2012 International Symposium on Nonlinear Theory and its Applications
NOLTA2012, Palma, Majorca, Spain, October 22-26, 2012

Propagation Waves in a Ladder of Coupled Simultaneous 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, we investigate a ladder of simultaneous oscillators with three or four LC resonators
coupled by inductors. By computer simulations, we observe a new type of waves where oscillation frequency
or/and oscillation amplitude of each resonator changes according to the wave propagation. Further, we confirm thatthe oscillation amplitude of each resonator depend on itspositions.
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 asymptotically stable and an proper initial condition, which is larger
than a critical value, is necessary to generate the oscillation. Such an oscillator is often called as hard oscillatoror said to have hard nonlinearity. Two identical oscillators with hard nonlinearities coupled by an inductor are investigated by Datardina and Linkens [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 oscillators having hard nonlinearity. They have also confirmed that four
diﬀerent modes coexist for some range of parameter values; zero, two single-modes, and a double-mode. In 1954,
Schaﬀner reported that an oscillator with two degrees offreedom could oscillate simultaneously at two diﬀerent frequencies when the nonlinear characteristics are describedby a fifth-power polynomial function [4]. Kuramitsu also
investigated the simultaneous oscillations for three or moredegrees case theoretically and confirmed the generation ofsimultaneous 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 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 the past studies, we investigated two or three simultaneous oscillators coupled by a resistor or inductors [8][9][10]. We could confirm that double-mode andsimultaneous oscillations are generated at the same timefrom inductively coupled simultaneous oscillators. We
also investigated the case that twenty simultaneous oscillators with two LC resonators are coupled by inductors ina one-dimensional array [13]. By computer simulations of
the circuit model, we discovered two kinds of waves; thefirst wave is a propagating change of phase states between
two horizontally adjacent resonators from in-phase to antiphase or from anti-phase to in-phase, which is the same as
the phase-inversion waves in [11][12]. the second wave isa propagating change of oscillation states from oscillation
death to oscillation or from oscillation to oscillation death.
In this study, we investigate twenty simultaneous oscillators with three or four LC resonators coupled by inductors
in a one-dimensional array. By computer simulation, we
discover a new type of propagation waves. In this wave,
oscillation frequency or/and oscillation amplitude of eachresonator changes according to the wave propagation. Further, we confirm that the oscillation amplitude of each resonator depend on its positions.

2. Circuit Model
The circuit model is shown in Fig. 1. In the circuit,
twenty simultaneous oscillators with three LC resonators
are coupled by inductors LC and 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√ three √resonators √( L1C1 , L2C2 and L3C3 ). The equations governing
the coupled oscillators are described by the following differential equations including twenty nonlinear functions iR j- 907 -the normalized circuit equations are given as follows.

Figure 1: Twenty simultaneous oscillators with three resonators coupled by inductors in a one-dimensional array.

( j = 1, 2, · · · , 20).
dv

C1 dtj1 = −i j1 − iR j − iC j + iC, j−1dv
C2 dtj2 = −i j2 − iR j − iC j + iC, j−1


dv


C3 dtj3 = −i j3 − iR j − iC j + iC, j−1




di

L1 dtj1 = v j1



di

L2 dtj2 = v j2




di


L3 dtj3 = v j3
 ( j = 1, 2, · · · , 20),(2)

1{L1 (im1 − im+1,1 )LC+L2 (im2 − im+1,2 )+L3 (im3 − im+1,3 )}(3)(m = 1, 2, · · · , 19),with iC0 = iC,20 = 0. The currents through the nonlinear
resistors iR j are given asiR j = iR (v j1 + v j2 + v j3 )( j = 1, 2, · · · , 20).

(4)

By using the following variables and parameters,
√√
√
g1
g1
C1
4x jk , i jk =y jk ,5g55g5 L1v jk =4

αC1 =C1
L1
L1, αL1 = , γ =,C2L2LC

C1
L1, αL2 = ,C3L3
√
√
√
L13g3
g1
ε = g1, β=, t = L1 C 1 τC1
g15g5( j = 1, 2, · · · , 20) (k = 1, 2, 3) ,αC2 =with yC0 = yC,20 = 0. The nonlinear function f (·), whichcorresponds to the v − i characteristics of the nonlinear resistors, is given as()β 3 1 5f (x) = ε x − x + x .
(8)35
3. Two kinds of waves [13]where iCm are the currents through the coupling inductors
and are given asiCm = dxj1= −y j1 − f (x j1 + x j2 + x j3 ) − yC j + yC, j−1dτ
dx j2= αC1 {−y j2 − f (x j1 + x j2 + x j3 ) − yC j + yC, j−1 }


dτ




dx j3




 dτ = αC2 {−y j3 − f (x j1 + x j2 + x j3 ) − yC j + yC, j−1 }


dy j1




= x j1



dτ




dy j2



= αL1 x j2



dτ




dy j3



= αL2 x j3
( j = 1, 2, · · · , 20),dτ
(6)where yCm corresponding to iCm is given as()ym2 − ym+1,2 ym3 − ym+1,3yCm = γ ym1 − ym+1,1 ++(7)αL1αL2(m = 1, 2, · · · , 19),(5)

