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

Influence of Coupling Strength on Synchronizationin Two Rings of Coupled van der Pol Oscillators
Daiki Nariai, Yoko Uwate and Yoshifumi Nishio
Tokushima University, Tokushima 770-8501, Japan* E-mail: {nariai, uwate, nishio}@tokushima-u.ac.jp
1. Introduction
In this study, we propose to the coupled oscillatory system. By computer simulations and circuit experiments,
we investigate synchronization phenomena observed in the proposed circuit system by changing one of thecoupling strength.
2. System modelFigure 1 shows two circuits and a model of thesystem. The circuit of a van der Pol oscillator is calledcircuit VDP. The circuit with a slightly modified VDP
structure is called NC. We use six van der Pol oscillatorsin this study. We use two ring circuits of van der Pol
oscillators. The three VDP of the first ring are connectedby resistors. The three NC of the second ring areconnected by inductors and resistors. When the two rings
are not connected, the oscillators in the first ring exhibit
in-phase synchronization and oscillators of the secondring exhibit three-phase synchronization. The first andFig1 Circuit model.
second rings are connected by resistors (R1, R2, R3). We
observe the synchronization phenomena of adjacentoscillators. We investigate how the synchronization phenomena change upon changing the value of R2.
A nonlinear resistor defined as follows:
𝑖𝑔𝑛 = −𝑔1𝑣𝑛 + 𝑔3 𝑣𝑛3 .
the normalized equations of the first ring are given as:𝑥̇ = 𝜀(1 − 𝑥𝑛2 )𝑥𝑛 − 𝑦𝑛 + 𝛼(−2𝑥𝑛 + 𝑥𝑖 + 𝑥𝑗 ) − 𝛾𝑛 (𝑥𝑛 − 𝑥𝑛+3 ){ 𝑛(𝑛 = 1,2,3)
𝑦̇𝑛 = 𝑥𝑛
and the normalized equations of the secondo ring are given as:
𝑥̇ 𝑛 = 𝜀(1 − 𝑥𝑛2 )𝑥𝑛 − 𝑦𝑎𝑛 − 𝑦𝑏𝑛 − 𝛾𝑛 (𝑥𝑛 − 𝑥𝑛−3 ){𝑦̇𝑎𝑛 = (𝑥𝑛 − 𝛽(𝑦𝑎𝑛 + 𝑦𝑏𝑖 ))

(1)
(2)

(3)

(𝑛 = 4,5,6)
𝑦̇ 𝑏𝑛 = (𝑥𝑛 − 𝛽(𝑦𝑏𝑛 + 𝑦𝑎𝑗 ))where n, i and j denote the number of the circuit (n = 1,2,3,4,5,6, i = 2,3,1,5,6,4 and j = 3,1,2,6,4,5). The
parameters ε is non-linear strength. The parameters α, β and γ denote the coupling strengths of the resistor R,resistor R′ and resistor Rn, respectively.
3. Results
The simulation results of the system model are
shown from Fig. 2 to Fig. 4. The value of theparameters is set to ε = 0.05, α = 0.05, β = 0.05.
Figures 2 - 4 show the graph of the relationshipbetween coupling strength and the phasedifference. We change the value of γ2 from 0.000
to 0.030 at intervals of 0.001. In Fig. 2, we fix thevalue of other coupling strengths and initial value.
The values of other coupling strengths are set to γ1= γ3 = 0.015. In Fig.3, The values of othercoupling strengths are set to γ1 = γ3 = 0.020. In
Fig. 4, the values of other coupling strengths areset to γ1 = γ3 = 0.025.
In Fig. 2, when γ2 is even smaller than 0.015,

