2013 International Workshop on Nonlinear Circuits,Communications and Signal Processing
NCSP'13, Isrand of Hawaii, Hawaii, USA. March 4-7, 2013

Investigation of Oscillatory Networks Including Two Types of Resistors
Takahiro Nagai, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University,
2-1 Minamijosanjima, Tokushima, 770-8506 Japan
Email: {nagataka, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract
In this study, we investigate synchronization phenomena observed in coupled oscillators by using two types of resistors.
Negative and normal resistors are used as coupling resistors inour circuit model. By using computer simulations and circuit
experiments, we obtain interesting synchronization. Furthermore, we extend four coupled oscillators and compare withprevious study.
1. Introduction

Figure 2: Coupled van der Pol oscillators. (a) Three coupledoscillators. (b) Four coupled oscillators.

There are a lot of synchronization phenomena in this world.
This is one of the nonlinear phenomena that we can often observe by natural beings which does collective actions. For example, firefly luminescence, cry of birds and frogs, applauseof many people, swing of the pendulum and so on. Synchronization phenomena have a feature that the set of small
power can produce very big power by synchronizing at a time.
Therefore studies of synchronization phenomena have been
widely reported not only engineering but also the physicaland biological fields [1]-[6]. Investigation of coupled oscillators attracts attention from many researchers because coupled oscillatory network produce interesting phase synchronization such as the phase propagation wave, clustering andcomplex patterns.
In our investigations, we use several van der Pol oscillators (see Fig. 1). Van der Pol oscillators have been coupledin various form and investigated about their synchronization
phenomena [7], [8]. For example, as shown in Fig. 2(a),the circuit model which coupled three van der Pol oscillators as a ring topology have been found the three phase synchronization (phase shift 120 degrees) by the interaction ofeach oscillator [9]. However, the three-phase synchronization
was always observed stably in that system. In our previousstudy, we have investigated several kinds of interesting synchronization phenomena in coupled oscillatory system which
has stronger frustrations as shown Fig. 2(b) [10]. By computer simulation, we observed that the phase difference between adjacent oscillators changed and the synchronization
was destroyed after the adjacent oscillators synchronize with
anti-phase (phase shift 180 degrees). In other words, this circuit model has the feature such as repeated anti-phase synchronous and asynchronous.
In this study, we investigate synchronization phenomena of
coupled oscillators using different types of resistors partially.
We focus on the amplitude of each oscillator and phase differences between adjacent oscillators by changing couplingstrength. By using averaging method, we compare between
simulation results and theoretical analysis. Also, we increase
number of oscillators and demonstrate the circuit experimentof this oscillatory system.
2. Circuit Model

Figure 1: Van der Pol oscillator.

Figure 3 shows the circuit model. First and second oscil-- 632 -We use the following normalizations:
√√√
g1
g1 C
t = LCτ , vk =x k , ik =yk ,3g3
3g3 L
√ε = g1Figure 3: Conceptual circuit model for three coupled oscillators.

L1, γ1 =Cr√

L1, γ2 =C
R√L,
C

where ε is the nonlinearity, γ1 is the coupling strength of negative resistor r and γ2 is the coupling strength of normal resistor R. In the computer simulations, we calculate the phasedifferences between adjacent oscillators to observe synchronization states.
3. Synchronization Phenomena

lators are coupled by negative resistor. And third and otheroscillators are coupled by normal resistor. In the case of thenegative resistor coupling, the adjacent oscillators are synchronized at anti-phase. While, we use the normal resistor
as the coupling resistor, the adjacent oscillators are synchronized at in-phase.
In the computer simulations, we assume that the vk − iRkcharacteristics of nonlinear resistor in each oscillator is givenby the following third order polynomial equation.
iRk = −g1 vk + g3 vk 3

(g1 , g3 > 0),

We calculate Eqs. (2)-(4) using the fourth-order RungeKutta method with the step size h = 0.001. We show thesimulation result of the synchronization phenomena in Figs. 4
and 5. In Fig. 4, we show the attractor of each oscillator. We
set the parameters as follows ε = 0.02, γ1 = 0.05 and γ2 =0.05. Obvious difference of the amplitude of each oscillator
appears in this parameters. Namely, oscillation death is observed in third oscillator. Phase differences between first andsecond oscillators shows anti-phase synchronization, because
third oscillator stops oscillating in order to match consistency.

(1)(k = 1, 2, 3).
The normalized circuit equations are expressed as:[First oscillator]
dx11


= ε(1 − x1 2 )x1 − y1

dτ3




 dy1 = x1 .
dτ+γ1 (x1 − x2 )−γ2 (x1 − x3 )Figure 4: Attractor between adjacent oscillators (horizontal
axis:voltage, vertical axis:electric current.
(2)[Second oscillator]
1dx2


= ε(1 − xk 2 )x2 − y2

dτ3




 dy2 = x2 .
dτ+γ1 (x2 − x1 )−γ2 (x2 − x3 )[Third oscillator]
1dx3


= ε(1 − x3 2 )x3dτ3
 dy3 = x3 .
dτ−y3−γ2 (2x3 − x1 − x2 )Figure 5: The time waveform of the each oscillator.

