Synchronization Phenomena in Three Oscillators Coupled by a Resonator
Chikayasu HIGASHI Yoshifumi NISHIO (Tokushima University)
1. Introduction
There have been many investigations of the mutual syn- chronization of oscillators [1]. Moro et al. confirmed that N oscillators with same natural frequencies mutually cou- pled by one resistor giveN-phase oscillations, whenN is a prime number. In this study, we observe synchronization of 3 oscillators coupled by a resonator.
2. Circuit Model
C
Cc Lc
L
v1 v2
v3
) (v1
ir i(v2)
r i(v3)
r
NR
Nonlinear Resistor
i1 i2
i3
NR NR
L L
: NR
C C
iLc
vCc
Figure 1: Circuit model (3 oscillators case).
The circuit model is shown in Fig. 1. The circuit equa- tions are described as follows,
Cdvk
dt =−ik−ir(vk) Ldik
dt =vk−vCc (k= 1,2,3) CcdvCc
dt =
∑3 j=1
ij−iLc
LcdiLc
dt =vCc
(1)
whereir(vk) indicates thev−icharacteristics of the nonlin- ear resistor and is approximated by the following function.
ir(vk) =−g1vk+g3vk3. (2) By using the following variables and parameters,
vk=
√g1
g3
xk, ik=
√Cg1
Lg3
yk
vCc=
√g1
g3
X, iLc=
√Cg1
Lg3
Y t=√
LCτ, “·” = d dτ
²=
√ L
Cg1, β= C
Cc, γ= L Lc
(3)
the normalized circuit equations are given as follows.
˙
xk=−yk+²(xk−x3k)
˙
yk=xk−X (k= 1,2,3) X˙ =β
( 3
∑
j=1
yj−Y ) Y˙ =γX
(4)
3. Synchronization Phenomena
As showing in Figs. 2(a)(b)(c), we can observe two pat- terns of oscillations in the same parameter, 3-phase oscil- lation and in-phase oscillation. When the parameters are changed, the three circuits synchronized in another type of in-phase oscillation as shown in Fig. (d). It is interesting that the oscillation frequency depends on the synchroniza- tion patterns.
v1
v2
v3
v1
v1
v2
v3
(a)
(b)
v2
v3
(c)
(d) v1
v2
v3
Figure 2: Time waveform of 3-phase oscillation and in-phase oscillation (numerical results). (a)(b)(c)²= β=γ= 1.0. (d)²= 1.0, β= 0.5, γ = 2.0.
Also, the circuit experimental results show similar phe- nomenon to the numerical results.
4. Conclusions
In this article, we presented synchronization phenomena of 3 oscillators coupled by a resonator. Our future work is to investigate these synchronization phenomena in more detail.
References
[1] Seiichiro MORO, Yoshifumi NISHIO and Shinsaku MORI, “Synchronization Phenomena in Oscillators Cou- pled by One Resistor,” IEICE Transactions on Fundamen- tals, vol. E78-A, no. 2, pp. 244-253, Feb. 1995.
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