2010 International Workshop on Nonlinear Circuits,Communication and Signal Processing
NCSP'10, Waikiki, Hawaii, March 3-5, 2010

Investigation of Synchronization Phenomena in Coupled Simultaneous Oscillators
Chikayasu Higashi and Yoshifumi Nishio
Department of Electrical and Electronic Engineering,
Tokushima University
2-1 Minami-Josanjima, Tokushima, 770-8506, Japan
Phone:+81-88-656-7470
Email: {higashi, nishio}@ee.tokushima-u.ac.jp

Abstract

observe various synchronization phenomena.

There have been many investigations of the mutual synchronization of oscillators, because mutual coupled oscillators
have various phase states. We paid our attentions to simultaneous oscillator, which consists of two resonators with different frequencies. In this study, we investigate synchronization phenomena in coupled two simultaneous oscillators withhard nonlinearities. By computer calculations and circuit experiments, we observe various synchronization phenomena.

2. Circuit Model
The circuit model is shown in Fig. 1. Two simultaneous
oscillators are coupled by a resistor R. The simultaneous oscillator consists of nonlinear resistor and two resonators. We
apply fifth-power nonlinearity to the nonlinear resistor. Theoscillator with hard nonlinearity has multiple stable states andcauses complicate phenomenon. In order to change the frequencies of resonators, we change the value of L.

1. Introduction
In our surroundings, there are a lot of synchronous phenomena. The synchronous luminescence of firefly group, cellof heart producing pulses at equal intervals and revolution ofthe moon etc. are good examples in which a synchronous
phenomenon is comprehensible. Namely, synchronization iscommon phenomenon in the field of natural science. We feelthat it is very interesting but the mystery. Because, it is often
diﬃcult to analyze these phenomena.
There have been many investigations of the mutual synchronization of oscillators, because mutual coupled oscillators have various phase states. We consider that it is very
important to analyze synchronization phenomena of coupling
Figure 1: Circuit model.
oscillators. We can analyze the synchronization phenomena of coupling oscillators more easily than that of firefly group by using computer calculations and circuit exper3. Circuit Equations
iments. Through these research, we expect that investigation of simple synchronization phenomena is the key that arThe circuit equations are described as follows,
rives at complicated synchronization phenomena in real nat
di1di2di3
di4

ural field.

v1 = L1 , v2 = L1 , v3 = L2 , v4 = L2 ,


dt
dt
dt
dt
J. Schaﬀner confirmed that an oscillator with two degree


dv11


of freedom can oscillate simultaneously at two diﬀerent freC=−i−(v+v−v−v)−i[v+v],
11324r
13dtR
quencies when the nonlinear characteristics is fifth-power [1].

1
 dv2
However, after their pioneering work, there have not been
= −i2 + (v1 + v3 − v2 − v4 ) − ir [v2 + v4 ],
C

dt
R


many researches on simultaneous oscillator except [2][3].
dv31C= −i3 − (v1 + v3 − v2 − v4 ) − ir [v1 + v3 ],In this study, we investigate synchronization phenomena


dt
R


in two coupled simultaneous oscillators with hard nonlinear1dv
 C 4 = −i4 + (v1 + v3 − v2 − v4 ) − ir [v2 + v4 ].
ities. By computer calculations and circuit experiments, wedt
R- 113 -(1)where ir [vk ] indicates the v − i characteristics of the nonlinear 4.1. Same frequencies
resistor and is approximated by Eq. (2). For circuit experiFirst, we consider the case of ω1 = ω2 = ω3 = ω4 .
ments, the fifth-power nonlinear resistor is realized as shownin Fig. 2.
• ω1 = ω2 = ω3 = ω4ir [v] = g1 v − g3 v3 + g5 v5 (g1 , g3 , g5 > 0)(2)

Figure 3 shows the computer simulation results and the corresponding circuit experimental results. As shown in Fig. 3,
we are able to observe in-phase oscillation of x1 − x2 − x3 − x4 .
This is the most basic synchronization of the circuit.