(3)

(4)

In this oscillatory system, the nonlinearity and couplingstrength influence synchronization phenomena. Here we fix
the parameter ε = 0.02. In case γ1 is larger than or equal toγ2 , first and second oscillator synchronize in anti-phase (seeFig. 6(a)). The other way, in the case of γ2 bigger, all of thephase differences between adjacent oscillators converge inphase synchronization because effect of normal resistor increases. Figure 6(b) shows in-phase synchronization using
the following parameters γ1 = 0.01 and γ2 = 0.05.

- 633 -Figure 8: The time waveform of four coupled oscillators.

(a)

(b)

Figure 6: Phase differences between adjacent oscillators.
(a) Anti-phase synchronization. (b) In-phase synchronization.
Figure 9: Lissajous figures.

4. Four Coupled van der Pol Oscillators
Next, we investigate synchronization phenomena in tetrahedral form as shown in Fig. 7. This oscillatory system exists
three triangles as shown in Fig. 3. These oscillators from firstto the third are coupled using same negative resistors but wecouple normal resistor R between fourth oscillator and others.
In the case of γ1 bigger than γ2 , we can observe threephase synchronization between first, second and third oscillators because fourth oscillator occurs the oscillation death.
Figure 8 shows the simulation results for ε = 0.02, γ1 = 0.05
and γ2 = 0.01. In the case of four coupled oscillators, we canfind out synchronization phenomena that three coupled oscil-lators do not produce. When fourth oscillator continues to oscillate but attenuated, the synchronization phenomena about
fourth oscillator are observed asynchronous. Also, first, second and third oscillator do not produce three-phase synchronization clearly because fourth oscillator effect other oscillators. Namely three-phase synchronization keeps vibrating
few degrees. We confirm that this vibration width increases
if amplitude of fourth oscillator is big. Figure 9 shows theasynchronous of simulation result, we set the parameters ε =0.10, γ1 = 0.10 and γ2 = 0.03.
5. Circuit Experiment
We perform the circuit experiment and compare with simulation results. All of these oscillators are used same value

Figure 7: Conceptual circuit model for tetrahedron form.

- 634 -Figure 10: The time waveform of the each oscillator.

[3] G. Abramson, V.M. Kenkre and A.R. Bishop, “Analyticsolutions for nonlinear waves in coupled reacting systems,” Physica A: Statistical Mechanics and its Applications, vol.305, no.3-4, pp.427-436, Mar. 2002.
(a)(d)(b)(e)[4] 4. Belykh, M. Hasler, M. Lauret and H. Nijmeijer, “Synchronization and graph topology,” International Journalof Bifurcation and Chaos, vol.15, no.11, pp.3423-3433,
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(c)[5] J. Cosp, J. Madrenas, E. Alarcon, E. Vidal and G. Villar, “Synchronization of nonlinear electronic oscillatorsfor neural computation,” IEEE Trans. Neural Networks,
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(f)

Figure 11: The phase differences between adjacent oscilla- [6] R. Stoop and C. Wagner, “Neocortex’s architecture optors. (a) 1st-2nd. (b) 1st-3rd. (c) 1st-4th. (d) 2nd-3rd.
timizes computation, information transfer and synchro(e) 2nd-4th. (f) 3rd-4th.
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elements C = 33[nF ], L = 30[mH]. Observed synchronization phenomena hardly decide two types of resistors r and R. [7] T. Suezaki and S. Mori, “Mutual synchronization of two
The experiment results in this study are found out several inoscillators,” Trans. IECE, vol.48, no.9, pp.1551-1557,
teresting patterns of synchronization phenomena. Figures 10Sep. 1965.
and 11 show the comparison chart of the simulation resultsof Fig. 9. In this circuit experiment, we use the following [8] T. Endo and S. Mori, “Mode analysis of a multimode ladder oscillator,” IEEE trans. Circuit Syst., vol23, pp.100resistors r = 3.3[kΩ] and R = 33[kΩ].
113, Feb. 1976.
6. Conclusion[9] Y. Uwate, Y. Nishio and R Stoop, “Synchronization inthree coupled van der Pol oscillators with different couThis paper has presented the synchronization phenomena
pling strength,” Proc. of NCSP’10, pp.109-112, Mar.
of coupled oscillators which is used two types of resistors.
2010.
In this circuit model, we have been able to observe several
patterns of synchronization phenomena. And the oscillator [10] T. Nagai, Y. Uwate and Y. Nishio, “Rotation of Phase
which is coupled by normal resistor R has occurred oscilDifference in Four Coupled Oscillators as a Regularlation death. These synchronization phenomena have been
Tetrahedron Form,” Proc. of ECCTD’11, pp.762-765,
observed by circuit experiments.
Aug. 2011.
AcknowledgmentThis work was partly supported by JSPS Grant-in-Aid for
Scientific Research 22500203.
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