Nonlinear Dynamics of Electronic Systems 2012, Wolfenbüttel

Two Kinds of Waves in a Ladderof 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{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 the circuit model thattwenty hard oscillators with two LC resonators are coupled byinductors. We observe two kinds of waves by the change ofinitial states. First waves are changing phase states between two
adjacent oscillators from in-phase to anti-phase or from antiphase to in-phase and continuously exist. We call this wave
as “phase inversion waves”. Second waves are the change ofoscillation states from oscillation death to oscillation or fromoscillation to oscillation death and exist. We name this wave as“oscillatory inversion wave”.

I. I NTRODUCTION
In the natural fields, various synchronization phenomena
exists. For example, firefly luminescence, swing of pendulums,
cardiac heartbeat, and so on, are well known as synchronization 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, isnecessary to generate the oscillation. Such an oscillator is often
called as hard oscillator or said to have hard nonlinearity.
[3] are investigated by Datardina and Linkens. They haveconfirmed 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 for some rangeof parameter values; zero, two single-modes, and a doublemode. In 1954, Schaffner reported that an oscillator withtwo degrees 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 for three ormore degrees case theoretically and confirmed the generationof simultaneous oscillation with three frequencies by circuit
experiments [5]. The simultaneous oscillations are definitely
one of the most common nonlinear phenomena observed invarious higher-dimensional systems in the natural sciencefields. However, after their pioneering works, as far as theauthors know, there have not been many researches clarifying the basic mechanism of the simultaneous oscillations
except [6][7]. In our past study, we investigated synchroniza-tion phenomena observed from two or three coupled hard
oscillators [8]-[10]. In that case, we could confirm that various
synchronization phenomena including in-phase and anti-phasesynchronizations, double-mode oscillation, and simultaneous
oscillation are generated.
In this study, we investigate the circuit model that twenty
hard oscillators with two LC resonators are coupled byinductors. We discover two kinds of waves. First waves arechanging phase states between two adjacent oscillators fromin-phase to anti-phase or from anti-phase to in-phase andcontinuously exist. Second waves are the change of oscillation
states from oscillation death to oscillation or from oscillation
to oscillation death and exist.
II. C IRCUIT M ODELThe circuit model is shown in Fig. 1. In the circuit, twenty
hard oscillators with two LC resonators are coupled byinductors 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 differential equations including
twenty nonlinear functions iRj (j = 1, 2, ...20).
dv
C1 dtj1 = −ij1 − iRj − iCj + iC,j−1




dvj2


 C2 dt = −ij2 − iRj − iCj + iC,j−1
di(2)
L1 dtj1 = vj1di
L2 dtj2 = vj2(j = 1, 2, ..., 20),
where iCn are the currents through the coupling inductors andare given as233

iCm =L1 (im1 − im+1,1 ) + L2 (im2 − im+1,2 )LC(m = 1, 2, ..., 19),(3)

Fig. 1.

Twenty coupled hard oscillators with two resonators.
Fig. 2. Time wave of the voltage of right resonator plus left resonator.

and we assume that iC0 and iC,20 and zero. The currents
through the nonlinear resistors iRn are given as{iRj = iR (vj1 + vj2 )(4)(j = 1, 2, ..., 20).
By using the following variables and parameters,√√
√
g1
g1
C1
44vjk =xjk , ijk =yjk ,5g5
5g5 L1C1
L1
L1, αL =, γ=,C2L2LC
√
√
√
L1
3g3
g1, β=, t = L1 C1 τε = g1
C1
g1
5g5(j = 1, 2, ..., 20) (k = 1, 2) ,αC =the normalized circuit equations are given as follows.
dxj1


= −yj1 − f (xj1 + xj2 ) − yCj + yC,j−1
dτ


dxj2


= αC {−yj2 − f (xj1 + xj2 ) − yCj + yC,j−1 }dτdyj1
= xj1
dτ
 dyj2 = αL xj2(j = 1, 2, ..., 20),dτ
where yC corresponds to iC and is given as()ym2 − ym+1,2yCm = γ ym1 − ym+1,1 +αL(m = 1, 2, ..., 19),

(5)