By using the following variables and parameters,√ √√
gC 4 g11
vk = 4x k , ik =yk ,
5g
L 5g55√√√
CL13g3g1,ε=gφ=,α=R1
LCg5g
1
15(3)

LLL111
β= , γ= , δ= ,L2L3L4


√d

,
 t = L1C τ, “ · ” =dτ(a)(b)
the normalized circuit equations are given as follows.
ẏ1 = x1 , ẏ2 = βx1 , ẏ3 = γx3 , ẏ4 = δx4 ,Figure 3: Time waveform. (a) Numerical result (β=γ=δ=1.0,
()α=0.1).
(b) Experimental result (L1 =L2 =L3 =L4 =10mH,
1 31 5
C=47nF,R
L =6.45kΩ, R=2.17kΩ).
ẋ=−y−α(x−x)−εx−φx+x,
11132413

3 13 5 13

)
(
 ẋ2 = −y2 + α(x13 − x24 ) − ε x24 − 1 φx3 + 1 x5 ,
3 24 5 24



()

4.2. Two diﬀerent frequencies
1 31 5ẋ=−y−α(x−x)−εx−φx+x,33132413
3 13 5 13• ω1 = ω2 , ω3 = ω4
)
(
1 51 3As shown in Fig. 4, we can observe two pairs of in-phase ẋ4 = −y4 + α(x13 − x24 ) − ε x24 − 3 φx24 + 5 x24 .
oscillations of x1 − x2 and x3 − x4 . It is interesting that the(4)oscillators synchronize with simultaneously oscillation. x1( xkl = xk + xl )and x3 do not synchronize. x3 and x4 synchronize although
they are not directly connected each other.

(a)

(b)

Figure 4: Time waveform. (a) Numerical result (β=1.0,γ=δ=2.0, α=0.1). (b) Experimental result (L1 =L2 =20mH,
L3 =L4 =10mH, C=47nF, RL =6.45kΩ, R=2.17kΩ).

Figure 2: Nonlinear resistor (fifth-power).

4. Synchronization Phenomena
Synchronization phenomena for changing frequencies
of four resonators of two oscillators are classified below. The frequencies of x1 , x2 , x3 , x4 are corresponding toω1 , ω2 , ω3 , ω4 , respectively.

• ω1 = ω3 , ω2 = ω4
Figure 5 shows two pairs of in-phase oscillations of x1 − x3and x2 −x4 when α is small. α is corresponding to the couplingstrength. On the other hand, as shown in Fig. 6, when α is- 114 -large, we are able to observe in-phase oscillation of x1 − x2 −x3 − x4 . It seems that the frequency for this case is between
the frequency of x1 − x3 in Fig. 5 and the frequency of x2 − x4in Fig. 5.

(a)

(a)

(b)

Figure 7:
Time waveform.
(a) Numerical result(β=2.0, γ=2.0, δ=1.0, α=0.1).
(b) Experimental result (L1 =L4 =20mH, L2 =L3 =10mH, C=47nF, RL =6.45kΩ,R=2.17kΩ).

(b)

Figure 5:
Time waveform.
(a) Numerical result(β=2.0, γ=1.0, δ=2.0, α=0.01). (b) Experimental re4.3. Three diﬀerent frequencies
sult (L1 =L3 =20mH, L2 =L4 =10mH, C=47nF, RL =6.45kΩ,R=12kΩ).
• ω1 = ω2 , ω3 , ω4 or ω1 , ω2 , ω3 = ω4
Figures 10 and 11 show three types of synchronizations. Figure 10 shows waveforms for ω1 = ω2 , ω3 , ω4 , Fig. 11shows waveforms for ω1 , ω2 , ω3 = ω4 . In both conditions, we are able to observe three types of synchronizations
to change α. As increasing α, we are able to observe independent oscillations, oscillation death, and distorted oscillations.

