Propagation Waves in a Ladder of Coupled Simultaneous Oscillators

Saori Fujioka†, Yang Yang‡, Yoko Uwate† and Yoshifumi Nishio†

† Dept. Electrical and Electronic Eng., Tokushima University2-1 Minami-Josanjima, Tokushima, Japan

Email: {saori, uwate, nishio}@ee.tokushima-u.ac.jp

‡ The Institute of Artificial Intelligence and Robotics., Xi’an Jiaotong UniversityNo. 28 Xianning-West-Road, Xi’an, China

Email: yyang@mail.xjtu.edu.cn

Abstract—In this study, we investigate a ladder of si-multaneous oscillators with three or fourLC resonatorscoupled by inductors. By computer simulations, we ob-serve a new type of waves where oscillation frequencyor/and oscillation amplitude of each resonator changes ac-cording to the wave propagation. Further, we confirm thatthe oscillation amplitude of each resonator depend on itspositions.

1. Introduction

In the natural fields, various synchronization phenomenaexists. For example, firefly luminescence, swing of pen-dulums, cardiac heartbeat, and so on, are well known assynchronization phenomena. Oscillators containing a non-linear resistor whosev − i characteristics are described byfifth-power nonlinear characteristics are known to exhibithard excitation [1][2]. Namely, the origin is asymptoti-cally stable and an proper initial condition, which is largerthan a critical value, is necessary to generate the oscilla-tion. Such an oscillator is often called as hard oscillatoror said to have hard nonlinearity. Two identical oscilla-tors with hard nonlinearities coupled by an inductor are in-vestigated by Datardina and Linkens [3]. They have con-firmed that nonresonant double-mode oscillations, whichcould not occur for the case of third-power nonlinearity,were stably excited in the coupled system oscillators hav-ing hard nonlinearity. They have also confirmed that fourdifferent modes coexist for some range of parameter val-ues; 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 fre-quencies when the nonlinear characteristics are describedby a fifth-power polynomial function [4]. Kuramitsu alsoinvestigated the simultaneous oscillations for three or moredegrees case theoretically and confirmed the generation ofsimultaneous oscillation with three frequencies by circuitexperiments [5]. The simultaneous oscillations are defi-nitely one of the most common nonlinear phenomena ob-served in various higher-dimensional systems in the naturalscience fields. However, after their pioneering works, as faras the authors know, there have not been many researches

clarifying the basic mechanism of the simultaneous oscil-lations except [6][7].

In the past studies, we investigated two or three si-multaneous oscillators coupled by a resistor or induc-tors [8][9][10]. We could confirm that double-mode andsimultaneous oscillations are generated at the same timefrom inductively coupled simultaneous oscillators. Wealso investigated the case that twenty simultaneous oscil-lators with twoLC resonators are coupled by inductors ina one-dimensional array [13]. By computer simulations ofthe circuit model, we discovered two kinds of waves; thefirst wave is a propagating change of phase states betweentwo horizontally adjacent resonators from in-phase to anti-phase or from anti-phase to in-phase, which is the same asthe phase-inversion waves in [11][12]. the second wave isa propagating change of oscillation states from oscillationdeath to oscillation or from oscillation to oscillation death.

In this study, we investigate twenty simultaneous oscilla-tors with three or fourLC resonators coupled by inductorsin a one-dimensional array. By computer simulation, wediscover a new type of propagation waves. In this wave,oscillation frequency or/and oscillation amplitude of eachresonator changes according to the wave propagation. Fur-ther, we confirm that the oscillation amplitude of each res-onator depend on its positions.

2. Circuit Model

The circuit model is shown in Fig. 1. In the circuit,twenty simultaneous oscillators with threeLC resonatorsare coupled by inductorsLC and each simultaneous oscil-lator consists of a nonlinear negative resistor, whosev − icharacteristics are described by a fifth-power polynomialfunction as

iR(v) = g1v− g3v3 + g5v5 (g1, g3, g5 > 0), (1)

and three resonators with different natural frequencies(√

L1C1,√

L2C2 and√

L3C3). The equations governingthe coupled oscillators are described by the following dif-ferential equations including twenty nonlinear functionsiR j

2012 International Symposium on Nonlinear Theory and its ApplicationsNOLTA2012, Palma, Majorca, Spain, October 22-26, 2012

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

( j = 1,2, · · · ,20).

