平成24年度 電気関係学会四国支部連合大会

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Two Types of Waves in a Ladder ofSimultaneous Oscillators Coupled by Inductors
Saori Fujioka†
Yang Yang‡†(Tokushima University)

Yoko Uwate†
Yoshifumi Nishio†‡(Xi’an Jiaotong University)

1. Introduction
In the natural ﬁelds, various synchronization phenomena exist. For example, ﬁreﬂy luminescence, swing ofpendulums, cardiac heartbeat, and so on, are well known
as synchronization phenomena. In this study, we investigate a ladder of simultaneous oscillators with two LC resonators coupled by inductors. By computer simulation,
we observe two types of waves.

2. Circuit Model
The circuit model is shown in Fig. 1. In the circuit,
twenty simultaneous oscillators with two 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 v 3 + g5 v 5(g1 , g3 , g5 > 0),

Figure 1: Twenty simultaneous oscillators with two resonators coupled by inductors.

white area shows anti-phase synchronization states or oscillation death. Second wave is a propagation change ofoscillation states from oscillation death to oscillation orfrom oscillation to oscillation death. We call this wave
as “oscillatory inversion wave”. An example of this typeof wave is shown in Fig. 2(b). This ﬁgure shows timewaves of the voltages of each resonators. Hence, whitearea simply shows the oscillation death. In this ﬁgure, wecan conﬁrm that the upper resonators oscillate only when
the lower resonators stop to oscillate, and vice verse.

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and two resonators with diﬀerent natural angular frequen√√cies ( L1 C1 and L2 C2 ). The equations governing thecoupled oscillators are described by the following diﬀerential equations including twenty nonlinear functions iRj(j = 1, 2, ..., 20).

 dx
j1

= −yj1 − f (xj1 + xj2 ) − yCj + yC,j−1

dτ



dxj2


= α {−y − f (x + x ) − y + yCdτ
dyj1


= xj1

dτ


 dyj2= αL xj2dτj2

j1j2

CjC,j−1 }(2)(j = 1, 2, ..., 20),and yC0 and yC,20 are zero; and the nonlinear functionf (·) which corresponds to the v − i characteristics of thenonlinear resistors is given as(f (x) = ε x −β 3 1 5x + x
35(b)).

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3. Simulation Results
In this article, we show only several computer simulated
results obtained by giving diﬀerent initial conditions forthe ﬁxed 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 types of waves. First wave is a propagation changeof phase states between two adjacent oscillators from inphase to anti-phase or from anti-phase to in-phase. An
example of this type of wave is shown in Fig. 2(a). Thisﬁgure shows time waves of the sum of the voltages of
two horizontally adjacent resonators. We call this wave
as “phase inversion waves”. In this ﬁgure, black area ofthe diagram shows in-phase synchronization states, while

Powered by TCPDF (www.tcpdf.org)(a)Figure 2:

Time waves of voltages.
(a) Upper:
xj1,1 +xj+1,1 and Lower: xj2 +xj+1,2 . (b) Upper: xj1 andLower: xj2 .

4. Conclusions
In this study, we have investigated the circuit modelthat twenty simultaneous oscillators with two resonators
are coupled by inductors LC . We could observe two typesof waves. First wave is a propagation change of phasestates between two adjacent oscillators from in-phase toanti-phase or from anti-phase to in-phase. Second wave
is a propagation change of oscillation states from oscillation death to oscillation or from oscillation to oscillation
death.