Synchronization of Two Different Hierarchical Networks
using Coupled Chaotic Circuits
Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: {uwate, nishio}@ee.tokushima-u.ac.jp

Abstract
We present results obtained for a new paradigm of coupledoscillatory networks, called competitive interaction networks.
In the proposed circuit system, there are two hierarchical networks including multi-directed connections. We investigate
what kind of strategy in network is effective for wining in thebattle. We focus on dependencies on relationship of couplingstrength and network structure.

it is very important to use mathematical model for complex
networks in order to understand the synchronization states byapproaching theoretical methods, we also need to consider
physical models for future engineering applications.
In this study, we focus on synchronization state observed in
two networks of chaotic circuits which are coupled in multidirection hierarchically. We analyze the role of synchronization by changing the competitive coupling strategies.
2. Network Model

1. Introduction

Figure 1 shows the hierarchical network model of thisThe chaotic circuits located at every layers correspondstudy.
Coupled oscillatory circuits provide simple models for deto
chesspiece.
scribing high-dimensional nonlinear phenomena occurring in
our everyday world. Synchronization, in particular, is one
Kingof the most important features that can be described andexplored with the help of oscillators, because, upon theirGeneral
coupling, strongly correlated rhythms among the oscillators
emerge, called synchronized states. Therefore, many reKnight
searchers have proposed different coupled oscillatory networks and have discovered many interesting synchronization
phenomena [1].
Pawn (Soldier)
Recently, synchronization in complex networks with different types of interactions has been extensively investigatedFigure 1: Hierarchical Network model.
for understanding important role played by the interactions.
This is because interactions in networks leads to the emerFigure 2 shows the proposed network model called comgence of key synchronization phenomena, especially compet- petitive interaction network. There are two hierarchical netitive coupling can be observed in real world networks. In our works (Group A and Group B) including multi-direction conresearch group, we have investigated synchronization phe- nections. Each network consists of 11 chaotic circuits andnomena in complex networks by using analog electrical oscil- the adjacent chaotic circuits are coupled by resistors. The
lators [2]. Giron et al. investigated synchronization phenom- coupling strength in the group and between the groups areena in plant communities where usually both facilitation and summarized in Table 1.
competition coexist, playing a key role in the structure and orWe define competitive coupling strategy between two hierganization of these communities. They showed that synchro- archical networks. The lowest five chaotic circuits (A7 - A11)
nization provides an efficient way to unveil how species shar- in Group-A and (B7 - B11) in Group-B attack to in front ofing facilitative interactions group of the plant community [3]. the chaotic circuits. We call this competitive coupling ”disHowever, a node in a complex network is expressed by tributed attack”.
a mathematical model in most studies of synchronization of
In the proposed network, each node is expressed by thecomplex networks with competitive interactions. Although, chaotic circuit. Figure 3 shows the chaotic circuit which is- 719 -A11

B7A10

B8A6B4A3A1

B2A5A9B9A8B10A2

B3A4

B6A7

B11

Group_A
coupling strength:

δa_up
δa_downδa_equal

B1

B5Figure 3: Chaotic circuit.

Group_B
coupling strength:

δab
δba

coupling strength:

δb_up
δb_downδb_equal

Figure 2: Competitive interaction network.
(a) Group-A.
Table 1: Definition of coupling strength.
GroupPlace
Name
From higher to lower layer
δa−up
In Group-A
From lower to higher layer δa−downSame layerδa−equal
From higher to lower layerδb−up
In Group-B
From lower to higher layer δb−downSame layerδb−equal
BetweenFrom Group-A to Group-B
δabGroup-A and B From Group-B to Group-A
δbaFigure 4: Attractors (α = 0.4, β = 20.0, γ = 0.5).

dxn


= zn
dτ



Xdyn(yn − yk )= αγyn − αf (yn − zn ) − αδdτk∈Sn
 dz n = f (yn − zn ) − xn .
dτwherethree-dimensional autonomous circuit proposed by Shinrikiet al. in Ref. [4]. This circuit is composed by an inductor, a negative resistance, two condensers and dual-directional
diodes. The circuit generates asymmetric attractor as shownin Fig. 4. In this simulation, the attractor of Fig. 4 (a) is used
to Group-A, and the attractor of Fig. 4 (b) is used to Group-B.
Next, we develop the expression for the circuit equations of
the circuit model as shown in Fig. 3. The i − v characteristicsof the nonlinear resistor are approximated by the following
three-segment piecewise-linear function,

 Gd (v1n − v2n − V ) (v1n − v2n > V )idn =0(|v1n − v2n | ≤ V )Gd (v1n − v2n + V ) (v1n − v2n < −V )(1)(b) Group-B.