4th International Forum on Advanced Technologies
March 8th-9 th, 2018,Tokushima, Japan
circuit 4 - circuit 6 exhibit in-phase synchronization.
When γ2 is from 0.021 to 0.03, the phase differenceof circuit 4 - circuit 5 is 85∘. The phase difference of
circuit 5 - circuit 6 is 65∘. The phase difference of
circuit 6 - circuit 4 is 155∘ . This result shows thephase difference changes depending on whether thevalue of γ2 is larger or smaller than 0.015.
In Fig. 3, when γ2 is even smaller than 0.020,
circuit 4 circuit 6 exhibit in-phase synchronization.
When γ2 is from 0.020 to 0.03, the phase differenceof circuit 4 - circuit 5 is 60∘. The phase difference of
circuit 5 - circuit 6 is 55∘. The phase difference of
circuit 6 - circuit 4 is 120∘ . This result shows thephase difference changes depending on whether thevalue of γ2 is larger or smaller than 0.020.
In Fig. 4, when γ2 is even smaller than 0.025,
circuit 4 circuit 6 exhibit in-phase synchronization.
When γ2 is from 0.025 to 0.03, the phase differenceof circuit 4 - circuit 5 is 30∘. The phase differenceof circuit 5 - circuit 6 is 25∘. The phase differenceof circuit 6 - circuit 4 is 55∘. This result shows thephase difference changes depending on whether thevalue of γ2 is larger or smaller than 0.025.
Figure 5 shows the time waveform of oscillatorsof the first ring and the second ring. The values ofcoupling strengths are set to γ1 =γ3 = 0.020 and γ2 =0.030. The oscillators of the first ring exhibit inphase synchronization and the oscillators of thesecond ring exhibit three-phase synchronization
gravitate toward in-phase synchronization.
From these results, we consider the synchronization
phenomena change depending on whether the value ofγ2 is larger or smaller than the values of γ1 and γ3.
When the value of γ2 is smaller than the values of γ1and γ3, circuits exhibit in-phase synchronization.
When the value of γ2 is larger than the values of γ1and γ3, the oscillators of the first ring exhibit in-phasesynchronization and the oscillators of the second ringexhibit three-phase synchronization gravitate toward
in-phase synchronization. When the value of γ2 islarger than the values of γ1 and γ3, the phasedifference of the oscillators of the second ring
becomes smaller by increasing γ1 and γ3. We canobtain new synchronization which is neither in-phasesynchronization nor three-phase synchronization.

𝒙𝟏,𝟐,𝟑,𝒙𝟒,𝟓,𝟔,4. Conclusion
We have proposed a system model using two rings of coupled three van der Pol oscillators coupled byresistors or inductors. We investigate how the synchronization changes upon changing the value of γ2. Weobserved various synchronization phenomena by varying the coupling strengths. We found two factors that
determine the phase difference of the second ring. The first factor is the synchronization phenomena change
depending on whether the value of γ2 is larger or smaller than the values of γ1 and γ3. The second factor is thevalues of the γ1 and γ3 decide the phase difference of the oscillators of the second ring. When the value of γ2 islarger than the values of γ1 and γ3, the oscillators of the first ring exhibit in-phase synchronization and theoscillators of the second ring exhibit three-phase synchronization gravitate toward in-phase synchronization.

4th International Forum on Advanced Technologies
March 8th-9 th, 2018,Tokushima, Japan
Acknowledgment
This work was partly supported by JSPS Grant-in-Aid for Scientific Research 16K06357.

References[1] Y. Uwate, Y. Nishio, “Two van der Pol Oscillators Coupled by Chaotically Varying Resistor”, Proceedingsof International Workshop on Nonlinear Dynamics of Electronic Systems (NDES’06), pp. 189-192, Jun.
2006.
[2] M. A. Barro´n, M. Sen, “Synchronization of four coupled van der Pol oscillators”,Nonlinear Dynamics, pp.
357-367, June 2009.
[3] K. Matsumura, T. Nagai, Y. Uwate, Y. Nishio, “Analysis of Synchronization Phenomenon in Coupled Oscillator Chains”, Proceedings of IEEE International Symposium on Circuits and Systems (ISCAS’12), pp.
620623, May 2012.
[4] K. Oi, K. Ago, Y. Uwate, Y. Nishio“Effect of the Hub in Complex Networks of Coupled Parametrically Excited Oscillators with Dispersion”, Proceedings of IEEE Workshop on Nonlinear Circuit Networks(NCN’15), pp. 11-14, Dec. 2015.

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