Oscillation Quenching in Coupled van der Pol
Oscillators with Different Frequencies
Yoko Uwate

Yoshifumi Nishio

Dept. of Electrical and Electronic Engineering,
Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan
uwate@ee.tokushima-u.ac.jp

Dept. of Electrical and Electronic Engineering,
Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japannishio@ee.tokushima-u.ac.jp

Abstract—In this study, we investigate oscillation quenchingin coupled van der Pol oscillators with different frequencies. The
network topology is inspired from real brain network. Here, weconsider the network which consists of two modules including
connector and provincial hubs. We confirm oscillation quenchingfrom the proposed system.

shown in Fig. 2. The van der Pol oscillators are coupled by aresistor.

75Keywords—oscillation quenching; coupled oscillatory network

8649I.

INTRODUCTION

3In our research group, we have focused on synchronization
phenomena observed from nonlinear oscillatory networks [1],[2]. This is because the results of synchronization in complex
networks are useful deeper understanding of control methods inpower networks, communication systems and so on. Also, they
can be used as an alternative approach, apart from existing ones,
for describing mode-locking phenomena in biological networks.
Therefore, we consider the schematic brain networks as one ofcomplex biological networks. Because, we would like topropose modeling of synchronization in brain by using coupledelectrical oscillatory circuits, in order to make clear themechanism of functional operation in brain.
Recently, the relationship between structural and functional
network in biological neural network has attracted theirattention from many researchers. Hartelt et al. has discovered
the network topology in the pre-BotC which consists ofdensely connected clusters with rare inter-cluster links. And,the hubs of the network behave quiescent output [3].

131Connector hub
122

11
Provincial hub

Fig. 1. Network model.

irnNR

iCv

(a) van der Pol oscillator.

LR(b) Coupling organization.

Fig. 2. Ocillator and coupling organization.
Next, we develop the expression for the circuit equation of
the network model. The normalized circuit equations governing
the circuit are expressed as

We apply this phenomenon for the coupled oscillatory
systems using electrical oscillator. Namely, one node is set tolower frequency than the others. We also investigate theinfluence of location of oscillator with different frequencies.
II.

10PROPOSED SYSTEM

A. Network Model
A network model composed of 13 nodes is shown in Fig. 1[4]. There are two important hubs in this network, “Connectorhub” and “Provincial hub”. The both hubs are high-degree
nodes. “Connector hub” shows a diverse connectivity byconnecting two sub-networks. “Provincial hub” primarily
connects nodes in the same sub-network. There are twomodules and the node is expressed by van der Pol oscillator as

978-1-5386-7960-9/18/$31.00 ©2018 IEEE(1)In these equations, γis the coupling strength, εdenotes
the nonlinearity of the oscillators. For the computer simulations,
we calculate circuit equations using the fourth-order RungeKutta method with the step size ｈ=0.005. The parameter ofthe nonlinearity of oscillator is fexed as ε=0.1.

180

ISOCC 2018

B. Adding Different Frecuency
In this simulation, one node of the network has lower
frequency than the others. The standard frequency is ω=1.0and the lower frequency is set to ω =0.47. In order toinvestigate influence of location of oscillator with different
frequencies, we focus on several nodes as follows.
Case-1) Connector hub (oscillator no. 6)
Case-2) Provincial hub (oscillator no. 9)
Case-3) Other node (oscillator no. 2)
Case-4) Other node (oscillator no. 10)

III.

OSCILLATION QUENCHING(a) Case-3: Oscillator no. 2.

Figure 4 shows the simulation result of amplitude changewith the coupling strength. From these results, we confirm
amplitude death at certain range. Furthermore, the range ofamplitude death of connector hub is smaller than the provincialhub.

(b) Case-4: Oscillator no. 10.
Fig. 5. Amplitude of other nodes.
IV.

CONCLUSIONSIn this study, we have investigated oscillation quenchingobserved in coupled oscillatory system with different
frequencies. We have confirmed amplitude death at certain
range with the coupling strength from the proposed system.
The range of amplitude death is smallest when the lower
frequency is set to the connector hub. In the future work, wewould like to investigate the relationship between theamplitude death and importance of the nodes.

(a) Case-1: Connector hub (oscillator no. 6).

REFERENCES
[1]

[2]

(b) Case-2: Provincial hub (oscillator no. 9).

[3]

Fig. 4. Amplitude of connector and provincial hubs.
[4]

Next, the simulation results of the amplitude change inCases 3 and 4 are shown in Fig. 5. By increasing the couplingstrength, we also observe the amplitude death in the both cases.
We can see that the range of amplitude death of Cases 3 and 4is longer than Cases 1 and 2.

978-1-5386-7960-9/18/$31.00 ©2018 IEEE

181

Y. Setou, Y. Nishio and A. Ushida, "Synchronization Phenomena in
Resistively Coupled Oscillators with Different Frequencies", IEICE
Transactions on Fundamentals, vol. E79-A, no. 10, pp. 1575-1580, Oct.
1996.
Y. Uwate and Y. Nishio, "Synchronization in Several Types of Coupled
Polygonal Oscillatory Networks", IEEE Transactions on Circuits andSystems I, vol. 59, no. 5, pp. 1042-1050, May 2012.
N. Hartelt, E. Skorova, T. Manzke, M. Suhr, L. Mironova, S. Kugler andS. L. Mironov, “Imaging of Respiratory Network Topology in Living
Brainstem Slices,” Mol. Cell. Neurosci. 37, pp. 425–431, 2008.
E. Bullmore and O. Sporns, “Complex Brain Networks: Graph
Theoretical Analysis of Structural and Functional Systems,” Nature,Reviews, Neuro, vol. 10, pp. 186-198, Mar. 2009.

ISOCC 2018