(2)r

C2V xn ,L
v1n = V yn , v2n = V zn ,rC2L
α=, β=Gd ,C1C2rrL
1Lg, δ =.
γ=C2
R C2
where n = 1, 2, 3, ..., 22 and Sn is set to nodes which areconnected to chaotic circuits. The nonlinear function f () corresponds to the i − v characteristics of the nonlinear resistors
consisting of the diodes and are described as follows: β(yn − zn − 1) (yn − zn > 1)0(|yn − zn | ≤ 1) (3)f (yn − zn ) =β(yn − zn + 1) (yn − znn < −1)t=pLC2 τ , in =For the computer simulations, we calculate Eq. (2) using
The normalized circuit equations governing the circuit are the fourth-order Runge-Kutta method with the step size h =expressed as follows.
0.005.

- 720 -3. Synchronization Phenomena

A11B7A10B8A6

3.1 Dependency on relationship of coupling strength

B4A3

As first simulation, we focus on the relationship of multidirected coupling strength by using same network structure.
Table 2 summarizes the setting of coupling strength values.
In Group-A, the coupling strength from higher layer to lower
layer has larger value than other direction coupling strength.
While in Group-B, the both direction couplings have samevalue. We investigate synchronization state by changing thecoupling strength between two groups.

A1B2A5A9B9A8B10B1B5A2B3A4B6A7B11Group_A

Group_B(a) δab = δba =0.1.

Table 2: Setting of coupling strength.
Coupling strengthValue
δa−up
0.5
In Group-Aδa−down0.1
δa−equal
0.3
δb−up
0.3
In Group-B
δb−down0.3
δb−equal
0.3
BetweenδabvariabilityGroup-A and B
δbavariability

A11

B7

A10

B8A6Group

B4A3

B2

A5A1

A9

B9A8

B10B1

B5A2

B3
A4

B6

A7

B11Group_A

Group_B(b) δab = δba =0.2.

Figure 5 shows the obtained attractors from the computer
simulations. When the coupling strength δab = δba is set
to 0.1 and 0.2, the amplitude of chaotic attractor located atlowest layer becomes small than the original attractor. In thecase of δab = δba =0.3, all chaotic circuits oscillate at upperside as shown in Fig. 5 (c). Namely, Group-A wins against
Group-B. We confirm that if the network structure of twogroups is same, the coupling strength from higher to lowerlayers should have larger value than other direction coupling
for wining at the battle.

A11B7A10B8A6B4A3A1B2A5A9B9A8B10B1

B5

A2

B3A4

B6A7

B11

Group_A

Group_B(c) δab = δba =0.3.

Figure 5: Attractors.
3.2 Dependency on network structure
Next, we consider dependency on network structure. In
this simulation, the relationship of the coupling strength issame in two groups. The network structure of two group has
different connection pattern. One connection is rewired inGroup-B. There are two types of rewiring connection.
Figure 6 shows the competitive interaction network whichGroup-B has the connection between top layer to third layer(1 layer skip connection). Figure 7 shows the competitive
interaction network which Group-B has the connection between top layer to bottom layer (2 layers skip connection).
For the simulations, the coupling strength in each group andbetween groups are changed. We focus on wining or losingof the networks.
The summarized simulation results of competitive interaction network Fig. 6 is shown in Fig. 8. When the couplingstrength from higher layer to lower layer has larger value thanthe other direction, the both networks can not win. In thecase of δa−up =0.34 and 0.32, wining group depends on thecoupling strength of between two groups. When the couplingstrength δab and δba is small, we obtain draw. By increasingδab and δba , we confirm Group-A wins at certain range. Afterthat Group-B wins with large value of δab and δba .
The summarized simulation results of competitive interaction network Fig. 7 is shown in Fig. 9. In this case, we obtain
the only two results; ”draw” and ”Group-B wind”. We can
not observe the situation that Group-A wins. From these results, we confirm that the network structure especially layerskipping connection has important role for wining at the battle.

- 721 -Figure 8: Wining or losing results (Group-B: 1 layer skip connection).

Figure 9: Wining or losing results (Group-B: 2 layers skip connection).
A11

B7A10

B8A6A9B9A8

B10A1

B1

B5A2
B6A7Group_A

B8B4
B2A5A9B9A8

B10B1B5A2

B3
A4A10A3B2A5B7A6

B4A3A1A11

1 layer skip connection

B3A4B6A7

B11Group_AGroup_B

2 layers skip connection

B11Group_BFigure 6: Competitive interaction network (Group-B: 1 layer Figure 7: Competitive interaction network (Group-B: 2 layers
skip connection).
skip connection).
References

4. Conclusions
In this study, we have investigated wining or losing observed in two networks of chaotic circuits which are coupledin multi-direction hierarchically. We confirmed that the network structure especially layer skipping connection has important role for wining at the battle.
In our future works, we would like to investigate effect ofcompetitive strategies and apply this model to more complexnetworks such as smart grid network and social network.

Acknowledgment
This work was partly supported by JSPS Grant-in-Aid for
Scientific Research 16K06357.

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Gomez-Gardenes, “Synchronization Unveils the Organizationof Ecological Network with Positive and Negative Interactions” Chaos, no. 26, 065302, 2016.
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