C1dvj1

dt = −i j1 − iR j − iC j + iC, j−1

C2dvj2

dt = −i j2 − iR j − iC j + iC, j−1

C3dvj3

dt = −i j3 − iR j − iC j + iC, j−1

L1di j1

dt = v j1

L2di j2

dt = v j2

L3di j3

dt = v j3

( j = 1,2, · · · ,20),

(2)

whereiCm are the currents through the coupling inductorsand are given as

iCm =1

LC{L1(im1 − im+1,1) +L2(im2 − im+1,2)

+L3(im3 − im+1,3)} (3)

(m= 1,2, · · · ,19),

with iC0 = iC,20 = 0. The currents through the nonlinearresistorsiR j are given as

iR j = iR(v j1 + v j2 + v j3) ( j = 1,2, · · · ,20). (4)

By using the following variables and parameters,

v jk =4

√g1

5g5x jk, i jk =

4

√g1

5g5

√C1

L1y jk,

αC1 =C1

C2, αL1 =

L1

L2, γ =

L1

LC,

αC2 =C1

C3, αL2 =

L1

L3,

ε = g1

√L1

C1, β =

3g3

g1

√g1

5g5, t =

√L1C1τ

( j = 1,2, · · · ,20) (k = 1,2,3) ,

(5)

the normalized circuit equations are given as follows.

dxj1

dτ= −y j1 − f (x j1 + x j2 + x j3) − yC j + yC, j−1

dxj2

dτ= αC1{−y j2 − f (x j1 + x j2 + x j3) − yC j + yC, j−1}

dxj3

dτ= αC2{−y j3 − f (x j1 + x j2 + x j3) − yC j + yC, j−1}

dyj1

dτ= x j1

dyj2

dτ= αL1x j2

dyj3

dτ= αL2x j3 ( j = 1,2, · · · ,20),

(6)whereyCm corresponding toiCm is given as

yCm = γ

(ym1 − ym+1,1 +

ym2 − ym+1,2

αL1+

ym3 − ym+1,3

αL2

)(7)

(m= 1,2, · · · ,19),

with yC0 = yC,20 = 0. The nonlinear functionf (·), whichcorresponds to thev− i characteristics of the nonlinear re-sistors, is given as

f (x) = ε

(x− β

3x3 +

15

x5

). (8)

3. Two kinds of waves [13]

In our past study, we investigated the twenty simultane-ous oscillators with twoLC resonators coupled by induc-tors and discovered coexisting two kinds of waves.

The first wave is a propagating change of phase states be-tween two horizontally adjacent resonators from in-phaseto anti-phase or from anti-phase to in-phase. An exam-ple of this type of waves is shown in Fig. 2. The ver-tical axes of this diagram are the sum of the voltages oftwo horizontally adjacent resonators. Hence, black area ofthe diagram shows in-phase synchronization states, whilewhite area shows anti-phase synchronization states or os-cillation death. From this figure, we can see that two phase-inversion waves propagates in the upper resonators, whileall the lower resonators stop the oscillation.

The second wave is a propagating change of oscillationstates from oscillation death to oscillation or from oscilla-tion to oscillation death. An example of this type of wavesis shown in Fig. 3. The vertical axes of this diagram corre-spond to the voltages of the resonators. Hence, white areasimply shows the oscillation death. From this figure, wecan confirm that the upper resonators oscillate only whenthe lower resonators stop to oscillate, and vice versa.

4. Propagation wave in threeLC resonators

In this study, we investigate the twenty simultaneous os-cillators with threeLC resonators coupled by inductors in

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Figure 2:Propagating change of phase states between two hor-izontally adjacent resonators.αC = 0.57, αL = 1.0, γ = 0.01,ε = 0.98 andβ = 2.73.

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

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 res-onators, and the lower 20 graphs are the lower resonators.

There are two characteristics of this wave. First, all theresonators always oscillate, although the oscillation ampli-tude becomes smaller for the area without the new wave.This means that oscillation death cannot be observed. Thesecond 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 strengthγ. Figure 5shows how the propagation speed changes for different val-ues ofγ. 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, namelyif we change the ratio of the natural frequencies of the res-onators, the amplitude of some of the new waves becomessmaller as shown in Fig. 6. The values of the other param-eters of Fig. 6 areαL1 = 1.0, αL2 = 1.0, γ = 0.2, ε = 0.35andβ = 3.3.

Figure 4:Propagation wave in threeLC resonators.αC1 = 0.81,αL1 = 1.0,αC2 = 0.9,αL2 = 1.0, γ = 0.3, ε = 0.35 andβ = 3.3.

5. Propagation wave in fourLC resonators

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

(a)

(b)

Figure 5:Propagation waves in threeLC resonators for differentcoupling strengths. (a)γ = 0.2. (b)γ = 0.38.

6. Conclusions

In this study, we investigated twenty simultaneous os-cillators with three or fourLC resonators coupled by in-

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(a)

(b)

Figure 6:Propagation waves in threeLC resonators for differentsets of natural frequencies. (a)αC1 = 0.1 andαC2 = 0.3. (b)αC1 =

0.2 andαC2 = 0.9.

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

Acknowledgment

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

References

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Figure 7:Propagation wave in fourLC 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 andβ = 3.3.

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