IEEE Workshop on Nonlinear Circuit Networks
December 3-4, 2021

Correlation between Chaos Propagation
and Features of Complex Networks
Naoto Yonemoto† , Kenta Ago‡ , Yoko Uwate† and Yoshifumi Nishio††Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: {yonex34, kentaago, uwate, nishio}@ee.tokushima-u.ac.jp‡Murata Manufacting Co., Ltd. Email: kentaago@gmail.com
Abstract—Recently, it is said that it is important to analyze
the damage of natural disasters and the process of spreading
infectious diseases in a networked way to take countermeasures.
The propagation of chaotic phenomena, in which complex waveforms are generated, are also focused, and chaotic propagation isstudied by coupling many chaotic circuits to construct a network.
In this study, the propagation path and the chaos generation
rate of each circuit are studied by computer simulation. The
correlation between the features of the network, such as thedegree and the shortest path length, and the chaos propagationis also investigated.

I. I NTRODUCTION
In recent years, the damage caused by natural disasters
and the process of spreading infectious diseases are studied
focusing on the fact that they are network-like [1]. In natural
disasters, damage spreads in the network-like way, with nodes
as compound disasters and edges as cascading disasters, resulting in a wide variety of mixed damage. In infectious diseases,
the structure of the network is that clusters are nodes, andedges extending from the nodes create new clusters. In orderto prevent such damage from increasing, it is important tofocus on the network and develop research on propagation.
It is also confirmed that propagation occurs in chaoticphenomena. The notion of propagation of chaos for large
systems of interacting particles originates in statistical physics
and has recently become a central notion in many areas ofapplied mathematics [2]. In addition, the propagation of chaos
is studied by using chaotic circuits, which can be clearly
described by physical laws, and coupling them with resistors
to construct a network.
Then, what factors affect chaos propagation? For example,
it could be the characteristics of the waveform, distance, path,
network structure, and regulation. They can be considered
as non-linearity, frequency, number of circuit to be passed,
presence of hubs, clustering coefficient, and coupling strength
in the networks coupled chaotic circuit.
The effects of frequency and coupling strength on phasedifference and propagation speed were studied by generating
chaos at one circuit in a ring coupled chaotic circuit [3], andthe effects of hub presence and path distance were studied in
multi-ring coupled chaotic circuits [4]. In addition, the effects
of clustering coefficients and generation points were studiedin complex networks [5]. As a result, it was found that thepropagation occurred only between circuits set to the same

frequency. In addition, it was found that propagation was easier
when the path distance was small. Furthermore, when there
were many hubs and the clustering coefficient was large, thepropagation rate became small.
In this study, the propagation paths and the rate that chaos isgenerating are investigated in random and scale-free networks.
Therefore, the order of propagation is shown by visualizing it
in color and calculating the rate that chaos generating in eachcircuit for a certain time. Furthermore, the correlation between
the results of propagation and network features is investigated.
II. S YSTEM M ODELThe chaotic circuit is shown in Fig. 1. This circuit consistsof a negative resistor, two inductors, a capacitor and dualdirectional diodes. This circuit is considered as a node in thecomplex networks.

i1i2

C

L2v

L1-r

Cnvd

Fig. 1. Chaotic circuit.

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:dxn= αxn + zndτ dyn = zn − f (yn )
dτ∑
dzn

γ(zn − zm )= −xn − βyn − dτj∈Sn(n, m = 1, 2, · · ·, N ).

(1)

In Eq. (4), N is the number of coupled chaotic circuits andγ is the coupling strength.

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The characteristic equation for dual-directional diodes
f (yn ) is described as follows:f (yn ) =12(yn +1
1− yn −δδ).

(2)

The parameters are described as follows:√√C
L1 CVx,i=V yn , v = V z ni=n21
L1L2√√CL1L1 C(3), β=, δ = rd,α=rLLL
122√√
1L1
 γ=, t = L1 C2 τ.
R C
In this chaotic circuit, we define αc to generate the chaotic
attractor (Fig. 2(a)) and αp is defined to generate the threeperiodic attractors (Fig. 2(b)). We calculate Eq. (4) using
the fourth-order Runge-Kutta method with the step size h =0.005 with the computer simulations. In this study, we set theparameters of the system as αc = 0.460, αp = 0.413, β = 3.0,
γ = 0.002 and δ = 470.0. After getting to the steady statethat all the circuits generate three-periodic attractors, chaos
propagation is simulated by generating chaos from one circuit.

complex networks of N nodes and E edges can be evaluatedby the typical three types structural metrics such as degree,
clustering coefficient and path length. First, degree k is thenumber of edges which is connected on a node. Second, path
length L shows the shortest path in the network between
two nodes. Third, clustering coefficient C shows the numberof actual links between neighbors of a node divided by thenumber of possible links between those neighbors.
TABLE IFEATURE QUANTITIES OF PROPOSED EACH NETWORK .

