2007 IEEE Workshop on Nonlinear Circuit Networks
NCN'07, Tokushima, Japan, Dec 7--8, 2007

Parametrically Excited van der Pol Oscillators
Coupled by a Resistor
Hironori Kumeno

Yoshifumi Nishio

Dept. of Electrical and Electronic Eng.,
Tokushima University,
Tokushima 770-8506, JAPAN
Email: kumeno@ee.tokushima-u.ac.jp

Dept. of Electrical and Electronic Eng.,
Tokushima University,
Tokushima 770-8506, JAPAN
Email: nishio@ee.tokushima-u.ac.jp

Abstract— In this study, we investigate synchronization ofparametrically excited van der Pol oscillators. By carrying out
computer calculations for two or three subcircuits case, weconfirm that various kinds of synchronization phenomena ofchaos are observed. In the case of two subcircuits, we confirm
that synchronization phenomena are related to phase differenceof the functions corresponding to the parametric excitation. Thenthe two subcircuits are synchronized at the opposite-phase. In thecase of three subcircuits, we confirm self-switching phenomenonof synchronization states when there is not phase difference ofthe functions corresponding to the parametric excitation. On theother hand, when there is phase difference, two of the threesubcircuits are synchronized at the opposite-phase.

I. I NTRODUCTION
Synchronization is one of the fundamental phenomena innature and it can be described by nonlinear oscillators thathas regular and chaotic states. Synchronization of van der
Pol oscillator is one of synchronization of natural rhythm
phenomena, and many researchers studied it. 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 thisstudy, we investigate synchronization of parametrically excited
van der Pol oscillators.
II. C IRCUIT MODEL
The circuit model used in this study is shown in Fig 1.
In our system n same parametrically excitaed van der Poloscillators are coupled by one resistor R. The circuit includes
a time-varying inductor L whose characteristics are given as
the following equation, and are shown as Fig. 1(b).
L = L0 γ(t).

(1)(a) Parametrically excited van der Pol oscillators coupled by aresistor.

(b) Time-varying inductor.
Fig. 1. Circuit model.

γ(τ ) is expressed in a rectangular wave as shown in Fig. 2,and its amplitude and angular frequency are termed α and ω.
The v − i characteristics of the nonlinear resistor are approximated by the following equation.
id = −g1 vk + g3 vk .
By changing the variables andrthe parameters,rpg1
Ct = L0 Cτ, vk =xk , δ =R,
gL
3 r
0r r
g1 C
L0yk , ε = g1,
ik =
g3 L0
C

(2)

(3)
Fig. 2.

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Rectangular wave.

the normalized circuit equations are given by the followingequations.
dxk
= ε(xk − x3k ) − yk dτn(4)
X1
 dyk

y.
=x−δkj dτγ(τ )j=1(a)(b)(c)
Fig. 3. One-parameter bifurcation diagram for rectangular wave. Horizontal
axis: x. Vertical axis: ε. α = 0.95 and ω = 1.50.

Figure 3 shows the bifurcation diagram observed fromthe isolated subcircuit. When parameter ε changes, periodicattractors, quasi-periodic attractors and chaotic attractors areconfirmed from the isolated subcircuit. Figure 4 shows examples of chaotic attractors and Poincaré maps observed fromthe isolated subcircuit. We define the Poincaré section as“ωτ = 2nπ”.

(d)(1)(2)(3)

Fig. 5. Attractors and phase differences. ε = 1.50, α = 0.95, ω = 1.50
and δ = 0.80. Phase difference of rectangular wave: (a) 0, (b) π/20, (c) π/4
and (d) π. (1) x1 versus y1 . (2) x2 versus y2 . (3) y1 versus y2 .

Fig. 5(a), when there is not phase difference, two subcircuits
are synchronized at the opposite-phase completely. However,as phase difference increases, two circuits become out ofsynchronization (see Figs. 5(b) and (c)).
Second, fix ε = 1.35, α = 0.95, ω = 1.50 and δ = 0.80 andvary phase difference of the rectangular wave. Two subcircuits
generate periodic attractor for there parameter values. Figure 6
shows computer calculated results. In the same way, when
there is not phase difference, two subcircuits are synchronizedat the opposite-phase completely as shown in Fig. 6(a). And
as phase difference increases, two circuits become out ofsynchronization (see Figs. 6(b), (c) and (d)). Additionally,
chaotic attractor is obtained as shown in Fig. 6(b). In spiteof isolated subcircuit conditions, chaotic attractors generation
or disappearance are obtained by phase difference.

(a)

(b)
Fig. 4. Examples of chaotic attractors and Poincaré maps observed fromeach subcircuit, α = 0.95 and ω = 1.50. (a) ε = 1.0. (b) ε = 1.5.

