IEEE Workshop on Nonlinear Circuit Networks
December 12-13, 2014

Synchronization Phenomenaof Parametrically Excited Oscillatorswith Small Mismatch in Random Network
Kosuke Ooi, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: {ooi,uwate,nishio}@ee.tokushima-u.ac.jp
Abstract—In this study, we investigate synchronization phenomena of parametrically excited van der Pol oscillators withsmall mismatch. By increasing the small mismatch, we observe
unsynchronous phenomena. Furthermore, we apply this circuit
model to ten coupled oscillators as random network model.

In this study, we investigate synchronization of this circuit
model to ten coupled oscillators as random network model
without hub.

I. I NTRODUCTION

UNDER PARAMETRIC EXCITATION

Synchronization phenomena is one of the fundamental phenomena in nature and it is observed over the various fields.
Studies on synchronization phenomena of coupled oscillators
are extensively carried out in various fields, physics [1],biology [2], engineering and so on. We consider that it isimportant to investigate the synchronization phenomena ofcoupled oscillators for the future engineering application. The
coupled van der Pol oscillator is one of coupled oscillators,
and synchronization generated in the system can model certain
synchronization of natural rhythm phenomena. The van der
Pol oscillator is studied well because it is expressed in simple
circuit. Parametric excitation circuit is one of resonant circuits,
and it is important to investigate various nonlinear phenomenaof the parametric excitation circuits for future engineering applications. In simple oscillator including parametric excitation,
Ref. [3] reports that the almost periodic oscillation occurs if
nonlinear inductor has saturation characteristic. Additionallythe occurrence of chaos is referenced in Refs. [4] and [5].
In our previous study, we have investigated synchronizationof parametrically excited van der Pol oscillators [6]. By
carrying out computer calculations for two or three subcircuits
case, we have confirmed that various kinds of synchronization phenomena of chaos are observed. In the case of twosubcircuits, the anti-phase synchronization is observed. Inthe case of three subcircuits, self-switching phenomenon ofsynchronization states is observed. Furthermore, we investigate
synchronization of parametrically excited van der Pol oscillators with small mismatch. The small mismatch is added to theamplitude of the function relating to parametrically excitation.
In the case of two subcircuits, we confirm that the twosubcircuits are synchronized at in-phase state when the adding
mismatch is small. By increasing the small mismatch, weobserve unsynchronous phenomena. Additionally, we applied
this circuit model to ten coupled oscillators as random networkmodel with hub [7].

II. VAN DER P OL OSCILLATOR
The circuit model of van der Pol oscillator under parametric
excitation is shown in Fig. 1.

iidnCv

L(a) Parametrically excited van der Pol oscillator(b) Time-varying inductor
Fig. 1. Circuit model.

The circuit includes a time-varying inductor L whose characteristics are given as the following equation. The timevarying inductor is shown as Fig. 2.

- 49 -L = L0 γ(τ ).

(1)

( )

921+

3181
70/

2/

Fig. 3. A random network model without hub.

γ(τ ) is expressed in a rectangular wave as shown in Fig. 2,
and its amplitude and angular frequency are termed α and ω,respectively. The v − i characteristics of the nonlinear resistor
are approximated by the following equation.

By changing the variables and the parameters,
√
√
g1

LCτ,
v
=xnt=
0n

g3
√ω=
ωL0 C0√ √g1 C
i=ynn

g3 L0√√LL10 ε = g1, δ=CR C

R5Fig. 2. Function relating to parametrically excitation.

id = −g1 vk + g3 vk .

64

101-TABLE I
SMALL MISMATCH

small mismatch
α1
α2
α3
α4
α5
α6
α7
α8
α9
α10(2)

w=1
0-0.009-0.006-0.001
0.009-0.00200-0.001
0.005

w=2-0.001
0.008-0.006-0.0050.007-0.01-0.001-0.004-0.01
0.009

w=30.009
0.008
0.009
-0.01
-0.01
-0.004-0.0030.005
0.009-0.009

w=4
0.003
-0.01
0.009
-0.0030.001-0.0020.007-0.006-0.01-0.009w=5-0.006-0.0090.01-0.007-0.001
0.004
0.009
0.007
0.005
-0.001(3)

The normalized circuit equations are given by the following
equations.
∑dxn


(xk − xn )= ε(xn − x3n ) − yn + δ
 dτ
k∈Sn
(4)
dyn1


=xndτ
γ(τ )

line denotes synchronous area and the upper area of each line
denotes unsynchronous area. By increasing the value of thecoupling strength, the synchronous area becomes large. We
can see that the synchronous area is determined by the small
mismatch pattern. Namely, several types of synchronization
states can be observed from ten coupled oscillators with thesmall mismatch.

where n = 1, 2, 3, ..., 10. Sn is the set of nodes which are
directly connected to the node n.
III. R ANDOM N ETWORK C ASE
We investigate synchronization phenomena of ten coupledoscillators as random network model without hub. Ten coupledoscillator model is shown in Fig. 3. In this figure, the model
does not have hub. Here, the small mismatch is added to α
which is corresponding to the amplitude of the function relating to parametrically excitation. The mismatch is generated byrandom and the range of the mismatch is set to [-0.01:0.01].
Table I shows the pattern of five types of the small mismatch
used in this computer simulations.
Figure 4 shows the simulation results. The horizontal axis
denotes the coupling strength δ, and the vertical axis denotes
α of the second oscillator. In this graph, the lower area of each- 50 -Fig. 4. Synchronous state with δ.

IV. C ONCLUSIONSIn this study, we have investigated synchronization phenomena of parametrically excited van der Pol oscillators with small
mismatch. In the case of we have investigated this circuit
model to ten coupled oscillators as random network model
without hub.
For the future work, we would like to consider the influenceof the small mismatch for synchronization state of more large
scale networks or different types of network.
R EFERENCES[1] I. Belykh, M. Hasler, M. Lauret and H. Nijmeijer, “Synchronization andgraph topology,” Int. J. Bifurcation and Chaos, vol.15, no.11, pp.34233433, Nov. 2005.
[2] J. Cosp, J. Madrenas, E. Alarcon, E. Vidal and G. Villar, “Synchronizationof nonlinear electronic oscillators for neural computation,” IEEE Trans.
Neural Networks, vol.15, no.5, pp.1315-1327, Sep. 2004.
[3] C. Hayashi, “Nonlinear Oscillations in Physical Systems,” Chap. 11,
McGraw-Hill, New York (1964).
[4] C. Hayashi, M. Abe, K. Oshima and H. Kawakami, “The method ofmapping as applied to the solution for certain types of nonlinear differential equations,” Ninth International Conference on Nonlinear Oscillations,
Kiev (Aug.-Sept.1981).
[5] M. Inoue, “A Method of Analysis for the Bifurcation of the Almost Periodic Oscillation and the Generation of Chaos in a Parametric Excitation
Circuit,” Trans. of IEICE, vol. J68-A, no. 7, pp. 621-626, 1985.
[6] H. Kumeno, Y. Nishio, “Synchronization Phenomena in Coupled Parametrically Excited van der Pol Oscillators,” Proc. NOLTA’08, pp. 128-131,
Sep. 2008.
[7] Kosuke Ooi, Yoko Uwate and Yoshifumi Nishio,“Synchronization and
Clustering in Coupled Parametrically Excited Oscillators with Small
Mismatch,” Proc. ISCAS’15 (submitted).

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