2013 International Symposium on Nonlinear Theory and its Applications
NOLTA2013, Santa Fe, USA, September 8-11, 2013

Asynchronous Oscillations of Double-Mode and N-Phase in a Ring ofSimultaneous Oscillators
Saori FUJIOKA† , Yang YANG‡ , Yoko UWATE† and Yoshifumi NISHIO†† Dept. Electrical and Electronic Eng., Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: {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, a ring of coupled simultaneous
oscillators with two LC resonators is investigated. By computer simulations, asynchronous oscillations of doublemode and N-phase are confirmed to be stably generated.
Careful observation reveals various interesting synchronization phenomena which have not reported before.

1. Introduction
In the natural fields, various synchronization phenomena
exists. For example, firefly luminescence, swing of pendulums, cardiac heartbeat, and so on, are well known assynchronization phenomena. Oscillators containing a nonlinear resistor whose v − i characteristics are described by
fifth-power nonlinear characteristics are known to exhibit
hard excitation [1][2]. Namely, the origin is asymptoticallystable and an proper initial condition, which is larger than acritical value, is necessary to generate the oscillation. Suchan oscillator is often called as hard oscillator or said to have
hard nonlinearity.
Two identical oscillators with hard nonlinearities coupled by an inductor [3] are investigated by Datardina andLinkens. They have confirmed that nonresonant doublemode oscillations, which could not occur for the case ofthird-power nonlinearity, were stably excited in the coupled system. They have also confirmed that four diﬀerentmodes coexist for some range of parameter values; zero,two single-modes, and a double-mode.
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 [4]. Kuramitsu also investigated the simultaneous oscillations forthree or more degrees case theoretically and confirmed thegeneration of simultaneous oscillation with three frequencies by circuit experiments [5]. The simultaneous oscillations are definitely one of the most common nonlinear phenomena observed in various higher-dimensional systems in
the natural science fields. However, after their pioneering
works, as far as the authors know, there have not been many

researches clarifying the basic mechanism of the simultaneous oscillations except [6][7].
In our past study, we investigated synchronization phenomena observed from two or more coupled hard oscillators [8]-[10]. In those studies, we could observe variousinteresting nonlinear phenomena including asynchronous
double-mode oscillations and wave propagations.
In this study, we investigate a ring of simultaneous oscillators with two LC resonators coupled by inductors.
By computer simulations, asynchronous oscillations ofdouble-mode and N-phase are confirmed to be stably generated. Careful observation reveals various interesting synchronization phenomena which have not reported before.
2. Circuit Model
The circuit model is shown in Fig. 1. In the circuit, N
simultaneous oscillators with two LC resonators are coupled by inductors LC as a ring. Each simultaneous oscillator consists of a nonlinear negative resistor, whose v − icharacteristics are described by a fifth-power polynomial
function asiR (v) = g1 v − g3 v3 + g5 v5(g1 , g3 , g5 > 0),(1)

andwith diﬀerent natural frequencies√ two resonators√( L1C1 and L2C2 ). The equations governing these coupled oscillators are described by the following diﬀerentialequations.
dv j1
C1= −i j1 − iR j − iC j + iC, j−1dt
dv j2= −i j2 − iR j − iC j + iC, j−1 C2 dt(2)


di j1




L1= v j1


dt


 L2 di j2 = v j2

( j = 1, 2, · · · , N),
dt
where iC0 = iCN . The currents through the coupling inductors iC j are given as- 459 -iC j =L1 (i j1 − i j+1,1 ) + L2 (i j2 − i j+1,2 )LC(3)where iN+1,k = i1k (k = 1, 2). The currents through thenonlinear resistors iR j are given asiR j = iR (v j1 + v j2 )By using the following variables and parameters,
√
√
√
g1
g1
C1
44v jk =x jk , i jk =y jk ,5g55g5 L1C1L1L1αC =, αL = , γ =,C2L2LC
√
√
√
L1
3g3
g1, β=, t = L1C1 τ,ε = g1
C1
g1
5g5

(4)
(a)