III. T WO SUBCIRCUITS CASE
In this section, we consider the case of n = 2. Only two
parametrically excited van der Pol oscillators are coupled by
one resistor. First, fix ε = 1.50, α = 0.95, ω = 1.50 andδ = 0.80 and vary phase difference of the rectangular wave.
Two subcircuits generate chaos for these parameter values.
Figure 5 shows computer calculated results. As shown in- 47 -IV. T HREE SUBCIRCUITS CASE
In this section, we consider the case of n = 3. Figure 7
shows computer calculated results for ε = 1.50, α = 0.95,
ω = 1.50, δ = 0.80 and there is not phase difference of therectangular wave. In this case, self-switching phenomenon ofsynchronizations is observed. As shown in Fig. 7(a), subcircuits 1 and 2 are synchronized at opposite-phase, subcircuits
1 and 3 are synchronized at opposite-phase. However, as timeadvances, pattern of synchronization changes. As shown in the

(a)

(b)

(a)

(c)

(d)
(1)

(2)

(b)

(3)

Fig. 6. Attractors and phase differences. ε = 1.35, α = 0.95, ω = 1.50
and δ = 0.80. Phase difference of rectangular wave: (a) 0, (b) π/20, (c) π/4
and (d) π. (1) x1 versus y1 . (2) x2 versus y2 . (3) y1 versus y2 .

Fig. 7(b), subcircuits 1 and 2 are synchronized at oppositephase, subcircuits 2 and 3 are synchronized at opposite-phase.
Furthermore, as time passes subcircuits 1 and 3 are synchronized at opposite-phase, subcircuits 2 and 3 are synchronizedat opposite-phase (see Fig. 7(c)). When the subcircuit parameters are regarded as constant, switching speed is related tothe coupling parameter δ. If the coupling parameter δ is small,bonding force is weak. So switching speed is fast. As the coupling parameter δ increases, switching speed becomes slow.
Additionally, self-switching phenomenon of synchronizations
is also confirmed when isolated subcircuits generate periodic
attractors for ε = 1.35. However, when the parameter ε issmall, self-switching phenomenon of synchronizations is notobserved. A pair of subcircuits are synchronized at oppositephase that are decided by initial values. Thus, generation of
the self-switching phenomenon of synchronizations is related
to the parameter ε, and its switching speed is related to thecoupling parameter δ. By comparison of attractors generated
from the isolated subcircuit, local amplitude of the attractor
for ε = 1.5 is larger than it for ε =1.0. We believe that thegeneration of the self-swiching phenomenon is related to thischange of the local amplitude.
Second, we consider the case that there is phase differenceof the rectangular wave. Fix ε = 1.50, α = 0.95, ω = 1.50
and δ = 0.30, and shift phases of the rectangular wave of
the subcircuits to 2π/3 and 4π/3. In this case, self-switching- 48 -(c)Fig. 7. Attractors and phase differences. ε = 1.50, α = 0.95, ω = 1.50
and δ = 0.80.

phenomenon of synchronizations is not observed. Figure 8
shows that three different types of synchronization states.
These three synchronization states can be obtained by giving
different pattern of phase shift of the rectangular wave. Two of
the three subcircuits are synchronized at the opposite-phase.
A pair of synchronized subcircuits is decided by the sequenceof phase shift of the rectangular wave.
V. C ONCLUSIONSIn this study, we investigated synchronization of parametrically excited vander Pol oscillators. By carrying out
computer calculations for two or three subcircuits case, weconfirmed that various kinds of synchronization phenomenaof chaos were observed. In the case of two subcircuits, weconfirmed that synchronization phenomena are related to phasedifference of the functions that corresponding to the parametric excitation. Then the two subcircuits are synchronizedat the opposite-phase. In the case of three subcircuits, justthree coupling van der Pol oscillators are synchronized at
the three-phase. However, three coupling parametric excited
van der Pol oscillators generate self-switching phenomenonof synchronization states when there is not phase differenceof the the functions corresponding to the parametric excitation.
On the other hand, when there is phase difference, two of thethree subcircuits are synchronized at the opposite-phase.
ACKNOWLEDGMENT
This work was partly supported by Yazaki Memorial Foundation for Science and Technology.
R EFERENCES[1] C. Hayashi, “Nonlinear Oscillations in Physical Systems,” Chap. 11,
McGraw-Hill, New York (1964).
[2] 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).
[3] 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.

(a)

(b)

(c)
Fig. 8. Attractors, phase differences and time series. ε = 1.50, α = 0.95,
ω = 1.50 and δ = 0.30. Phase differences of rectangular wave are 2π/3 and4π/3.

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