FeatureThe number of nodes
The number of edges
Avg. degree
Avg. path length
Avg. clustering coefficient

BA scale-free
2546
3.680
2.363
0.165

ER random
25463.680
2.453
0.114

Figure 5 shows degree distribution of each network. In thisgraph, the vertical axis denotes the probability of existence ofdegree and the horizontal axis denotes the value of degree.

zzx(a) Chaos.

x(b) Three-period.

Fig. 2. Attractors of chaotic circuit.
Fig. 5. Degree distribution of each network.

III. C OMPLEX NETWORK MODEL
Figures 3 and 4 show the proposed different type networkmodels. The characteristic of the complex network in Fig. 3 isthe BA scale-free topology. The characteristic of the complex
network in Fig. 4 is the ER random topology. In each model,
each chaotic circuit is coupled by one resistor R. We use25 coupled chaotic circuits and 46 resistors in each networkmodel.

Fig. 3. BA scale-free network.

IV. R ESULTS
The order of chaos propagation and the rate that chaos ispropagating in each circuit are investigated in the BA scalefree and the ER random. We explain the simulation methodof the chaos propagation. First, the three-periodic attractors
are generated with the same initial values of xn , yn and zn
for all the circuits. Then, the chaos is generated at node 0. In
other words, the parameter αp = 0.413 of the circuit at node
0 is changed to αc = 0.46. The chaos propagation for 900,000
counts is simulated. At that time, the threshold width was set
in the plane of x1 ≥ 0, x2 = 0 to determine the three-periodic
attractor and the chaotic attractor. When the three-periodic
attractor drawn in black changes to the chaotic attractor, thechaotic attractor and waveform are drawn in blue. When it
becomes the three-periodic attractor again, the three-periodic
attractor is drawn in red.

Fig. 4. ER random network.

Moreover, the feature quantities of the proposed each network are summarized in Table I. Topological structures in

A. Attractors and wave forms
At about 900,000 counts after the generation of chaos, theattractors that the chaos propagated in the BA scale-free are- 66 -shown in Fig. 6(a), and those in the ER random are shown in
Fig. 6(b). For 900,000 counts after the generation of chaos,wave forms that the chaos propagated in the BA scale-free areshown in Fig. 7(a), and those in the ER random are shown inFig. 7(b).
01234012345678956789101112131410111213141516171819151617181920212223242021222324(a) BA scale-free.

(b) ER random.

(b) ER random.
Fig. 7. Wave forms of xn .

Number of node

Number of nodeFig. 6. Attractors.

Rate of periodic attractor [%]Rate of periodic attractor [%](a) BA scale-free(b) ER random

Number of node(a) BA scale-free.

After the chaos propagated to most of the circuits, some ofthem returned to the three-periodic attractors. However, only
node 5 in Fig. 6(b) remained the initial three-periodic attractor.

Number of nodeFig. 8. Rate of the three-periodic attractor in the initial state.

B. The Order of chaos propagation
The rate is calculated by counting the time when chaos
is generating and the time when chaotic attractors become
periodic attractor again. From those results, the rate of thethree-periodic attractor in the initial state are shown in Fig. 8,
and it indicates that the chaos propagates in the order ofdecreasing value.
The start of propagation in the BA scale-free took longer
than in the ER random. However, when the chaos propagatedto the hub, it propagated quickly in large numbers in Fig. 8.
The rate of time that each circuit is generating chaos areshown in Fig. 9. As well as the previous studies [5], the ratethat chaos is generating in BA scale-free is smaller than thatin ER random.

Rate of chaotic attractor [%]Rate of chaotic attractor [%](a) BA scale-free(b) ER random

Fig. 9. Rate of the chaotic attractor.

