IEEE Workshop on Nonlinear Circuit Networks
December 9-10, 2011

Double-Mode Simultaneous Oscillationin Coupled Hard Oscillators
Saori Fujioka, Yoko Uwate and Yoshifumi Nishio
Dept. Electrical and Electronic Eng.,
Tokushima University
2-1 Minami-Josanjima,Tokushima,Japan
Email: saori@ee.tokushima-u.ac.jp

Abstract—In this study, synchronization phenomenon observed
from two inductively coupled simultaneous oscillators can exhibit
both in-phase and anti-phase synchronizations. We can confirm
the phenomenon that double-mode and simultaneous oscillationis generated at the same time.

nonlinear functions iR1 and iR2 .
dv11

C1= −i11 − iR1 − iC ,

dt



dv12


C2= −i12 − iR1 − iC ,


dt




dv13



C3= −i13 − iR1 − iC ,

dt





dv21



C1= −i21 − iR2 + iC ,


dt






dv22


C2= −i22 − iR2 + iC ,

dt




dv

 C3 22 = −i23 − iR2 + iC ,dtI. Introduction
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 [1]. Kuramitsu
investigated the simultaneous oscillations for three or moredegrees case theoretically and confirmed the generation oftriple mode oscillation by circuit experiments [2]. Simultaneous oscillations are definitely one of the most common
nonlinear phenomena observed in various higher-dimensional
systems in the natural fields. However, after their pioneering
works, as far as the authors know, there have not been many
researches clarifying the basic mechanism of the simultaneousoscillations except [3][4].
In our past study, we have reported synchronization phenomena observed from two resistively coupled simultaneousoscillators [5] and two inductively coupled simultaneous oscillators [6]. In this study, we investigate two inductively coupled simultaneous oscillators with more degrees of freedom.
We confirm the generation of various interesting modes ofoscillations from the circuits.

iC =L2di12= v12dt

L3

di13= v13dt

L1

di21= v21dt

L2

di22= v22dt

L3

di23= v23dt(2)

L1 (i11 − i21 ) + L2 (i12 − i22 ) + L3 (i13 − i23 )LC(3)The currents through the nonlinear resistors iR1 and iR2 aregiven as
 iR1 = iR (v11 + v12 + v13 )(4)
 iR2 = iR (v21 + v22 + v23 )By using the following variables and parameters,

The circuit model is shown in Fig. 1. In the circuit, twosimultaneous oscillators with three LC resonators are coupled
by an inductor LC and each simultaneous oscillator consistsof a nonlinear negative resistor, whose v − i characteristics aredescribed by a fifth-power polynomial function as(g1 , g3 , g5 > 0)

di11= v11dt

where iC is the current through the coupling inductor and isgiven as

II. Circuit modeliR (v) = g1 v − g3 v3 + g5 v5

L1√vmn =- 75 -C1,
C2

αC2 =
√
ε = ga

√

g1
xmn ,
5g5

αC1 =(1)and three resonators with diﬀerent natural frequencies. Theequations governing the coupled oscillators are described by
the following 12th-order diﬀerential equations including two4L1,
C1

imn =αL1 =C1,
C3β=

3g3
g1

4

g1
5g5

L1,
L2

αL2 =
√√

γ=C1ymn ,
L1

L1,
LC

L1,
L3

g1,
5g5

t=

√

L1C1 τ,(5)

III. Synchronization Phenomena
We investigate the phenomenon by changing the parameters.

Fig. 1.

Circuit diagram.

the normalized circuit equations are given as follows.
 dx11
= −y11 − f (x11 + x12 + x13 ) − yCdτdy11= x11
dτ


dx12


= αC1 {−y12 − f (x11 + x12 + x13 ) − yC }dτ

dy12


= αL1 x12
dτ


dx13



= αC2 {−y13 − f (x11 + x12 + x13 ) − yC }


dτ



dy13



 dτ = αL2 x13

dx21



= −y21 − f (x21 + x22 + x23 ) − yC


dτ


dy21
= x21
dτdx22= αC1 {−y22 − f (x21 + x22 + x23 ) − yC }dτ
dy22
= αL1 x22


dτ

dx23



= αC2 {−y23 − f (x21 + x22 + x23 ) + yC }

dτ



dy23



= αL2 x23dτ
where yC corresponds to iC and is given as()y12 − y22 y13 − y23
yC = γ y11 − y21 ++αL1αL2Fig. 2. Lissajous figures of six resonators. Horizontal: xmn . Vertical: ymn .
αC1 = 0.5, αL1 = 0.3, αC2 = 0.4, αL2 = 0.2, γ = 0.01, ε = 0.055 and β = 3.14.

