IEEE Workshop on Nonlinear Circuit Networks
December 13-14, 2013

Wave Phenomena Observed
from a Ring of Simultaneous Oscillators
Saori FUJIOKA∗ , Yang YANG† , Yoko UWATE∗ and Yoshifumi NISHIO∗∗ Dept. Electrical and Electronic Eng., Tokushima University

2-1 Minami-Josanjima, Tokushima, Japan{saori, uwate, nishio}@ee.tokushima-u.ac.jp† The Institute of Artificial Intelligence and Robotics., Xi’an Jiaotong University
No. 28 Xianning-West-Road, Xi’an, China
Email: yyang@mail.xjtu.edu.cn

Abstract—In this study, we investigate a wave phenomena
observed from a ring of simultaneous oscillators with two LC
resonators coupled by inductors. We discover the generation ofasynchronous oscillations of double-mode and N -phase synchronization by computer simulation.

I. I NTRODUCTION
In the natural environment, several synchronization phenomena exist. For example, firefly luminescence, swing ofpendulums, cardiac heartbeat, and so on, are well known assynchronization phenomena.
Oscillators containing a nonlinear resistor whose v − icharacteristics are described by fifth-power nonlinear characteristics are known to exhibit hard excitation [1][2]. Such anoscillator is often called as hard oscillator or said to have hard
nonlinearity.
In 1954, Schaffner reported that an oscillator with twodegrees of freedom could oscillate simultaneously at two
different frequencies when the nonlinear characteristics aredescribed by a fifth-power polynomial function [3]. Kuramitsu
also investigated the simultaneous oscillations for three ormore degrees case theoretically and confirmed the generationof simultaneous oscillation with three frequencies by circuit
experiments [4]. The simultaneous oscillations are definitely
one of the most common nonlinear phenomena observed invarious higher-dimensional systems in the natural sciencefields. However, after their pioneering works, as far as theauthors know, there have not been many researches clarifying the basic mechanism of the simultaneous oscillations
except [5][6].
Two identical oscillators with hard nonlinearities coupled
by an inductor [7] are investigated by Datardina and Linkens.
They have confirmed that nonresonant double-mode oscillations, which could not occur for the case of third-power
nonlinearity, were stably excited in the coupled system. They
have also confirmed that four different modes coexist for somerange of parameter values; zero, two single-modes, and adouble-mode.
In this study, we investigate a ring of simultaneous oscillators with two LC resonators coupled by inductors. By

computer simulations, asynchronous oscillations of doublemode and N -phase are confirmed to be stably generated. 1
II. C IRCUIT M ODELThe circuit model is shown in Fig. 1. In the circuit, N simultaneous oscillators with two LC resonators are coupled byinductors LC as a ring. Each simultaneous oscillator consistsof a nonlinear negative resistor, whose v − i characteristics aredescribed by a fifth-power polynomial function asiR (v) = g1 v − g3 v 3 + g5 v 5(g1 , g3 , g5 > 0),(1)√and two√ resonators with different natural frequencies ( L1 C1and L2 C2 ). The equations governing these coupled oscillators are described by the following differential equations.
dvj1


= −ij1 − iRj − iCj + iC,j−1C1dtdv
 C2 j2 = −ij2 − iRj − iCj + iC,j−1dt(2)
dij1
L=v
1j1dtdi L2 j2 = vj2(j = 1, 2, · · · , N ),dt
where iC0 = iCN . The currents through the coupling inductors
iCj are given as
iCj =L1 (ij1 − ij+1,1 ) + L2 (ij2 − ij+1,2 )LC(3)where iN +1,k = i1k (k = 1, 2). The currents through thenonlinear resistors iRj are given as
iRj = iR (vj1 + vj2 )(4)

1 We have already presented this result in: IEICE Technical Report on
Nonlinear Problems (NLP), no. NLP2013-110, pp. 215-218, Oct. 2013, andProc. of NOLTA, pp. 459-462, Sep. 2013.

-4-Fig. 1.

Ring of coupled simultaneous oscillators with two resonators.

III. S IMULATION RESULTS FOR FIVE COUPLEDOSCILLATORSIn this article, we show several computer simulation results
observed from coupled simultaneous oscillators. The results
obtained from 5 coupled oscillators are given in this section.
A typical example of the observed results is shown in Fig. 2.
Figure 2(a) shows attractors observed at 10 resonators. Upper
figures correspond to the attractors observed from upper
resonators and they have a shape of torus, namely upper
resonators exhibit double-mode oscillations. On the other
hand, lower figures correspond to lower resonators and they
seem to have single frequency. Figure 2(b) shows the phaserelationship between horizontally adjacent resonators. Upper
figures are almost synchronized at in-phase. While lower
figures are also almost synchronized but with some phasedifferences. Figure 2(c) shows the phase relationship between
vertically resonators and that upper and lower resonators
are not synchronize. Figure 2(d) shows the time waveforms.
From this figure we can observe that the upper resonators
generate double-mode oscillations with their envelopes synchronized in 5-phase. Further, oscillations themselves of theupper resonators are synchronized in in-phase even though
their amplitudes fluctuate periodically. Moreover, we can seethat the lower resonators generate 5-phase synchronizations
with almost constant amplitudes. We should also note thatthe fundamental oscillation frequencies of upper and lower
resonators are different, namely they are asynchronous.
IV. C ONCLUSIONS(a)(b)(c)(d)
Fig. 2. Typical example of obtained results from 5 coupled oscillators

for αC = 0.8, αL = 1.0, γ = 0.3, ε = 0.3, and β = 3.5.
(a) Attractors observed at 10 resonators. (b) Phase relationshipbetween horizontally adjacent resonators. (c) Phase relationship between vertically adjacent resonators. (c) Time waveforms of thevoltages.

In this study, we investigated a ring type of simultaneousoscillators with two LC resonators coupled by inductors. By
computer simulations, asynchronous oscillations of doublemode and N -phase were confirmed to be stably generated. We
also investigated the effect of changing some important circuit
parameters and larger rings of oscillators, but we could not
introduce these observed phenomena, because of the limitationof the page space. They will be presented at the conference.
R EFERENCES[1] C. Hayashi, Nonlinear oscillations in physical systems, Princeton Univ. Press,
p. 367, 1984.
[2] V.I. Arnold, Geometrical methods in the theory of ordinary differential equations,
Springer-Verlag, pp. 270-272, 1988.

-5-[3] J. Schaffner, “Simultaneous oscillations in oscillators,” IRE Trans. Circuit Theory,
Vol. 1, pp. 2-81, Jun. 1954.
[4] M. Kuramitsu and F. Takase, “Averaged potential analysis of multi mode oscillatorswith hard operating conditions,” IEICE Technical Report on NLP, Vol. 81, No. 13,
pp. 1-10, Sep. 1981.
[5] M. Matsuki and S. Mori, “Asynchronous excitation phenomena in oscillatory
circuit containing periodically operating analog switch,” Transactions of IEICE,
Vol. J77-A, No. 3, pp. 1694-1700, Mar 1994.
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resistance oscillatory circuit containing periodically operating analog switch,”
Transactions of IEICE, Vol. J77-A, No. 3, pp. 399-407, Mar 1994.
[7] S.P. Datardina and D.A. Linkens, “Multimode oscillations in mutually coupled
van der Pol type oscillators with fifth-power nonlinear characteristics,” IEEETrans. Circuits Syst., Vol. 25, No. 5, pp. 308-315, May 1978.