2011 International Workshop on Nonlinear Circuits,Communication and Signal Processing
NCSP'11, Tianjin, China, March 1-3, 2011

Synchronization Phenomena in Simultaneous Oscillators
Coupled by an Inductor
Yoshifumi Nishio (1) , Yang Yang (2) and Yoko Uwate (3)Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan
Tel: +81-88-656-7470, Fax: +81-88-656-7471
Email: (1) nishio@ee.tokushima-u.ac.jp, (2) yyang 993@hotmail.com, (3) uwate@ee.tokushima-u.ac.jp

Abstract
In this study, synchronization phenomena observed from two
inductively coupled simultaneous oscillators are reported.
Since inductively coupled oscillators can exhibit both inphase and anti-phase synchronizations, the circuit generatevarious synchronization patterns. Computer simulations confirm that 9 different synchronization patterns appear in thecircuits.

oscillators exhibit in-phase synchronization only, the variety
of synchronization patterns were poor. In this study, we investigate synchronization phenomena observed from two inductively coupled simultaneous oscillators. Since inductively
coupled oscillators can exhibit both in-phase and anti-phasesynchronizations [13], we expect the generation of various
synchronization patterns. Computer simulations confirm that
9 different synchronization patterns, including the state where
all oscillations stops, appear in the circuits.

1. Introduction

2. Circuit model

In 1954, Schaffner reported that an oscillator with two deThe circuit model is shown in Fig. 1.
grees of freedom could oscillate simultaneously at two different frequencies when the nonlinear characteristics are described by a fifth-power polynomial function [1]. Kuramitsu
investigated the simultaneous oscillations for three or moredegrees case theoretically and confirmed the generation ofR2
triple mode oscillation by circuit experiments [2]. Simul- R1
21
1121
11
taneous oscillations are definitely one of the most common
nonlinear phenomena observed in various higher-dimensional
11
11
systems in the natural fields. However, after their pioneering
works, as far as the authors know, there have not been many
researches clarifying the basic mechanism of the simultaneous oscillations except [3][4].
1222
1222
On the other hand, studies on coupled oscillators become2222
more and more popular not only in the field of physics but
also in the engineering and the biological fields. For example, synchronization of complex networks is one of the hot
topics in the engineering field, because the outcome could be
used to develop better schemes attaining synchronization ofcommunication networks [5]-[8]. In the biological field, neuFigure 1: Circuit diagram.
ronal oscillations are said to play a role in a variety of brain
operations [9]-[11].
In our previous study, we reported some synchronization
In the circuit, two simultaneous oscillators are coupled byphenomena observed from two resistively coupled simulta- an inductor LC and each simultaneous oscillator consists ofneous oscillators [12]. However, because resistively coupled a nonlinear negative resistor, whose v − i characteristics are- 147 -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 ω1 =1/ L1 C1 and ω2 = 1/ L2 C2 .
The equations governing the coupled oscillators are described by the following eighth-order differential equations
including two nonlinear functions iR1 and iR2 .
dv11


= −i11 − iR1 − iCC1

dt




di11


= v11L1


dt




dv12


C2= −i12 − iR1 − iC


dt





di12


 L2 dt = v12

dv21


C1= −i21 − iR2 + iC


dt





di21


L1= v21

dt



dv22


 C2= −i22 − iR2 + iC

dt

 L2 di22 = v22dt(2)the normalized circuit equations are given as follows.
dx11


= −y11 − f (x11 + x12 ) − yC

dτ

dy11


= x11

dτ




dx12


= αC {−y12 − f (x11 + x12 ) − yC }

dτ



 dy12 = x
12
 dτ
dx21


= −y21 − f (x21 + x22 ) + yCdτdy21


= x21
dτ
dx22
= αC {−y22 − f (x21 + x22 ) + yC }dτ dy22 = x22dτ
where yC corresponds to iC and is given asµ¶y12 − y22
yC = γ y11 − y21 +αL(7)

and the nonlinear function f (·) which corresponds to the v −icharacteristics of the nonlinear resistors is given asµ¶
β 3 1 5f (x) = ε x − x + x .
35

where iC is the current through the coupling inductor and isgiven as
L1 (i11 − i21 ) + L2 (i12 − i22 )
iC =LC(6)(8)

3. Synchronization Phenomena(3)

Figure 2 shows computer simulated results. The coupledoscillators exhibit various synchronization patterns for differand the currents through the nonlinear resistors iR1 and iR2 ent initial conditions. We can see from Fig. 2(b) that x11 andx21 are synchronized at in-phase while x12 and x22 are synare given as
chronized at anti-phase. We also found that the oscillation(frequency of the pair x11 and x21 is different from that of theiR1 = iR (v11 + v12 )(4) other pair x12 and x22 .
.
iR2 = iR (v21 + v22 )We tried to give different sets of initial conditions to thecircuit in order to confirm the generation of all possible combinations of synchronization patterns. Figure 3 shows the obBy using the following variables and parameters,
served 9 different synchronization patterns. The upper figuresrshow the synchronization states between the upper resonators
rr
g1
g1
C1
while the lower figures show the synchronization states bevmn = 4
xmn , imn = 4
ymn ,
5g5
5g5 L1tween the lower resonators. So, for example, Fig. 3(a) shows
the generation of the state where the upper resonators are inC1L1
L1
αC =, αL =, γ=,(5) phase and the lower resonators are also in-phase. While, forC2L2LC
the case of Fig. 3(c), the uppers are in-phase but the lowers
rrp
are anti-phase. In Figs. 3(e) and (f), the lower resonators stop
L13g3
g1
ε = g1, β=, t = L1 C1 τ,their oscillations, while the uppers stop in Figs. 3(g) and (h).
C1
g1
5g5

