2018 International Symposium on Nonlinear Theory and Its Applications,NOLTA2018, Tarragona, Spain, September 2-6, 2018

Synchronization of Coupled van der Pol Oscillators with Distributed Hub Nodes
Shuhei Hashimoto, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: {s-hashimoto, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract
In this study, we focus on network structure with a hub whichis consisted of Scale-free network. We consider the synchronization of coupled van der Pol oscillator network whose hubnodes are split two nodes. We investigate synchronization
phenomena of network which a hub node replace by two distributed nodes connected by a negative resistor. Furthermore,
we compare synchronization phenomena of this network ofcoupled van der Pol oscillators and chaotic circuits.

1. Introduction
Synchronization phenomena can be observed everywherein 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 areinteresting. In addition, complex networks attract attention
from various fields. The feature of networks is characterized
by the degree distribution, the path length and the clustering
coefficient.
Numerous researches on complex networks have been
made so far [1]-[7]. Especially, there are many studies 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 the networks.
We pay our attention on the behavior of the networks
whose hub nodes are split into several nodes. Namely, whensuch a hub node is split into several nodes, how other nodes
connected to the hub node change their behavior? Furthermore, when such split hub nodes agree with or oppose to eachother, how the whole network behaves to cooperate or to compete? In our previous study, we investigated synchronization
phenomena of complex networks by using coupled chaotic
circuits and evaluate it by using synchronization rate between
a pair of the chaotic circuits [8]. We used the model which wereplace a hub node with two distributed hub nodes connectedby a negative resistor in previous study. In this study, weinvestigate synchronization phenomena by using same net-work model and change the circuit from chaotic circuit tovan der Pol oscillator. That is because chaotic circuit behave
very complex, so we need to research with a simpler oscillator. Furthermore, we compare synchronization phenomena ofcoupled van der Pol oscillators and chaotic circuits.

2. Circuit model
Figure 1 shows van der Pol oscillator. This circuit consists of a nonlinear resister, an inductor and a capacitors. In
this study, we use a network in which each node is replaced
by a van der Pol oscillator. The circuit equation is shown inEq. (1).

Figure 1: Circuit model.

diLn
L= vndtdvCn1 X
(vn − vk ) − ig n − iLn C1 dt = R

(1)

k∈Sn

where n, k = 1, 2, 3, · · · and Sn is the set of nodes which aredirectly connected to the node n. Nonlinear resistor defined
as follows:- 542 -ign = −g1 vn + g3 v3 3 .

(2)By changing the variables and parameters,srrgg1 CL1
 vn =xn , iLn =yn,  = g1,3g3
3g3 LCr√
1 Ld
 δ=, t = LCτ, ” · ” =R C
dt

(3)

the normalized equations of this circuit are given as follows:X ẋn = (xn − xn 3 ) − yn − δ(xn − xk )
k∈Sn



(4)

ẏn = xn .

The parameter δ corresponds the coupling strength between the circuits. The parameter  is the strength of nonlinear.

Figure 2: Basic network model.
3. Network model
We made the underlying network which is 10 van der Pol
oscillators and each circuit connected by a resistor. This network model and the degree distribution of this network which
imitate Scale-free network are shown in Fig. 2.
In this study, we proposed network model which a hub
node replaced with two distributed nodes. The example ofsplitting hub is shown in Fig. 3. Right network model in Fig. 4
is the network that applied method shown in Fig. 3 to VDP1
as a hub of the basic network. Further, distributed hub nodes
are connected by a negative resistor and other nodes are connected by a positive resistor each other.

4. Simulation Results
In this section, we show the synchronization phenomenaof coupled chaotic circuits, which is a previous study, andthat of coupled van der Pol oscillators of this study. Further,
the coupling strength other than between distributed nodes
is defined as δ and the coupling strength between distributed
nodes is defined as δ 0 .

Figure 3: Splitting a node.
4.1 Synchronization phenomena of coupled chaotic circuits in previous study [8]In previous study, we used chaotic circuit which proposedby Shinriki et al. [9][10]. The chaotic circuit is shown inFig. 5.
And Fig. 6 shows synchronization phenomena of previous
study. We evaluate synchronization by phase difference due
to compare previous study and this study. In Fig. 6, the coupling strength δ is 0.1 and the coupling strength δ 0 is −0.2.
We can confirm that the couplings which connected to distributed hub nodes are asynchronous and the coupling between distributed hub nodes is asynchronous. Furthermore,
the couplings between nodes which do not connect to distributed hub node seems asynchronous. However, from previous study, the couplings between nodes which do not connect
to distributed hub node are rather synchronous by using syn-- 543 -Figure 4: Network model with distributed hub nodes.
Figure 7: Voltage difference of network coupled chaotic circuits in previous study.
chronization rate. Figure 7 shows voltage difference of previous study. Figure 7(a) corresponds the couplings between
nodes which do not connect to distributed hub node.

