2017 International Symposium on Nonlinear Theory and Its Applications,NOLTA2017, Cancun, Mexico, December 4-7, 2017

Modeling of Competitive 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—Coupled oscillatory circuits provide simple
models for high-dimensional nonlinear real-world phenomena. Synchronization, in particular, is one of themost important features that can be described and exploredwith the help of oscillators, because, upon their coupling,strongly correlated rhythms emerge among the oscillators,
called synchronized states. Here, we present results obtained for a new paradigm of coupled oscillatory networks,
called competitive interaction networks.

in two networks of chaotic circuits which are coupled in
one direction hierarchically. We analyze the role of synchronization by changing the competitive coupling strategies. A schematic competitive network is shown in Fig. 1.

1. Introduction
Coupled oscillatory circuits provide simple models fordescribing high-dimensional nonlinear phenomena occurring in our everyday world. Synchronization, in particular, is one of the most important features that can be described and explored with the help of oscillators, because,
upon their coupling, strongly correlated rhythms among theoscillators emerge, called synchronized states. Therefore,
many researchers have proposed different coupled oscillatory networks and have discovered many interesting synchronization phenomena [1], [2].
Recently, synchronization in complex networks with different types of interactions has been extensively investigated for understanding important role played by the interactions. This is because interactions in networks leads
to the emergence of key synchronization phenomena, especially competitive coupling can be observed in real world
networks. In our research group, we have investigatedsynchronization phenomena in complex networks by using analog electrical oscillators [3]-[6]. Giron et al. investigated synchronization phenomena in plant communities where usually both facilitation and competition coexist,
playing a key role in the structure and organization of thesecommunities. They showed that synchronization provides
an efficient way to unveil how species sharing facilitativeinteractions group of the plant community [7].
However, a node in a complex network is expressed bya mathematical model in most studies of synchronizationof complex networks with competitive interactions. Although, it is very important to use mathematical model
for complex networks in order to understand the synchronization states by approaching theoretical methods, we alsoneed to consider physical models for future engineering applications.
In this study, we focus on synchronization state observed

Figure 1: A schematic competitive network.

2. Network Model
Figure 2 shows the proposed network model of thisstudy. There are two hierarchical networks (Group A andGroup B) including the directed and undirected connections. Each network consists of 11 chaotic circuits and theadjacent chaotic circuits are coupled by resistors. The coupling strength in the group is defined as δin and the couplingstrength between the groups is defined as δout . The coupling methods of the directed and undirected connections
are shown in Fig. 3.

Figure 2: Network model.
We define two types of competitive coupling between
two hierarchical networks. One is the concentrated attack
which the lowest five chaotic circuits (A7 - A11) in GroupA attack to only one chaotic circuit (B9) in Group B. Another one is the distributed attack which the lowest five- 600 -chaotic circuits (A7 - A11) in Group B attack to in frontof the chaotic circuits (B7 - B11) one by one in Group B.

 dxn



= zn


dτ



X
 dyn(yn − yk )=αγy−αf(y−z)
−αδ

nnn

dτ

k∈S
n

dzn
= f (yn − zn ) − xn .
dτFigure 3: Coupling method.

(2)

whereIn the proposed network, each node is expressed by thechaotic circuit. Figure 4 shows the chaotic circuit which isthree-dimensional autonomous circuit proposed by Shinriki et al. in Refs. [8], [9]. This circuit is composed by aninductor, a negative resistance, two condensers and dualdirectional diodes. The circuit generates asymmetric attractor as shown in Fig. 5. In this simulation, the attractorof Fig. 5 (a) is used to Group-A, and the attractor of Fig. 5(b) is used to Group-B.

rC2
V xn , v1n = Vyn , v2n = Vzn ,Lrrr
LLLC2
1
α=, β=Gd , γ =g, δ =.
C1C2C2R C2t=

pLC2 τ, in =where n = 1, 2, 3, ..., 22 and S n 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)f (yn − zn ) = (3)
 β(y − z + 1) (y − z < −1)
nnnnn

For the computer simulations, we calculate Eq. (2) using
the fourth-order Runge-Kutta method with the step size h =0.005.
3. Synchronization Phenomena

Figure 4: Chaotic circuit.

3.1. Attractors(a) Group-A.

(b) Group-B.

Figure 5: Attractors (α = 0.4, β = 20.0, γ = 0.5).
Next, we develop the expression for the circuit equationsof the circuit model as shown in Fig. 4. The i − v characteristics of the nonlinear resistor are approximated by thefollowing three-segment piecewise-linear function,

Figure 6 shows the simulation results of the obtained attractors by changing the coupling strength δout . When thecoupling strength δout is set to 0.1, the attractor of the chaoscircuit (B9) in Group-B which is attacked from Group-B
is hardly damaged. And, the attractors of the chaos circuits (A7-A11) located lowest layer in Group-A become
bit small from original attractor.
In the case of δout = 0.3, the chaos circuits (A7-A11)
located lowest layer in Group-A are all damaged. And, thechaos circuits (A6, B6) located in second lowest layer inGroup-A and B are also damaged.
As increasing the coupling strength, the chaos circuits
located in lowest and second lowest layer in Group-A and
B are damaged. However, the chaos circuits (B8, B10) located in lowest layer in Group-B is recovered thier attractor.
3.2. Synchronization Ratio in Each Group
G (v − v2n − V) (v1n − v2n > V)
 d 1n0(|v1n − v2n | ≤ V)
idn = 
 G (v − v + V) (v − v < −V)
d1n2n1n(1)2nIn this section, we fix a certain time interval as τ =100, 000 and step size h = 0.005. We calculate the synchronization states observed from 22 coupled chaotic circuits by using the following equation,

The normalized circuit equations governing the circuit
are expressed as follows.

- 601 -|yn − yk | = 0.01(k ∈ S n ).

(4)

(a) δ=0.1.

Figure 7: Synchronization ratio.

3.3. Synchronization Ratio in Each Edge
Finally, we investigate synchronization ratio in eachedge. The definition of edge number is shown in Fig. 8.
Figure 9 shows the simulation results with the competitive
coupling strength δout . When δout is smaller than 0.1, there
is big difference of synchronization ratio between Group-Aand B around e12 to e19 which are located the lowest andsecond lowest layers.

(b) δ=0.3.

(c) δ=0.5.

Figure 8: Definition of edge number.

Figure 6: Attractors.

Figure 7 shows the coupling strength dependency ofglobal synchronization in each group. The horizontal axis
denotes the coupling strength δout and the vertical axis denotes the synchronization ratio. The blue line with circle
mark shows the synchronization ratio of Group-A. The redline with cross mark shows the synchronization ratio ofGroup-B. When the competitive coupling strength is small,the synchronization ratio of Group-A is higher than GroupB. The both synchronization ratio decrease with δout , andthe two graphs are crossed when δout is 0.3. After that, thesynchronization ratio of Group-B is bit higher than GroupA.
From this result, we confirm that the concentrated attack is effective when the competitive coupling strength issmall. While, the distributed attack is effective when thecompetitive coupling strength becomes strong.

4. Conclusions
In this study, we have investigated synchronization state
observed in two networks of chaotic circuits which are coupled in one direction hierarchically. We analyze the roleof synchronization by changing the competitive couplingstrategies. By using the computer simulations, we confirm
that the concentrated attack is effective when the competitive coupling strength is small. While, the distributed attack is effective when the competitive coupling strength becomes strong.
In our future works, we would like to investigate effect ofcompetitive strategies and apply this model to more complex networks such as smart grid network and social network.

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Figure 9: Synchronization ratio in each edge.
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