2009 International Symposium on Nonlinear Theory and its Applications
NOLTA'09, Sapporo, Japan, October 18-21, 2009

Synchronization in Coupled van der Pol Oscillators
Involving Periodically Forced Capasitors
Hironori Kumeno† and Yoshifumi Nishio††Dept. of Electrical and Electronic Eng.,
Tokushima University,
Tokushima, 770-8506 JAPAN
Email: kumeno@ee.tokushima-u.ac.jp
Abstract—In this study, synchronization phenomena
observed from coupled van der Pol oscillators involving
periodically forced capacitors are investigated. Firstly, we
confirm effects of parametric excitation on synchronization. Next, we investigate behavior of parametric excitationin coupled system. By carrying out computer calculations
for two to five subcircuits cases, various interesting synchronization phenomena of chaos are confirmed.

Figure 1: Circuit model.
1. Introduction
Synchronization is one of the fundamental phenomena innature, and one of typical nonlinear phenomena. Therefore,studies on synchronization phenomena of coupled oscillators are extensively carried out in various fields, physics[1], biology [2], engineering and so on. However, issues
that should be investigated for synchronization remain inexistence in spite of many researching. In particular, itis necessary to investigate synchronization phenomena in
special conditions. There is parametric excitation that increases amplitude of oscillation by periodic changing of aparameter in the system. Parametric excitation circuit is
one of resonant circuits, and it is important to investigate
various nonlinear phenomena of the parametric excitation
circuits for future engineering applications. In simple oscillator including parametric excitation, Ref. [3] reports thatthe almost periodic oscillation occurs if nonlinear inductor
has saturation characteristic. Additionally the occurrenceof chaos is referenced in Refs. [4] and [5].
In the past we have investigated synchronization phenomena in coupled van der Pol oscillators involving timevarying inductors [1]. In this study, for specify the effectof parametric excitation on synchronization, we focus ona parametric excitation which is generated by a capacitor
periodically forced. Then, we investigate synchronization
phenomena observed from coupled van der Pol oscillators
involving periodically forced capacitors. By carrying out
computer calculations for two to five subcircuits cases, various interesting synchronization phenomena of chaos are
confirmed.

Figure 2: Subcircuit corresponds to van der Pol oscillator
involving periodically forced capasitor.
2. Circuit model
The circuit model used in this study is shown in Fig 1.
In our system n identical parametrically excited van der
Pol oscillators are coupled by one resistor R. The subcircuit which is parametrically excited van der Pol oscillator
consists of an inductor, a nonlinear resistor and a timevarying capacitor, which is periodically forced, and realized as Fig. 2. This circuit exhibits bifurcation phenomena
whose diagram is shown in Fig. 3. Figure 4 shows examples of attractors obtained from the subcircuit. In Fig. 2, thedashed-line box area corresponds to the time-varying capacitor which exhibits periodic rectangular characteristics
by an externally switch which gives periodic force. The
characteristics of the time-varying capacitor are given as
following equation.
C = C0 γ(t).

(1)

γ(τ) is expressed in a rectangular wave as shown in Fig. 5,
and its amplitude and angular frequency are termed α and- 226 -

Figure 5: Function relating to parametrically excitation.
Figure 3: One-parameter bifurcation diagram for α = 0.8,
ω = 1.1 and varying ε.

(a)

(b)

(c)
Figure 4: Attractors and Poincaré maps. α = 0.8 and ω =1.1. (a) ε = 0.10. (b) ε = 0.12. (c) ε = 0.75.
ω, respectively. The v − i characteristics of the nonlinear
resistor are approximated by the following equation.
id = −g1 vk + g3 vk .