C. Correlation
We investigate the correlation between the propagation ofchaos and the features of the networks. First, the correlation
between the rate of three-periodic attractor in initial state, thatis mean, the order that the chaos is propagating, and the shortest path length from node 0 is investigated in Figs. 10-11(a).
Secondly, the correlation between the order of propagation andthe degree is investigated. The results are shown in Figs. 10-- 67 -Degree

Shortest path length

DegreeShortest path length11(b) and Table II. In addition, the correlation between the ratethat chaos is generating and the features of those networks isinvestigated. The results are shown in Figs. 12-13 and Table
III.

Rate of chaotic attractor [%](a) Shortest path length

Rate of chaotic attractor [%](b) Degree

Fig. 13. Correlation between rate chaos is generating and features of ER
Random.

Rate of periodic attractor [%](a) Shortest path length

Rate of periodic attractor [%](b) Degree

TABLE IIIC ORRELATION BETWEEN RATE CHAOS IS GENERATING
AND FEATURES OF EACH NETWORK .

DegreeShortest path length

Fig. 10. Correlation between order chaos propagate and features of BA Scalefree.

Rate of periodic attractor [%](a) Shortest path length

BA scale-free
0.366- 0.725

ER random0.455- 0.581

DegreeInvestigating the correlation between the order of propagation and the features of the network, we found that thelarger degree of the BA scale-free, the stronger the tendency
to propagate. On the other hand, the ER random tended topropagate more easily when the shortest path length was
shorter than that of the BA scale-free, although the correlation
was weak.
Shortest path length

ER random- 0.480
0.848

V. C ONCLUSION

Rate of periodic attractor [%](b) Degree

TABLE IIC ORRELATION BETWEEN ORDER CHAOS PROPAGATEAND FEATURES OF EACH NETWORK .

Rate of chaotic attractor [%](a) Shortest path length

BA scale-free- 0.360
0.907

It was found that the larger the degree of both networks,
the stronger the tendency for chaos to keep generating. On
the other hand, the correlation with the shortest path length
was similar to the result of the propagation order.

Fig. 11. Correlation between order chaos propagate and features of ERRandom.

FeatureShortest path lengthDegree

FeatureShortest path length
DegreeRate of chaotic attractor [%](b) Degree

Fig. 12. Correlation between rate chaos is generating and features of BA
Scale-free.

In this study, the order of chaos propagation and the rate thatchaos is generating in complex networks were investigated.
The correlations between these results and the features ofcomplex networks were compared between ER random andBA scale-free. In the BA scale-free, the larger the degree, thefaster the order of propagation, and the higher the rate thatchaos is generating. In addition, the ER random was morecorrelated with the shortest path length than the BA scale-free.
In the future, we will investigate whether the same trend canbe obtained when the generation points are changed according
to the degree. In addition, we will investigate the effect in alarge scale network with 100 circuits. We are also considering
to investigate the effect of reducing the parameter of the chaosof the generating point to the three-period attractor.
R EFERENCES[1] Yoshiaki Kawata, “Response to social disaster through phase transition
Cases of COVID-19 and heavy rainfall disaster”, Journal of societal safety
sciences,11,37-56 (2021-03-31)[2] Louis-Pierre Chaintron and Antoine Diez, “Propagation of chaos: a reviewof models, methods and applications”, Mathematics, Physics. Published
28. June. 2021.
[3] Yoko Uwate and Yoshifumi Nishio, “Chaos Propagation in a Ring of
Coupled Circuits Generating Chaotic and Three-Periodic Attractors”,
Proceedings of IEEE Asia Pacific Conference on Circuits and Systems(APCCAS’12), pp. 643-646, Dec. 2012.
[4] Takahiro Chikazawa, Yoko Uwate and Yoshifumi Nishio, “Chaos Propagation in Coupled Chaotic Circuits with Multi-Ring Combination”,
Proceedings of IEEE Asia Pacific Conference on Circuits and Systems(APCCAS’16), pp. 65-68, Oct. 2016.
[5] Takahiro Chikazawa, Yoko Uwate and Yoshifumi Nishio, “Spread ofChaotic Behavior in Scale-Free and Random Networks”, Proceedings ofAsia Pacific Conference on Postgraduate Research in Microelectronics
and Electronics (PrimeAsia’17), pp. 21-24, Oct. 2017.

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