Figure 5: Mori-Shinriki circuit.

Figure 8 shows the result of this simulation. In Fig. 8, thecoupling strength δ is 0.0002 and the coupling strength δ 0
is −0.02. We can confirm that couplings which connected
to distributed hub nodes are asynchronous. Furthermore, thecouplings between nodes which do not connect to distributed
hub node are almost synchronous and the coupling between
distributed hub nodes is anti-phase synchronization. This result is similar to the result of section 4.1.
From the result of section 4.1 and 4.2, we summarize therekationship between synchronization phenomena and network model. Figure 9 shows network model visualizing
synchronization phenomena. We can confirm that network
is classified three groups which are synchronous, anti-phasesynchronous and asynchronous.

Figure 6: Phase difference in previous study network.
Figure 8: Phase difference of coupled van der Pol oscillators.
4.2 Synchronization phenomena of coupled van der Poloscillators
In this section, we observed synchronization phenomenaof coupled van der Pol oscillators. As mentioned above,
we evaluate synchronization phenomena by phase difference.

5. Conclusion
In this study, we have investigated synchronization phe-- 544 -ternational Workshop on Nonlinear Circuits, Communications and Signal Processing (NCSP’14), pp. 417-420,Mar. 2014.
[5] K. Ago, Y. Uwate, Y. Nishio, “Investigation of Partical Synchronization in Coupled Chaotic Circuit Networkwith Local Bridge”, Proc. of IEEE Workshop on Nonlinear Cirsuit Networks (NCN’14), pp.63-66, Dec. 2014.

Figure 9: Network model visualizing synchronization phenomena.
nomena of coupled van der Pol oscillators. We used network whose hub node is replaced with two distributed hub
nodes and distributed hub nodes are connected negative resistor. Furthermore, we compared synchronization phenomenaof coupled van der Pol oscillators and that of coupled chaotic
circuits which is already researched in previous study.
In the result of synchronization phenomana of the coupled van der Pol oscillators, we can observe synchronization phenomena changes due to differences in the structureof the coupling. At first, the coupling between distributed
hub nodes which connected by negative resistor is anti-phasesynchronization. Then the couplings which connected to distributed hub nodes are asynchronous and the couplings between nodes which do not connect to distributed hub node
are almost synchronous. This result is similar to the resultof coupled chaotic circuits of our previous study. However,
it is necessary to investigate more detailed conditions of thecoupling strength and initial value.
Ackowledgment[6] T. Chikazawa, Y. Uwate, Y. Nishio, “Chaos Propagation in Coupled Chaotic Circuits with Multi-RingCombination”, Proc. of IEEE Asia Pacific Conferenceon Circuits and Systems (APCCAS’16), pp. 65-68, Oct.
2016.
[7] S. Hashimoto, T. Chikazawa, Y. Uwate, Y. Nishio,“Synchronization Phenomena in Complex Networksof Coupled Chaotic Circuits with Different DegreeDistribution”, Proc. of RISP International Workshop on
Nonlinear Circuits, Communications and Signal Processing (NCSP’17), pp. 333-336, Feb. 2017.
[8] S. Hashimoto, Y. Uwate, Y. Nishio, “Synchronization
Phenomena of Chaotic Circuit Networks with Distributed
Hub Including Positive and Negative Coupling”, Proceedings of IEEE International Symposium on Circuitsand Systems (ISCAS’18), May 2018.
[9] M. Shinriki, M. Yamamoto and A. Mori, “Multimode Oscillations in a Modified van der Pol Oscillator Conraining
a Positive Nonlinear Conductance”, Proc. IEEE, vol. 69,
pp. 394-395, 1981.
[10] N. Inaba, T. Saito and S. Mori, “Chaotic Phenomena ina Curcuit with a Negative Resistance and an Ideal Switchof Diodes”, Trans. of IEICE, vol.E70, no. 8, pp. 744-754,
1987.

This work was partly supported by collaborative research
project with Hitachi.
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