III. S IMULATION R ESULTS
In this article, we show only several computer simulated
results obtained by giving different initial conditions for thefixed parameters as αC = 0.51, αL = 1.0, γ = 0.01, ε = 0.98
and β = 2.73. As a result, we can discover two kinds of waves.
First waves are changing phase states between two adjacentoscillators from in-phase to anti-phase or from anti-phase toin-phase and continuously exist. We call this wave as “phaseinversion waves”. Second waves are the change of oscillation
states from oscillation death to oscillation or from oscillation
to oscillation death and exist. We call this wave as “oscillatoryinversion wave”. In the case of the phase inversion waves, weobserve various patterns are as follows. In figures 2, 3 and4, show time waves of the voltages of right resonators plus
left resonators. From these figures, we can see that the upper
resonators oscillate when the lowers are oscillation death. On
the other hands, when the above resonators are oscillation
death, are oscillate.

(6)

(7)

and yC0 and yC,20 are zero; and the nonlinear function f (·)
which corresponds to the v − i characteristics of the nonlinear
resistors is given as)(1β.
f (x) = ε x − x3 + x5(8)35Fig. 3. Time wave of the voltage of right resonator plus left resonator.

234

(a)
Fig. 4. Time wave of the voltage of right resonator plus left resonator.

The case of the oscillatory inversion wave, we observevarious patterns of the waves. In figures 5, 6 and 7, (a)shows time waves of forty xmn ; (b) shows time waves ofthe voltages of right resonators plus left resonators. In thiscase, oscillation of the resonators continuously propagate. To
make it more clearly, figure 8 shows the enlargement of apart of time waves on the oscillatory inversion wave. This
figure is separated into four groups of lines. The top four linesshows time waves of the voltage of right resonator plus left
resonator among the above resonators. The second group oflines shows time wave of the voltage of right resonator plus
left resonator among the blow resonators. The third group oflines shows time wave of four xm1 of the upper resonators.
The fourth group of lines shows time wave of four xm2 of
the lower resonators. From two lower groups of lines in thefigure, we can see propagation of oscillation from a resonatorto adjacent resonators. Here we focus on the relationshipbetween the upper resonators and the lower resonators. Whenan upper resonator is oscillation death, the lower resonatorblow the upper resonators is oscillation death. Whereas, when
an upper resonator is oscillation death, the lower resonatorblow the upper resonator oscillate. Moreover, upper and lower
resonators oscillate with different frequencies. The two uppergroups of lines show the sum of amplitudes of adjacentresonators. Thus, if adjacent resonators are synchronized in inphase, the sum is the double of the amplitude of the resonator.
Whereas, if adjacent resonators are synchronized in anti-phase,the sum is zero. From the top group of lines in the figure, wecan see that amplitudes tend toward zero. Namely, adjacentresonators among the upper resonators tend toward anti-phasesynchronization. On the other hand, from the second groupof lines in the figure, we can see that amplitudes of thesums of adjacent resonators are double of the amplitude of
the resonator. Namely, adjacent resonators among the lower
resonators tend toward in-phase synchronization.

(b)
Fig. 5. (a) Time waves of forty xmn . (b) Time wave of the voltageof right resonator plus left resonator.

IV. C ONCLUSION
In this study, we have investigated the circuit model thattwenty hard oscillators with two resonators are coupled byinductors LC . We could observe two kinds of waves. First
waves are changing phase states between two adjacent oscillators from in-phase to anti-phase or from anti-phase to inphase and continuously exist. We call this wave as “phaseinversion waves”. Second waves are the change of oscillation
states from oscillation death to oscillation or from oscillation
to oscillation death and exist. We call this wave as “oscillatoryinversion wave”. In the oscillatory inversion wave, adjacentresonators among the upper resonators tend toward anti-phasesynchronization. On the other hand, adjacent resonators among
the lower resonators tend toward in-phase synchronization.

235

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(a)

(a)

(b)

(b)

(a) Time waves of forty xmn . (b) Time wave of the voltageof right resonator plus left resonator.

Fig. 6.

Fig. 7. (a) Time waves of forty xmn . (b) Time wave of the voltageof right resonator plus left resonator.

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Fig. 8.

Enlargement of a part of time waves of Fig.7