases. In scale free network, we confirmed that chaos
propagation is more difficult when we set the initial chaos
position in high degree node. Additionally, we have comparedscale free and random network for investigation spread ofchaotic behavior [4]. Even though we increase the valueof degree in initial chaos position, the ratio of propagation
changes only a little. In detail of this result, we explain thisresult in section (III).
However, these previous study have been reported by static
network model. Therefore it is necessary to investigate thespread of chaotic behavior in dynamic network model. Example of dynamic network model is temporal networks [5].
Figure 1 shows the example of temporal networks. In addition,
various research of propagation by using temporal networks
have been reported by many researchers [6].

Fig. 1. System model of temporal networks.

In this study, we propose ring network model by 10 coupledchaotic circuits. In ring network of initial state, we fixed that
one circuit is set to generate chaotic attractor and the other circuits are set to generate three-periodic attractors. Additionally,we switch the network topology such as temporal networks.
We propose two different switching coupling patterns. We
define these patterns that we cut two edges at every switchingtime. In these conditions, we investigate the spread of chaotic
behavior in switching ring network topology.
II. C IRCUIT MODEL
The chaotic circuit is shown in Fig. 2. This circuit consistsof a negative resistor, two inductors, a capacitor and dualdirectional diodes. This chaotic circuit is called Nishio-Inabacircuit.
The circuit equations of this circuit are described as follows:- 111 -i2i1L1-r

L2Czz
Cnv

xvd(a)Chaotic attractor

x(b)Three-periodic attractor

Fig. 3. Attractor.
Fig. 2. Chaotic circuit.

di L1 dt = v + ri




di
L2 = v − vddt





dv



 C dt = −i1 − i2 ,III. S TATIC NETWORK MODEL IN PREVIOUS STUDY [4]In this section, we investigate the ratio of propagation instatic complex model.
(1)

The characteristic of nonlinear resistance is described asfollows:()rdV
Vvd =i2 +− i2 −.
(2)
2rdrd
By changing the variables and parameters,√√
CL1 C
V x n , i2 =V yn , v = V z ni1 =

L
L
12
√
√

CL1L1 C, β=, δ = rd,α=r
L1L2L2


√


√


 γ = 1 L1 , t = L1 C2 τ,R C
The normalized circuit equations are given as follows:dxi

= αxi + zi dτ dyi = zi − f (yi )dτ∑dzi
=−x−βy−γ(zi − zj )
ii
dτ
j∈Sn


(i, j = 1, 2, · · ·, N ).

A. System model
Figures 3 and 4 show the proposed different type networkmodels in our previous study [6]. The characteristic of ModelA is the scale-free network topology. The characteristic ofModel-B is the random network topology. In each model,
each chaotic circuit is coupled by one resistor R. We use
25 coupled chaotic circuits and 34 resistors in each networkmodel. Moreover, these proposed network model is staticnetwork model.
Figure 5 shows degree distribution of each network. In thisgraph, the vertical axis denotes the number of nodes and thehorizontal axis denotes the value of degree.

(3)
2223

1721

2124

191792415

181016201816252212132315118101425

214

911

6

1719

58

4

75

3

420

3

1

2

13
6

12(4)
Fig. 4. Model-A (Scale-free network). Fig. 5. Model-B (Random network).

In Eq. (4), N is the number of coupled chaotic circuits andγ is the coupling strength. f (yi ) is described as follows:()11
1f (yi ) =yi +− yi −.
(5)2δδIn Eq. (4), N is the number of coupled chaotic circuitsand γ is the coupling strength. We define αc to generate thechaotic attractor (see Fig. 3(a)) and αp is defined to generate
the three-periodic attractors (see Fig. 3(a)). For the computer
simulations, we calculate Eq. (4) using the fourth-order RungeKutta method with the step size h = 0.01. In this study, we
set the parameters of the system as αc = 0.460, αp = 0.418,
β = 3.0 and δ = 470.0.

- 112 -Fig. 6. Degree distribution of each network..

B. Simulation result
We investigate the ratio of propagation when we change thevalue of degree in the initial chaos position according to degree
distribution. Moreover, in scale-free and random networks, wecompare their differences of simulation results. For example,
in model-A, when the value of degree k in the initial chaos
position is 4, we set the chaotic attractor in 9th or 13th node.
Also, in model-B, when the value of degree k in the initialchaos position is 4, we set the chaotic attractor in 9th or 17thnode.
The simulation results of ratio of propagation according todegree distribution are shown in Figs. 6 and 7.
Furthermore, we average the ratio of propagation in eachnode under the same condition. In addition, we investigate theratio of propagation in the static state when we fix couplingstrength as γ = 0.001, 0.005. Here, we define the ratio of
propagation as number of chaotic circuits of whole networkat steady state.

From the result, in scale-free network, chaos propagation
become to more difficult, when we increase the value ofdegree in initial chaos position. On the other hand, in random
network, even though we increase the value of degree in initialchaos position, the ratio of propagation changes only a little.
IV. DYNAMIC NETWORK MODEL
In this section, we investigate the spread of chaotic behavior
in dynamic model.
A. Switching coupling patternFigure 9 shows the proposed ring network model in thisstudy. In the proposed model, each chaotic circuit is coupledby one resistor R. We use 10 coupled chaotic circuits and 10edges.
1st
10th

2nd
3rd

9th

4th

8th
7th

5thR

6th

Fig. 9. Ring network model.