(2)

By changing the variables and the parameters,r
g1
C
t = L0Cτ, vk =xk , δ =
R,
g
L
3
0
r
r r
g1
C
L0
ik =
yk , ε = g1
,
g3 L0
C
p

r

3. Synchronous effects of parametric excitation
Before investigating synchronization phenomena in the
coupled parametrically excited van der Pol oscillators, itis necessary to investigate synchronization phenomena in
non-coupled chaotic circuits which are identically forced.
Namely, it is necessary to investigate synchronous effects
that all capacitors of the chaotic circuits are changed at thesame time. In this section, for investigate the synchronous
effects we focus on the two identical chaotic circuits. Figure 6 shows chaotic attractors, time series and rectangularwave corresponding to the periodically forced capacitor. In
Fig. 6, a chaotic attractor, shown as Fig. 6(a), consists ofupper rotationg orbit (Fig. 6(b)) and lower rotating orbit(Fig. 6(c)) that corresponding to the region (b’) and the region (c’) in time series, respectively. From this figure, aperiod of the chaotic subcircuit is determined by the periodic force and according to its period. Additionally, theupper rotating orbit and the lower rotating orbit are symmetric about the coordinate origin. From the above things,
the periodically forced chaotic circuits can be synchronizedat the in phase or at the opposite phase or be self-switchingat these in and opposite phase and it is decided by overlapping of the upper rotating areas and lower rotating areas between the two chaotic subcircuits. It means, sampling synchronization states every periods, the chaotic subcircuits
which the capacitors are governed by one force seem to besynchronized at the in-phase or at the opposite-phase.

(3)

the normalized circuit equations are given by the following
equations.

o
dxk
1 n



=
ε(xk − xk3 ) − yk



dτ
γ(τ)



n

X


dyk



=
x
−
δ
y j.
k


 dτ
j=1

(a)

(b)

(c)

(4)
(b’)

In the following computer calculations, the parametervalues are fixed as ε = 0.75, α = 0.80 and ω = 1.10 and (4)is calculated by using the Runge-Kutta method with step
size ∆t = 0.01.

(c’)

Figure 6: Attractors, time series and rectangular wave relating to parametrically excitation ε = 0.75, α = 0.80 andω = 1.10.

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Figure 7: Three types of synchronization by parametric excitation.
4. Synchronization phenomena
In this section, we investigate synchronization phenomena by carrying out computer calculations for two to fivesubcircuits cases.
4.1. Two subcircuits caseIn this subsection, we consider the case of N = 2,
namely only two van der Pol oscillators involving periodically forced capacitors are coupled by one resistor. In thiscase, opposite-phase synchronization of two subcircuits isobserved. Figure 8 shows computer calculated results. As
shown in the figure, with γ increases, chaotic signals obtained from two subcircuits become to be synchronized at
the opposite-phase. When γ = 0.01, two chaotic signals
seem to be completely synchronized at the opposite-phase.

one. Figure 10 shows a computer calculated simulation forobserved attractors in some steady synchronization states
when initial values of two subcircuits are set as the samevalues and initial values of the rest subcircuit is set as thealmost inverse about the coordinate origin. In the figure,two attractors synchronized at the in-phase are also smaller
than the other one.
The coupled chaotic circuits used in this study are expected to minimize the current through the coupling resistor. Additionally, subcircuits are synchronized at the inphase or the opposite-phase by the effect of parametric excitation. Thus, the two attractors synchronized at the inphase are small for minimizing current through the coupling resistor. However, chaos attractor does not have periodic orbit. Additionally, three subcircuits can not be synchronized completely without the special case that all initial values are same. Thus sometimes two solutions of the
subcircuits synchronized at the opposite-phase approacheach other. In that time, synchronization states switch toother synchronization state.

Figure 9: Self-switching phenomena of in-phase synchronization and opposite-phase synchronization. ε = 0.75,
α = 0.80, ω = 1.10 and γ = 0.10.