(a)

(b)

4.4. Four diﬀerent frequencies

Figure 6:
Time waveform.
(a) Numerical result• ω1 , ω2 , ω3 , ω4(β=2.0, γ=1.0, δ=2.0, α=8.0).
(b) Experimental result (L1 =L3 =20mH, L2 =L4 =10mH, C=47nF, RL =6.45kΩ,Shown in Fig. 12(a), we are able to observe four independentR=200Ω).
oscillations when α is small. On the other hand, when α islarge, the oscillations are distorted as shown in Fig. 12(b) orsome resonators stop to oscillate.
• ω1 = ω4 , ω2 = ω3
Figure 7 shows two pairs of in-phase oscillations of x1 − x4
and x2 − x3 . x1 and x2 do not synchronize. Although two different frequencies are set to diﬀerent positions, the oscillators
can synchronize. For this case, synchronous state does notchange for diﬀerent α.
• ω1 = ω3 = ω4 , ω2 or ω1 = ω2 = ω3 , ω4

5. Conclusions
In this article, we have presented synchronization phenomena in coupled simultaneous oscillators. We focused on theoscillators with the fifth-power nonlinear characteristics, andinvestigated synchronization phenomena with coupled multiple frequencies. We investigated synchronization phenomena by computer calculations and circuit experiments. Bychanging the frequencies of the resonators and the couplingstrength, we observed various synchronous states.

Figure 8 shows in-phase oscillations of x1 − x3 − x4 and x1 −x2 − x3 when α is small. The amplitude of x4 is about twiceof x1 or x3 , as shown in Fig. 8(a). Similarly, the amplitude ofx2 is about twice of x1 or x3 , as shown in Fig. 8(c). On theReferences
other hand, Fig. 9(a) shows the distorted oscillations when αis large. However the top of the oscillators (x1 + x3 ) and (x2 + [1] J. Schaﬀner, “Simultaneous Oscillations in Oscillators,”
IRE Trans. on Circuit Theory, vol.1, pp.2-8, June. 1954.
x4 ) synchronize in-phase oscillation, as shown in Fig. 9(b).

- 115 -(a)
(a)

(b)

(b)

(c)

Figure 10: Change in waveform by changing α. (β=1.0,γ=2.0, δ=4.0, α=0.01). (a) Independent oscillation (α=0.01).
(b) Oscillation death (α=0.1). (c) Distorted oscillation
Figure 8:
Time waveform.
(a) Numerical result(α=8.0).
(β=0.5, γ=1.0, δ=1.0, α=0.1).
(b) Experimental result (L1 =L3 =L4 =10mH, L2 =20mH, C=47nF, RL =6.45kΩ,R=2.17kΩ). (c) Numerical result (β=1.0, γ=1.0, δ=0.5,α=0.1).
(d) Experimental result (L1 =L2 =L3 =10mH,
L4 =20mH, C=47nF, RL =6.45kΩ, R=2.17kΩ).
(c)

(d)

(a)

[2] M. Matsuki, S. Mori, “Asynchronous Simultaneous Oscillations in Negative Resistance Oscillatory Circuit Containing Periodically Operating Analog Switch,” IEICE
technical report, Circuits and systems, vol.101, pp.81-87,June. 1993.
[3] M. Matsuki, S. Mori, “Asynchronous Excitation Phenomena in Negative Resistance Oscillatory Circuit Containing Periodically Operating Analog Switch,” IEICE
technical report, Nonlinear problems, vol.94, pp.51-58,May. 1994.

(a)

(b)

Figure 11: Change in waveform by changing α. (β=0.5,γ=4.0, δ=4.0). (a) Independent oscillations (α=0.01). (b)Oscillation death (α=0.1). (c) Distorted oscillations (α=8.0).

(a)

(b)

Figure 9: Time waveform. (a) Numerical result (β=0.5,γ=1.0, δ=1.0, α=0.1). (b) Top of the oscillator (x1 + x3 ) and(x2 + x4 )

(c)

(b)

Figure 12: Time waveform. (β=1.5, γ=2.0, δ=4.0). (a) Independent four oscillations (α=0.01). (b) Distorted oscillations(α=8.0).

- 116 -