(5)
(b)

the normalized circuit equations are given as follows.
 dxj1dτdx j2 dτdy j1dτ
dy j2dτ=−y j1 − f (x j1 + x j2 ) − yC j + yC, j−1=

αC {−y j2 − f (x j1 + x j2 ) − yC j + yC, j−1 }

=x j1

=

αL x j2(c)(6)

Figure 2: Typical example of obtained results from 5 coupledoscillators for αC = 0.8, αL = 1.0, γ = 0.3, ε = 0.3, and β = 3.5.
(a) Attractors observed at 10 resonators. Horizontal: x jk . Vertical: y jk . (b) Phase relationship between horizontally adjacentresonators. Horizontal: x j+1,k . Vertical: x jk . (c) Phase relationship between two resonators in each oscillator. Horizontal: x j2 .
Vertical: x j1 .

( j = 1, 2, · · · , N),

where yC0 = yCN . yC j corresponds to iC j and is given as()y j2 − y j+1,2
yC j = γ y j1 − y j+1,1 +(7)αLwhere yN+1,k = y1k (k = 1, 2). The nonlinear function f (·),
which corresponds to the v − i characteristics of the nonlinear resistors, is given as()β1f (x) = ε x − x3 + x5 .
(8)35

3. Simulation results for five coupled oscillators
In this article, we show only several computer simulation
results because of the limitation of page space. Firstly, theresults obtained from 5 coupled oscillators are given in thissection.
A typical example of the observed results is shown inFig. 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 oftorus, namely upper resonators exhibit double-mode oscillations. On the other hand, lower figures correspond tolower resonators and they seem to have single frequency.
Figure 2(b) shows the phase relationship between horizontally adjacent resonators. Upper figures show that upper resonators are almost synchronized at in-phase. Whilelower figures show that lower resonators are also almost
synchronized but with some phase diﬀerences. Figure 2(c)shows the phase relationship between two resonators ineach oscillator and that upper and lower resonators are notsynchronized, namely asynchronous simultaneous oscillations occur.
Figure 3 shows the time waveforms of the voltages of 10resonators. Figure 3(b) is the horizontally magnified figureof Fig. 3(a). From these figures we can observe that theupper resonators generate double-mode oscillations with
their envelopes synchronized in 5-phase. Further, oscillations themselves of the upper resonators are synchronized
in in-phase even though their amplitudes fluctuate periodically. Moreover, we can see that the lower resonators generate 5-phase synchronizations with almost constant amplitudes. We should also note that the fundamental oscillation frequencies of upper and lower resonators are diﬀerent, namely they are asynchronous. As far as we know, thiscomplicated phenomenon has never reported before. We
believe that revealing the generation mechanism and conditions of such unknown phenomenon is important to understand highly nonlinear phonomena in natural fields andexploiting them for future engineering applications.
3.1. Changing coupling strength
Next, we investigate the eﬀect of changing the couplingstrength γ. The other parameters are fixed as the same inFigs. 2 and 3.
Figure 4 shows how the time waveforms of the voltagesof 10 resonators change as γ increases. The beat oscillations of the upper resonators become weaker and resulted
in simple in-phase synchronizations as Fig. 4(c). While- 460 -Figure 1: Ring of coupled simultaneous oscillators with two resonators.

(a)

(a)

(b)

(b)

Figure 3: Time waveforms of the voltages of 10 resonators for
αC = 0.8, αL = 1.0, γ = 0.3, ε = 0.3, and β = 3.5.

the lower resonators become to show weak double-mode
as Fig. 4(b), but resulted in almost oscillation-death as
Fig. 4(c).

(c)
Figure 4: Time waveforms of the voltages of 10 resonators aschanging coupling strength. (a) γ = 0.35. (b) γ = 0.4. (c) γ = 0.5.