- 148 -y11y21x11y12

4. Conclusions
In this study, we have investigated the generation of synchronization phenomena observed from two inductively coupled simultaneous oscillators. By computer simulations wehave confirmed that 9 different synchronization patterns appeared in the circuits.
Our future research topics include theoretical analysis andcircuit experiments, and the investigation for the cases ofmore than two resonators in each oscillator and more than
three coupled oscillators.

x21
y22

x12

Acknowledgement

x22

This work was partly supported by CASIO Science Promotion Foundation.

(a)Referencesx11[1] J. Schaffner, “Simultaneous oscillations in oscillators,”IRE Transactions on Circuit Theory, vol. 1, pp. 2–8, Jun.
1954.

x21

[2] M. Kuramitsu and F. Takase, “Averaged potential analysis of multimode oscillators with hard operating conditions,” IEICE Technical Report on NLP, vol. 81, no. 13,
pp. 1–10, Sep. 1981.

x12x22τ(b)[3] M. Matsuki and S. Mori, “Asynchronous simultaneous oscillations in negative resistance oscillatory circuit
containing periodically operating analog switch,” IEICE
Technical Report on CAS, vol. 101, pp. 81–87, Jun. 1993.

√Figure 2: Computer simulated results. αC = 1.0, αL = 2,
γ = 0.01, ε = 0.1 and β = 3.14. (a) Lissajous figures of [4] M. Matsuki and S. Mori, “Asynchronous excitation phenomena in negative resistance oscillatory circuit containfour resonators. Horizontal xmn and vertical ymn . (b) Timeing periodically operating analog switch,” IEICE Techniwaveforms of xmn .
cal Report on NLP, vol. 94, pp. 51–58, May 1994.

[5] C.W. Wu, Synchronization in complex networks of nonlinear dynamical systems, World Scientific, Oct. 2007.
Finally, in the case of Fig. 3(i), all the resonators stop oscillating. Please note that the upper pair and the lower pair[6] W. Wei, W. Zhou and T. Chen, “Cluster synchronizaoscillate asynchronously in Figs. 3(a)-(d).
tion of linearly coupled complex networks under pinning
The result is encouraging because our next target is the incontrol,” IEEE Transactions on Circuits and Systems I,
vestigation of the synchronization patterns when the numvol. 56, no. 4, pp. 829–839, Apr. 2009.
ber of resonators increases. We expect the generation ofmuch larger number of synchronization patterns, if all pos- [7] J. Lu, G. Chen and M. Bernardo, “On some recent adsible combinations appear even for the case of larger numbervances in synchronization and control of complex netof resonators. However, we should notice the fact that the osworks,” Proceedings of ISCAS’10, pp. 3773–3776, May
cillation amplitudes of the resonators become smaller when2010.
the asynchronous oscillations appear. This may means thatthe simultaneous oscillations of many resonators are not easy [8] J. Lu, X. Yu and D. Hill, “A brief overview of some recent advances in pinning control of complex networks,”
to be synchronized. The detailed investigation of such cases
Proceedings of NOLTA’10, pp. 398–401, Sep. 2010.
will be reported elsewhere.

- 149 -

x21

x21

x21x11(a)x12(b)
x21

x22x22x12x22x12(g)x11x11x22x12(f)x21x11x22x12(d)x21x11x22x12(c)x21x11x11x22

x22x12(e)x11x11x22

x21

x21

x12x12(h)(i)√Figure 3: Different synchronization patterns. αC = 1.0, αL = 2, γ = 0.01, ε = 0.1 and β = 3.14. Upper figures:
horizontal x11 and vertical y21 . Lower figures: horizontal x12 and vertical y22 .

[9] R.T. Canolty, E. Edwards, S.S. Dalal, M. Soltani, [12] C. Higashi and Y. Nishio, “Investigation of synchronizaS.S. Nagarajan, H.E. Kirsch, M.S. Berger, N.M. Barbarotion phenomena in coupled simultaneous oscillators,”
and R.T. Knight, “High gamma power is phase-locked toProceedings of NCSP’10, pp. 113–116, Mar. 2010.
theta oscillations in human neocortex,” Science, vol. 313,
[13] T. Suezaki and S. Mori, “Mutual synchronization ofno. 5793, pp. 1626–1628, Sep. 2006.
two oscillators,” Transactions of IECE, vol. 48, no. 9,[10] P. Lakatos, G. Karmos, A.D. Mehta, I. Ulbert andpp. 1551–1557, Sep. 1965.
C.E. Schroeder, “Entrainment of neuronal oscillations asa mechanism of attentional selection,” Science, vol. 320,no. 5872, pp. 110–113, Apr. 2008.
[11] G. Yu and J.-J. Slotine, “Visual grouping by neural oscillator networks,” IEEE Transactions on Neural Networks, vol. 20, no. 12, pp. 1871–1884, Dec. 2009.

- 150 -