2012 International Workshop on Nonlinear Circuits,Communications and Signal Processing
NCSP'12, Honolulu, Hawaii, USA. March 4-6, 2012

Double-Mode Simultaneous Oscillationin Three Coupled Hard 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, synchronization phenomena observed fromthree inductively coupled hard oscillators with two resonators
are investigated. Various interesting synchronization phenomena including in-phase and anti-phase synchronizations
are generated by giving different initial conditions. Especially, we can confirm that double-mode and simultaneous
oscillations are generated at the same time.
1. Introduction
There are a lot of synchronization phenomena in the natural environment. For example, swing of the pendulum, fireflyluminescence, cardiac heartbeat, and so on, are well known assynchronization phenomena. Oscillators containing a nonlinear resistor whose v − i characteristics are described by fifthpower nonlinear characteristics are known to exhibit hard excitation [1][2]. Namely, the origin is asymptotically stable
and an proper initial condition, which is larger than a criticalvalue, is necessary to generate the oscillation. Such an oscillator is often called as hard oscillator or said to have hard nonlinearity. Datardina and Linkens have investigated two identical oscillators with hard nonlinearities coupled by an inductor [3]. 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 forsome range of parameter values; zero, two single-modes, anda double-mode. In 1954, Schaffner reported that an oscillator
with two degrees of freedom could oscillate simultaneously
at two different frequencies when the nonlinear characteristics are described 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 aredefinitely one of the most common nonlinear phenomena observed in various higher-dimensional systems in the naturalscience 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 have reported synchronization phenomena observed from two resistively coupled
hard oscillators [8] and two inductively coupled hard oscillators [9].
In this study, three inductively coupled hard oscillators including two resonators are investigated. By computer simulaitons, we can confirm that various synchronization phenomena including in-phase and anti-phase synchronizations
can be generated in the coupled circuits. Especially, wecan observe an interesting phenomenon that double-modeand simultaneous oscillations are generated at the same time.
Namely, the neighboring oscillators generate double-modeoscillations, while the above and the below resonators generate simultaneous oscillations.
2. Circuit model
The circuit model is shown in Fig. 1. In the circuit, three
hard oscillators with two LC resonators are coupled by inductors LC and each hard oscillator consists of a nonlinear
negative resistor, whose v − i characteristics are described 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 the coupled oscillators
are described by the following 12th-order differential equa-- 72 -the normalized circuit equations are given as follows.
dx11


= −y11 − f (x11 + x12 ) − yC1

dτ


dx12


= αC {−y12 − f (x11 + x12 ) − yC1 }dτ
 dx21 = −y21 − f (x21 + x22 ) + yC1 − yC2

dτ



dx22


= αC {−y21 − f (x21 + x22 ) + yC1 − yC2 }dτdx31= −y31 − f (x31 + x32 ) + yC2dτdx32
= αC {−y31 − f (x31 + x32 ) + yC2 }
dτ


dym1


= xm1
(m = 1, 2, 3)
dτ


 dym2 = αL xm2(m = 1, 2, 3)dτ

Figure 1: Three coupled hard oscillators with two resonators.

tions including three nonlinear functions iR1 , iR2 and iR3 .
L1 didt11 = v11
C1 dvdt11 = −i11 − iR1 − iC1 ,

C2 dvdt12 = −i12 − iR1 − iC1 ,L2 didt12 = v12





 C1 dv21 = −i21 − iR2 − iC2 + iC1 , L1 di21 = v21
dt
dt

where yC corresponds to iC and is given as()y12 − y22
yC1 = γ y11 − y21 +αL()
y22 − y32
yC2 = γ y21 − y31 +,αLC2 dvdt22 = −i22 − iR2 − iC2 + iC1 , L2 didt22 = v22







C dv31 = −i31 − iR3 + iC2 ,L1 didt31 = v31 1 dt
C2 dvdt32 = −i32 − iR3 + iC2 ,L2 didt32 = v32 ,

(7)

(8)
(9)

and the nonlinear function f (·) which corresponds to the v −i(2) characteristics of the nonlinear resistors is given as
where iC1 and iC2 are the current through the coupling in()β
1ductors and are given asf (x) = ε x − x3 + x5 .
(10)
35iC1

=iC2

=

L1 (i11 − i21 ) + L2 (i12 − i22 )LCL1 (i21 − i31 ) + L2 (i22 − i32 )
.
LC

(3)