3.2. Changing ratio of natural frequencies
Next, we investigate the eﬀect of changing the ratio of
the natural frequencies of the resonators αC . The other parameters are fixed as the same in Figs. 2 and 3.
Figure 5 shows how the time waveforms of the voltagesof 10 resonators change as αC increases (equivalently C2becomes closer to C1 ). The beat frequency of the upper resonators changes according to the ratio of natural frequencies. While the 5-phase oscillations of the lower resonators
are not aﬀected by the change of the ratio of the naturalfrequencies.

voltages observed from the ring of 15 simultaneous oscillators. As expected, we could observe asynchronous oscillations of double-mode in the upper resonators and N-phasein the lower resonators from both rings. We also confirmedthat the envelopes of the double-mode oscillations made 7phase and 15-phase synchronizations.
5. Conclusions

4. Extension to larger rings of oscillators
Finally, we show some simulation results observed fromlarger rings of oscillators.
Figure 6 shows a typical example of asynchronous oscillations observed from a ring of 7 coupled simultaneous
oscillators. While, Fig. 7 shows time waveforms of theIn this study, we investigated a ring of simultaneous
oscillators with two LC resonators coupled by inductors.
By computer simulations, asynchronous oscillations ofdouble-mode and N-phase were confirmed to be stably
generated. We also investigated the eﬀect of changing
some important circuit parameters.

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(a)
(a)
(b)

(c)
(b)

(c)

(d)

Figure 5: Time waveforms of the voltages of 10 resonators as

Figure 6: Asynchronous oscillations of double-mode and 7phase obtained from 7 coupled oscillators for αC = 0.8, αL = 1.0,
γ = 0.3, ε = 0.3, and β = 3.5. (a) Attractors observed at 14
resonators. (b) Phase relationship between horizontally adjacentresonators. (c) Phase relationship between two resonators in eachoscillator. (d) Time waveforms of the voltages of 14 resonatorschanging ratio of natural frequencies. (a) αC = 0.5. (b) αC = 0.7.
(c) αC = 0.85.

We could not introduce some statistical analysis of theobserved phenomena, because of the limitation of the pagespace. They will be presented at the conference, if accepted.
Our future works include the investigation for the case of
three or more resonators in each oscillator and the circuit
experiments.

[7] M. Matsuki and S. Mori, “Asynchronous excitation phenomena in negative resistance oscillatory circuit containing periodically operating analogswitch,” IEICE Technical Report on NLP, Vol. 94, pp. 51-58, May 1994.
[8] Y. Nishio, Y. Yang and Y. Uwate, “Synchronization phenomena in simultaneous oscillators coupled by an inductor,” Proceedings of NCSP’11, pp. 147150, Mar. 2011.
[9] S. Fujioka, Y. Yang, Y. Uwate and Y. Nishio, “Double-mode simultaneous oscillation in three coupled hard oscillators,” Proceedings of NCSP’12,pp. 72-75, Mar. 2012.

Acknowledgment
This work was partly supported by JSPS Grant-in-Aidfor Young Scientists 23700269.

[10] S. Fujioka, Y. Yang, Y. Uwate and Y. Nishio, “Two kinds of waves in a ladderof coupled simultaneous oscillators,” Proceedings of NDES’12, pp. 232-236,
Jul. 2012.

References[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.
[3] S.P. Datardina and D.A. Linkens, “Multimode oscillations in mutually coupled van der Pol type oscillators with fifth-power nonlinear characteristics,”
IEEE Trans. Circuits Syst., Vol. 25, No. 5, pp. 308-315, May 1978.
[4] J. Schaﬀner, “Simultaneous oscillations in oscillators,” IRE Trans. Circuit
Theory, Vol. 1, pp. 2-81, Jun. 1954.
[5] M. Kuramitsu and F. Takase, “Averaged potential analysis of multi mode
oscillators with hard operating conditions,” IEICE Technical Report on NLP,Vol. 81, No. 13, pp. 1-10, Sep. 1981.
[6] M. Matsuki and S. Mori, “Asynchronous simultaneous oscillations in negative resistance oscillatory circuit containing periodically operating analog
switch,” IEICE Technical Report on CAS, Vol. 101, pp. 81-87, Jun. 1993.

Figure 7: Asynchronous oscillations of double-mode and 15-

phase obtained from 15 coupled oscillators for αC = 0.8, αL =1.0, γ = 0.3, ε = 0.3, and β = 3.5.

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