In our past study, we investigated the twenty simultaneous oscillators with two LC resonators coupled by inductors and discovered coexisting two kinds of waves.
The first wave is a propagating change of phase states between two horizontally adjacent resonators from in-phaseto anti-phase or from anti-phase to in-phase. An example of this type of waves is shown in Fig. 2. The vertical axes of this diagram are the sum of the voltages of
two horizontally adjacent resonators. Hence, black area ofthe diagram shows in-phase synchronization states, while
white area shows anti-phase synchronization states or oscillation death. From this figure, we can see that two phaseinversion waves propagates in the upper resonators, while
all the lower resonators stop the oscillation.
The second wave is a propagating change of oscillation
states from oscillation death to oscillation or from oscillation to oscillation death. An example of this type of waves
is shown in Fig. 3. The vertical axes of this diagram correspond to the voltages of the resonators. Hence, white area
simply shows the oscillation death. From this figure, wecan confirm that the upper resonators oscillate only when
the lower resonators stop to oscillate, and vice versa.
4. Propagation wave in three LC resonators
In this study, we investigate the twenty simultaneous oscillators with three LC resonators coupled by inductors in- 908 -Figure 2: Propagating change of phase states between two horizontally adjacent resonators. αC = 0.57, αL = 1.0, γ = 0.01,ε = 0.98 and β = 2.73.

Figure 4: Propagation wave in three LC resonators. αC1 = 0.81,αL1 = 1.0, αC2 = 0.9, αL2 = 1.0, γ = 0.3, ε = 0.35 and β = 3.3.

5. Propagation wave in four LC resonators

Figure 3: Propagating change of oscillation states. αC = 0.57,
αL = 1.0, γ = 0.01, ε = 0.98 and β = 2.73.

We also investigate the twenty simultaneous oscillatorswith four LC resonators coupled by inductors in a onedimensional array. We can confirm the generation of thesame propagation waves as those observed from the caseof three LC resonators. Figure 7 shows an example of thePropagation wave in four LC resonators.

a one-dimensional array. We can observe a new type ofwave propagation phenomenon which is diﬀerent from thewaves observed from the twenty simultaneous oscillatorswith two LC resonators.
Figure 4 shows an example of the new propagation wave.
The vertical axes of this diagram correspond to the voltagesof the resonators; the top 20 graphs are the voltages of theupper resonators, the middle 20 graphs are the middle resonators, and the lower 20 graphs are the lower resonators.

(a)There are two characteristics of this wave. First, all theresonators always oscillate, although the oscillation amplitude becomes smaller for the area without the new wave.
This means that oscillation death cannot be observed. The
second characteristic is that the propagation waves in thethree region (upper, middle and lower) look like the same.
Furthermore, we can observe that the propagation speedof the waves depends on the coupling strength γ. Figure 5
shows how the propagation speed changes for diﬀerent values of γ. The values of the other parameters of Fig. 5 areαC1 = 0.81, αL1 = 1.0, αC2 = 0.9, αL2 = 1.0, ε = 0.35 andβ = 3.3. Moreover, if we change the value of parameterof αC corresponding to the ratio of the capacitors, namely
if we change the ratio of the natural frequencies of the resonators, the amplitude of some of the new waves becomes
smaller as shown in Fig. 6. The values of the other parameters of Fig. 6 are αL1 = 1.0, αL2 = 1.0, γ = 0.2, ε = 0.35
and β = 3.3.

(b)Figure 5: Propagation waves in three LC resonators for diﬀerent
coupling strengths. (a) γ = 0.2. (b) γ = 0.38.

6. Conclusions
In this study, we investigated twenty simultaneous oscillators with three or four LC resonators coupled by in-- 909 -(a)Figure 7: Propagation wave in four LC resonators. αC1 = 0.65,

αL1 = 1.0, αC2 = 0.71, αL2 = 1.0, αC3 = 0.81, αL3 = 1.0, γ = 0.2,
ε = 0.08 and β = 3.3.

IEICE Technical Report on NLP, Vol. 81, No. 13, pp. 1-10,Sep. 1981.

(b)
Figure 6: Propagation waves in three LC resonators for diﬀerent
sets of natural frequencies. (a) αC1 = 0.1 and αC2 = 0.3. (b) αC1 =0.2 and αC2 = 0.9.

ductors in a one-dimensional array. By computer simulation, we discovered a new type of propagation waves whereoscillation frequency or/and oscillation amplitude of eachresonator changed according to the wave propagation. Further, we confirmed that the oscillation amplitude of eachresonator depended on its positions.

[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.
[7] 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.
[8] C. Higashi and Y. Nishio, “Investigation of synchronization
phenomena in coupled simultaneous oscillators,” Proceedings of 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.

Acknowledgment
This work was partly supported by JSPS Grant-in-Aidfor Scientific Research 22500203.
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