4th International Forum on Advanced Technologies
March 8th-9 th, 2018,Tokushima, Japan

Synchronization of Coupled Chaotic Circuit Networkwith Positive and Negative Coupling
Shuhei Hashimoto , Yoko Uwate and Yoshifumi Nishio
Tokushima University, Tokushima 770-8501, Japan* E-mail: {s-hashimoto, uwate, nishio}@tokushima-u.ac.jp
1. Introduction
Synchronization phenomena can be observed everywhere in our life. For example, we can conﬁrm
metronome, ﬂashing ﬁreﬂy lights, beating rhythm of the heart and so on. Especially, synchronization
phenomena of oscillatory network are interesting. In addition, complex networks attract attention from variousﬁelds. Numerous researches on complex networks have been made so far [1]-[7]. Especially, there are manystudies related with hub nodes and these are very interesting. Because, in the real world, large-scale complex
networks often include huge nodes with a large number of edges, and such “hub” nodes play key roles in thenetworks.
In this study, we focus on network structure with a hub which is consisted of Scale-free network. We consider
the synchronization of coupled chaotic circuit network whose hub nodes are split two nodes and we investigate
two network models. In one of them, we replace a hub node with two distributed nodes connected by a positiveresistor. In another, we replace a hub node with two distributed nodes connected by a negative resistor. Fromthe simulation results, we can conﬁrm that connection between hubs plays an important role in network
structure.
2. System model
The chaotic circuit model which proposed by Shinriki et al. [8][9] is shown in Fig. 1. This circuit consists of anegative resistor, an inductor, two capacitors and dual-directional diodes. In this study, we use a network inwhich each node is replaced by a chaotic circuit. The normalized equation of this chaotic circuits is shown in
Eq. (1).

Fig. 1. Circuit model.

Fig.

2.

Attractors

withthedifferentparameter.

(1)

(2)

Equation (2) shows parameters of this circuit and Fig. 2 shows chaotic attractor obtained from this circuit.
The parameter δ corresponds the coupling strength between the circuits. The parameter α is the strength ofnegative resistor.
In this study, we propose network model which a hub node replaced with two distributed nodes. The exampleof splitting hub is shown in Fig. 4. Right network model in Fig. 5 is the network that applied method shown inFig. 4 to CC1 as a hub of the basic network. Furthermore, we investigated synchronization of network in two
different ways of connecting. In the ﬁrst model, distributed hubs are connected by a positive resister and othernodes are connected by a positive resistor each other. In the second model, distributed hubs are connected by anegative resistor and other nodes are same at the ﬁrst model.

4th International Forum on Advanced Technologies
March 8th-9 th, 2018,Tokushima, Japan

Fig. 4. Splitting a node.

Fig. 5. Network model with distributed hub.

3. Simulation result
Deﬁnition of synchronization in this study is determined a voltage difference waveform. We deﬁne
synchronization as the following Eq. (3).

(3)
Fig. 6. A difference waveform
Figure 6 is a different waveform which was observed in this simulation. The two lines in Fig. 6 correspondthreshold which is given Eq. (3). It is determined that wave within two lines which is the threshold issynchronization. Therefore, we propose and investigate the synchronization probability denoted thesynchronization rate during a certain time interval. In this research, we ﬁx a certain time interval as(τ=1,000,000 and step=0.02τ) and investigate the synchronization rate in the entire network of 10 coupledchaotic circuits.
3-1. Comparing network with distributed hub and basic network
In this simulation, we use two bifurcation parameters of chaotic circuit γ. Attractors of these parameters areshown in Fig. 2. We compared synchronization rate of basic network and network with distributed hub nodes
which connected by positive resistor. Where the coupling strength other than between distributed hub nodes isdeﬁned as δ and the coupling strength between distributed nodes is deﬁned as δ′. In basic network, wemeasured synchronization rate by changing the coupling strength δ = 0.01 to δ = 0.1. In network with
distributed hub, we measured synchronization rate by changing the coupling strength δ = 0.01 to δ = 0.01 andthe coupling strength δ′ = 0.02 to δ′ = 0.2. Then we calculated difference of synchronization rate of basicnetwork and network with distributed hub nodes in each coupling strength δ. Figure 7 shows average ofsynchronization rate of the coupling strength δ′ in two parameters γ = 0.5 and γ = 0.65. In Fig. 7, we canobserved that synchronization rate is almost unchanged, even if a hub is replaced with two distributed hubnodes.

Fig. 7. Average of difference in synchronization rate.

Fig. 8. Synchronization rate of each group.

4th International Forum on Advanced Technologies
March 8th-9 th, 2018,Tokushima, Japan
3-2. Synchronization Phenomena of network with distributed hub which connected bynegative resistor
In this section, we observe synchronization phenomena of network with distributed hub nodes whichconnected by negative resistor. We measure synchronization rate the network by changing the couplingstrength δ′= −0.02 to δ′= −0.2 in the coupling strength δ = 0.1. Then, we grouped edges that are similar inconnection and we averaged synchronization rate for each group. The ﬁrst group is edges connected to CC1
which is distributed hub. The second group is edges connected CC11 which is distributed hub. The third groupis edges between the nodes connected to CC1 of distributed hub. The fourth group is edges between the nodes
connected to CC11 of distributed hub. The ﬁfth group is edges between node connected to CC1 and nodeconnected to CC11, that is, connection between nodes which are coupled with nodes that are in asynchronousrelationship. Figure 8 shows averages of synchronization rate of each group in the coupling strength δ′=−0.02 to δ′= −0.2. In Fig. 8, we can observed that synchronization rates of the ﬁrst group and the secondgroup are similar and that of the third group and the fourth group are similar. Synchronization rates of the ﬁrst
group and the second group are lower as the negative coupling is stronger. Synchronization rates of the thirdgroup and the fourth group are almost unchanged even if the negative coupling changed. Synchronization ratesof the ﬁfth group are lower as the negative coupling is stronger, although how to coupling of ﬁfth group issimilar to the third group and the fourth group. This results are considered that ﬁfth group is connections
between nodes which are coupled with nodes that are in asynchronous relationship.
4. Conclusion
In this study, we have investigated synchronization phenomena of network whose hub node is replaced withtwo distributed hub nodes. Furthermore, we connected distributed hub nodes in two ways. In the ﬁrst model,distributed hub nodes connected by a positive resistor. In the second model, distributed hub nodes connectedby a negative resistor. In the result of the ﬁrst model, when we compared synchronization rate of undelying
network and network with distributed hub, we can observed that synchronization rate is almost unchanged,
even if a hub is replaced with two distributed hub nodes. In the simulation of the second model, we compare
edges grouping similar connection. From the simulation result, synchronization rates of group which isconnection between nodes which are coupled with nodes that are in asynchronous relationship are lower as thenegative coupling is stronger.
Acknowlegement
This work was partly supported by JSPS Grant-in-Aid for Scientific Research 16K06357.
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