(6)

Fig. 3. Phase diﬀerences of six resonators. αC1 = 0.5, αL1 = 0.3, αC2 = 0.4,αL2 = 0.2, γ = 0.01, ε = 0.055 and β = 3.14.

(7)

and the nonlinear function f (·) which corresponds to the v − icharacteristics of the nonlinear resistors is given as)(1β(8)
f (x) = ε x − x3 + x5
35Figure 2 shows the computer simulated results. The coupledoscillators exhibit various synchronization patterns for diﬀerent initial conditions. Figure 3 shows the phase diﬀerences ofvoltages x11 -x13 and x21 -x23 . We can see from those figuresthat each phase diﬀerence is asynchronous. Hence, we can saythat double-mode and simultaneous oscillation is generated at
the same time. Namely, the relationship between right andleft oscillators is double-mode, while the relationship between
above and below oscillators is asynchronous. Figure 4 shows
the time waveforms of the six voltages. The time waveformsof x11 and x21 are double-mode, while x12 and x22 , x13 and x23
are also double-mode. We can confirm that the envelopes ofthe double-mode oscillations are synchronized in anti-phase.

- 76 -Fig. 4. Time waveforms of xmn . αC1 = 0.5, αL1 = 0.3, αC2 = 0.4, αL2 = 0.2,
γ = 0.01, ε = 0.055 and β = 3.14.
Fig. 6. Phase diﬀerences of six resonators. αC1 = 0.5, αL1 = 0.3, αC2 = 0.4,αL2 = 0.2, γ = 0.1, ε = 0.055 and β = 3.14.

Fig. 7. Time waveforms of xmn . αC1 = 0.5, αL1 = 0.3, αC2 = 0.4, αL2 = 0.2,
γ = 0.1, ε = 0.055 and β = 3.14.
Fig. 5. Lissajous figures of six resonators. Horizontal: xmn . Vertical: ymn .
αC1 = 0.5, αL1 = 0.3, αC2 = 0.4, αL2 = 0.2, γ = 0.1, ε = 0.055 and β = 3.14.

Figures 5, 6, and 7 show another example of the steady statesobtained for diﬀerent coupling parameter. The figure of x11 x21 is synchronized in in-phase. But, the figure of x12 -x21 issynchronized in anti-phase, and the figure of x13 -x23 is doublemode.
IV. Conclusions
In this study, we have investigated the generation of synchronization phenomena observed from two inductively coupled simultaneous oscillators with three resonators. We could
confirm the phenomenon that double-mode and simultaneous
oscillation have generated at the same time. Namely, therelationship between right and left oscillators is double-mode,
while the relationship between above and below oscillators isasynchronous.

[3] 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.
[4] M.Matsuki and S.Mori, “Asynchronous excitation phenomena in negative resistance oscillatory circuit containing periodically operating
analog switch,” IEICE Technical Report on NLP, Vol.94, pp.51-58, May
1994.
[5] C.Higashi and Y.Nishio, “Investigation of synchronization phenomena
in coupled simultaneous oscillators,” Proceedings of NCSP’10, pp.113116, Mar. 2010.
[6] Y.Nishio, Y.Yang and Y.Uwate, “Synchronization phenomena in simultaneous oscillators coupled by an inductor,” Proceedings of NCSP’11,pp.147-150, Mar. 2011.

References[1] J.Schaﬀner, “Simultaneous oscillations in oscillators,” IRE Transactions
on Circuit Theory, Vol.1, pp.2-81, Jun. 1954.
[2] M.Kuramitsu and F.Takase, “Averaged potential analysis of multimodeoscillators with hard operating conditions,” IEICE Technical Report onNLP , Vol.81, No.13, pp.1-10, Sep. 1981.

- 77 -