(a)

(b)
Figure 8: Synchronization of two chaos. ε = 0.75, α =0.80 and ω = 1.10. (a) γ = 0.001. (b) γ = 0.01.
4.2. Three subcircuits caseIn this subsection, we consider the case of N = 3. In
this case, self-switching phenomena of in-phase synchronization and opposite-phase synchronization can be observed. Figure 9 shows the computer calculated result. In
Fig. 9, the upper figure, the middle figure and the lower
figure show attractors of the subcircuits, phase differences
between subcircuits and time series of the subcircuits, respectively. As shown in time series of Fig. 9, the synchronization states which subcircuits are synchronized at thein-phase or opposite-phase are switching with time.
Three subcircuits generate the same shape of chaotic attractors in Fig. 9 for long time observation. However, inshort time observation during synchronization states are
steady, the attractors are not the same. Two attractors
synchronized at the in-phase are smaller than the otherFigure 10: Synchronization of three chaos when initial values of two subcircuits are set as same values. ε = 0.75,
α = 0.80, ω = 1.10 and γ = 0.10.
4.3. Four subcircuits caseIn this subsection, we consider the case of N = 4. In
this case, two pairs of the opposite-phase synchronization
can be observed. Figure 11 shows the computer calculated results. In Fig. 11, (a) and (b) are obtained by using different initial values. In Fig. 11(a), two of four subcircuits are synchronized at the opposite phase. Also theremain two subcircuits are synchronized at the opposite
phase. Then, two opposite-phase synchronizations switch- 228 -

between in and opposite phases. While, in Fig. 11(b),the synchronization state between two opposite-phase synchronizations is steady. For instance, subcircuits 2 and 3
seem to be synchronized at the in-phase. The synchronization states of four coupled subcircuits become either theabove two state because of parametric excitation that all
subcircuits are synchronized at in-phase or opposite-phaselocally.

(a)

(a)

(b)
(1)
(2)
(3)
(4)
Figure 11: Synchronization of four chaos. ε = 0.75, α =0.80, ω = 1.10 and γ = 0.70. (1) x1 versus y1 . (2) y1 versus
y2 . (3) y2 versus y3 . (4) y3 versus y4 .
4.4. Five subcircuits caseIn this subsection, we consider the case of N = 5. Figure 12 shows the computer calculated results. In this case,two kinds of synchronization phenomena are observed.
The one is self-switching phenomena of all subcircuits (seeFig. 12(a)). It can be observed when coupling intensity γ
is small. In Fig. 12(a), two subcircuits are synchronizedat the in-phase, while the remain three subcircuits are synchronized at the in-phase. Then, the above two groups aresynchronized at the opposite-phase. Though, as time advances one of the subcircuits which belongs to the threesubcircuits group switch to another group. By the way, increasing coupling intensity, another synchronization can beobserved as shown in Fig. 12(b). In Fig. 12 (b), subcircuits 1 and 5 are synchronized at the in-phase and subcircuits 2 and 4 are also synchronized at the in-phase. Then,the above two pairs are synchronized at the opposite-phase.
Besides, synchronization states between remain subcircuitand others switch from the in-phase to the opposite-phaseor vice versa with time.
5. Conclusions
In this study, we have investigated synchronization phenomena in coupled van der Pol oscillators involving periodically forced capacitors. Firstly, we confirm effects of parametric excitation on synchronization that all subcircuits aresynchronized at in-phase or opposite-phase locally. Next,
we investigate behavior of parametric excitation in coupledsystem. By carrying out computer calculations for two
to five subcircuits case, various interesting synchronization phenomena of chaos are confirmed. In two subcircuits
case, opposite-phase synchronization is observed. In threesubcircuits case, self-switching of in-phase and oppositephase synchronization is observed. In four subcircuits case,(b)
Figure 12: Synchronization of five chaos. ε = 0.75, α =0.80 and ω = 1.10. (a) γ = 0.17. (a) γ = 0.50.
two pairs of opposite-phase synchronization is observed. In
five subcircuits case, two different types of synchronization
phenomena are observed. One of the synchronization phenomena is self-switching of all subcircuits. Another one isin and opposite phase synchronization and self-switching.
Acknowledgments
This work was partly supported by Yazaki Memorial
Foundation for Science and Technology.
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