Fig. 7. Ratio of propagation according to degree distribution in Model-

A (scale-free network).

We propose two different switching coupling patterns inFig. 10. In this study, we change the switching coupling
patterns at every τ = 10, 000. Moreover, in every switching,
we cut the two edges.
In pattern-A, all edges are cut form τ = 0 to τ = 10, 000.
Next switching, we cut two edges from 4th to 5th node andfrom 7th to 8th node. Next switching, we cut two edges from5th to 6th node and from 6th to 7th node. Finally, we cut twoedges from 4th to 5th node and from 7th to 8th node.
In pattern-B, all edges are cut form τ = 0 to τ = 10, 000.
Next switching, we cut two edges from 3rd to 4th node andfrom 8th to 9th node. Next switching, we cut two edges from5th to 6th node and from 6th to 7th node. Finally, we cut twoedges from 3rd to 4th node and from 8th to 9th node.
B. Simulation results

Fig. 8. Ratio of propagation according to degree distribution in Model-B (Random network).

We investigate the chaos propagation by changing theswitching coupling patterns. In the initial states of each pattern,
one circuit is set to generate chaotic attractor and the othercircuits are set to generate three-periodic attractors. We set theinitial chaos position in 6th node and the other node is fixed
as three-periodic state. In addition, we investigate the chaos
propagation in the static state when we fix coupling strength
as γ = 0.010. Figure 11 shows the appearances how to spreadfrom three-periodic state to chaotic state when we change theswitching coupling according to pattern-A. Moreover, Fig. 12shows the appearances how to spread from three-periodic
attractor to chaotic attractor when we change the switching
coupling according to pattern-B.

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1st

1st

10th

2nd

9th

3rd

8th

4th

1st

1st

10th

2nd

3rd

10th

2nd
9th

3rd

8th

4th

10th

2nd
9th

3rd

8th

4th

9th

Pattern-A
4th
7th

5th

7th

5th

6th

6th

1st
3rd

8th

4th

6th

1st

1st

10th

2nd
9th

8th

10th

2nd
9th

3rd

8th

4th

7th

5th

6th

1st

10th

2nd
3rd

7th

5th

10th

2nd
9th

3rd

8th

4th

9th

Pattern-B
4th
7th

5th

7th

5th

6th

6th

6th

6th

τ=10,000

τ=0

8th
7th

5th

7th

5th

τ=20,000

τ=30,000

Fig. 10. Switching coupling pattern at every τ = 10, 000.

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

10th

V. C ONCLUSIONS

τ=0

In this study, we have investigated the spread of chaotic
behavior in switching ring network topology by coupledchaotic circuits. By the computer simulations, we confirmedthat the three-periodic state is affected from the chaotic state.
Moreover, we have observed the spread of chaotic behavior
by the influence of switching pattern. From the simulation
result, we consider that the spread of chaotic behavior depends
on previous switching coupling pattern. Even if the edge
temporarily connect to other nodes, it is assumed that weobserve the chaos propagation by changing the switching
coupling pattern.
For the future works, we propose the different switchingpattern such as random or systematic. Furthermore, we investigate the spread of chaotic behavior when we change theswitching coupling patterns at every more short τ .

τ=10,000

τ=20,000

τ=30,000

Fig. 11. spread of chaotic behavior in pattern-A.

1st

2nd

3rd

4th

5th

6th

7th

8th

9th

10th

τ=0

τ=10,000

τ=20,000

τ=30,000R EFERENCESFig. 12. spread of chaotic behavior in pattern-B.

From each simulation result, we have observed the chaos
propagation. In switching time from τ = 10, 000 to τ =20, 000, in each switching pattern, each three-periodic attractors which is connected to 6th node are changed tochaotic state by the influence of 6th node. In pattern-A from
τ = 20, 000 to τ = 30, 000, 5ht and 7th node change tothree-periodic state because the edges from 5th to 6th nodeand from 6th to 7th node are cut. However, in pattern-B from
τ = 20, 000 to τ = 30, 000, despite the fact that the bothside edges of 6th node do not connect to other nodes, weobserved that three-periodic state node change to chaotic state.
Moreover, in switching time from τ = 30, 000 to τ = 40, 000of pattern-B, only 4th, 5ht, 7th and 8th node keep at chaotic
state because the both side edges of 5ht, 6th and 7th node areconnected.

[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] T. Chikazawa, Y. Uwate and Y. Nishio, “Investigation of Spreading
Chaotic Behavior in Coupled Chaotic Circuit Networks with VariousFeatures,” Proceedings of RISP International Workshop on Nonlinear
Circuits, Communications and Signal Processing (NCSP’17), pp. 337340, Feb. 2017.
[4] T. Chikazawa, Y. Uwate and Y. Nishio, “Spread of Chaotic Behavior in
Scale-Free and Random Networks,” Proceedings of Asia Pacific Conference on Postgraduate Research in Microelectronics and Electronics(PrimeAsia’17), pp. 21-24, Oct. 2017.
[5] P. Holme, J Saramaki, “Temporal networks,” Physics Reports, vol. 519,
Issue 3, pp. 97-125, Oct 2012.
[6] B. Adhikari, Y. Zhang, A. Bharadwaj, B. A. Prakash,“Condensing
Temporal Networks using Propagation,” Proceedings of the 2017 SIAM
International Conference on Data Mining, pp. 417-425, Apr 2017.

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