3. Synchronization Phenomena(4)

The proposed circuits exhibit various types of synchronizations including double-mode and oscillation death. In thisarticle, we show only several comupter simulated results obtaind by giving different initial conditions for the fixed parameters as αC = 0.64, αL = 0.6, γ = 0.01, ε = 0.005
and β = 3.2. Namely, all the following results coexist as
well as some more unshown synchronization modes. The detailed explanation of all phenomena and their analysis will bepresented elsewhere.
Figures 2, 3, 4, and 5 show four different kinds of synchronization modes observed from the circuit. In these figures, (a)show the Lissajous figures of six resonators. The positions of
the six figures correspond to the positions of the resonators
in Fig. 1. The horizontal axes are xmn (voltages across capacitors) and the vertical axes are ymn (currents through inductors). Figures (b) show the phase differences between resonators. The left four figures show the relationships between
horizontal neighbors, while the right three figures show thosebetween vertical neighbors. Both horizontal and vertical axes
are xmn (voltages across capacitors). Figures (c) are the time
waveforms of six xmn (voltages across capacitors).

The currents through the nonlinear resistors iR1 , iR2 and iR3are given as iR1 = iR (v11 + v12 )iR2 = iR (v21 + v22 )iR3 = iR (v31 + v32 ).

(5)

By using the following variables and parameters,√
vmn =4√√g1
g1
C14xmn , imn =ymn ,5g5
5g5 L1C1
L1L1, αL =, γ=,C2L2LC
√
√√
L13g3
g1ε = g1, β=, t = L1 C1 τ .
C1
g15g5αC =(6)- 73 -

(a)

(a)

(b)

(b)

(c)

(c)

Figure 2: In-phase and in-phase simultaneous oscillation. (a) Lis-Figure 3: Double-mode and in-phase simultaneous oscillation.
(a) Lissajous figures of six resonators. (b) Phase differences between
six resonators. (c) Time waveforms of six resonators.

sajous figures of six resonators. (b) Phase differences between sixresonators. (c) Time waveforms of six resonators.

From these three types of figures, we can say that simultaneous oscillations are generated in Figs. 2, 3, and 5. Also, theupper resonators in Figs. 3 and 5 and the lower resonators inFigs. 4 and 5 exhibit double-mode oscillations. On the othr
hand, the upper resonators in Fig. 4 stops their oscillations, although their lower pairs oscillate in double-mode. From theseobservations, we can conclude these four oscillation modes asfollows:
Figure 2: Simultaneous oscillation of in-phase (upper) andin-phase (lower).
Figure 3: Simultaneous oscillation of double-mode (upper)
and in-phase (lower).
Figure 4: Single oscillation of oscillation-death (upper) anddouble-mode (lower).
Figure 5: Simultaneous oscillation of double-mode (upper)and double-mode (lower).

Especially, we are interested in the last synchronization
mode; double-mode and double-mode simultaneous oscillation, where the relationships between the horizontal neighbors are double-mode, while the relationships between thevertical neighbors are asynchronous. Because we believe thatsuch a phenomenon exists in natural nonlinear systems with alarge number of degrees of freedom and that its analysis contributes to unveil various unknown mysterious phenomena in
the nature.
4. Conclusions
In this study, we have investigated the generation of various synchronization phenomena observed from three inductively coupled hard oscillators with two resonators. We could
confirm the phenomenon that double-mode and simultaneous
oscillation have been generated at the same time. Namely, the- 74 -

(a)

(a)

(b)

(b)

(c)

(c)

Figure 4: Oscillation-death and double-mode oscillation. (a) Lissajous figures of six resonators. (b) Phase differences between sixresonators. (c) Time waveforms of six resonators.

Figure 5: Double-mode and double-mode simultaneous oscillation.

relationships between the horizontal neighbors are doublemode, while the relationships between the vertical neighbors
are asynchronous.

[5] M. Kuramitsu and F. Takase, “Averaged potential analysis ofmultimode oscillators with hard operating conditions,” IEICE
Technical Report on NLP, Vol.81, No.13, pp.1-10, Sep. 1981.

References[6] M. Matsuki and S. Mori, “Asynchronous simultaneous oscillations in negative resistance oscillatory circuit containing periodically operating analog switch,” IEICE Technical Reporton CAS, Vol.101, pp.81-87, Jun. 1993.

[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 inmutually 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
Transactions on Circuit Theory, Vol.1, pp.2-81, Jun. 1954.

(a) Lissajous figures of six resonators. (b) Phase differences between
six resonators. (c) Time waveforms of six resonators.

[7] M. Matsuki and S. Mori, “Asynchronous excitation phenomena in negative resistance oscillatory circuit containing periodically operating analog switch,” IEICE Technical Report onNLP, Vol.94, pp.51-58, May 1994.
[8] C. Higashi and Y. Nishio, “Investigation of synchronization
phenomena in coupled simultaneous oscillators,” Proceedingsof NCSP’10, pp.113-116, Mar. 2010.
[9] 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.

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