2008 International Symposium on Nonlinear Theory and its Applications
NOLTA'08, Budapest, Hungary, September 7-10, 2008

Synchronization Phenomena
in Coupled Parametrically Excited van der Pol Oscillators
Hironori Kumeno† and Yoshifumi Nishio††Dept. of Electrical and Electronic Eng.,
Tokushima University, Tokushima 770-8506, JAPAN
Email: kumeno@ee.tokushima-u.ac.jp, nishio@ee.tokushima-u.ac.jp
Abstract—In this study, we investigate synchronizationof parametrically excited van der Pol oscillators. In thecase of two subcircuits, we confirm that the two subcircuits are synchronized at the opposite-phase without phasedifference in the functions corresponding to the parametric excitation. In the case of three subcircuits, we confirm
self-switching phenomenon of synchronization states.
1. Introduction
Synchronization 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 canmodel certain synchronization of natural rhythm phenomena. The van der Pol oscillator is studied well because itis expressed in simple circuit. 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 Ref. [4] and [5].
In this study, we investigate synchronization of parametrically excited van der Pol oscillators. By carrying out
computer calculations for two or three subcircuits case, weconfirm that various kinds of synchronization phenomenaof chaos are observed. In the case of two subcircuits, theopposite-phase synchronization is observed. In the case of
three subcircuits, self-switching phenomenon of synchronization states is observed.
2. Circuit modelThe circuit model used in this study is shown in Fig 1.
In our system n same parametrically excited van der Pol
oscillators are coupled by one resistor R. The circuit includes a time-varying inductor L whose characteristics are(a) Parametrically excited van der Pol oscillators
coupled by a resistor.

(b) Time-varying inductor.

Figure 1: Circuit model.
given as the following equation. The time-varying inductor
is shown as Fig. 1(b).
L = L0 γ(t).
(1)
γ(τ) is expressed in a rectangular wave as shown in Fig. 2,and its amplitude and angular frequency are termed α andω, respectively. The v − i characteristics of the nonlinear
resistor are approximated by the following equation.
id = −g1 vk + g3 vk .
By changing the variables and the parameters,
rr
p
g1
C
t = L0Cτ, vk =xk , δ =R,
g3 r
L0r r
g1
C
L0
ik =yk , ε = g1,
g3 L0
C

(2)

(3)

the normalized circuit equations are given by the following
equations.

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(a)Figure 2: Function relating to parametrically excitation.

(b)

(c)
(1)

Figure 3: One-parameter bifurcation diagram of isolated
subcircuit. Horizontal axis: x. Vertical axis: ε. α = 0.95
and ω = 1.50.
dxk



= ε(xk − xk3 ) − yk
dτ


n(4)
X

dyk
1



=xk − δy j.



γ(τ)
 dτ
j=1(2)(3)

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

Figure 5 shows computer calculated results. As shown in
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 of synchronization (see Figs. 5(b) and (c)).

Figure 3 shows the bifurcation diagram observed fromthe isolated subcircuit. When parameter ε changes, periodic attractors, quasi-periodic attractors and chaotic attractors are confirmed from the isolated subcircuit. Figure 4 shows examples of chaotic attractors and Poincarémaps observed from the isolated subcircuit. We define thePoincaré section as ωτ = 2nπ.

(a)(b)(a)
(c)
(1)

(b)
Figure 4: Examples of chaotic attractors and Poincaré maps
observed from subcircuit. α = 0.95 and ω = 1.50. (a)ε = 1.0. (b) ε = 1.5.
3. Two subcircuits caseIn this section, we consider the case of n = 2. Only two
parametrically excited van der Pol oscillators are coupledby 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.

(2)

(3)

Figure 6: Attractors and phase differences. ε = 1.35,
α = 0.95, ω = 1.50 and δ = 0.80. Phase difference ofrectangular wave: (a) 0, (b) π/20 and (c) π/4. (1) x1 versus
y1 . (2) x2 versus y2 . (3) y1 versus y2 .
Second, fix ε = 1.35, α = 0.95, ω = 1.50 and δ = 0.80
and vary phase difference of the rectangular wave. Twosubcircuits generate periodic attractor for there parametervalues. Figure 6 shows computer calculated results. In
the same way, when there is not phase difference, two subcircuits are synchronized at the opposite-phase completely
as shown in Fig. 6(a). And as phase difference increases,
two circuits become out of synchronization (see Figs. 6(b),(c) and (d)). Additionally, chaotic attractor is obtained asshown in Fig. 6(b).

- 129 -4. Three subcircuits caseIn 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.

(1)

(2)
Figure 8: time series of self-switching. ε = 1.50, α = 0.95
and ω = 1.50. (1)δ = 0.80. (2)δ = 20.00.

(a)shows time series of amplitude difference between subcircuits. In Fig. 10, large amplitude shows opposite-phasesynchronization. If the coupling parameter δ is small (seeFig. 10(a)), switching speed is fast. As the coupling parameter δ increases, switching speed becomes slow (seeFig. 10(b)). Additionally, self-switching phenomenon ofsynchronizations is also confirmed when the isolated subcircuits generate periodic attractors for ε = 1.35. However,when the parameter ε is small as ε = 1.00, self-switching
phenomenon of synchronizations is not observed. And apair of subcircuits which depend on initial values are synchronized at opposite-phase. Thus, generation of the selfswitching phenomenon of synchronizations is related to theparameter ε, and its switching speed is related to the coupling parameter δ.

(b)(c)
Figure 7: Attractors and phase differences. ε = 1.50, α =0.95, ω = 1.50 and δ = 0.80.
In this case, self-switching phenomenon of synchronizations is observed. In Fig. 7, the upper line shows attractors of the subcircuits and the lower line shows phase differences between subcircuits. And the states of synchronization (a), (b) and (c) switch with time. Figure 8 showstime series of self-switching phenomenon of synchronizations. As shown in Fig. 7(a) and Fig. 8(a), subcircuit 2
synchronizes with other two subcircuits at opposite-phase.
However, as time advances, the amplitude of either subcircuit 1 or subcircuit 3 becomes small (it is subcircuit
3 in Fig. 8(2)), and the pattern of synchronization finallychanges to another pattern. After the switching, the othersubcircuit which synchronized to subcircuit 2 at oppositephase becomes to synchronizes with the other two subcircuits at opposite-phase as shown in the Fig 7(b) andFig. 8(b). Furthermore, as time passes next switchings occur in a chaotic way.
Switching speed is related to the coupling parameter
δ. Figure 9 shows the mean value and the standard deviation of the sojourn time of self-switching. Figure 10From the preceding section, two subcircuits are synchronized at opposite-phase for no phase difference of the rectangular wave. Thus, the effect of parametric excitation isstrong, and opposite-phase synchronization is stable for nophase difference of the rectangular wave. However in caseof three subcircuits, there is no stable pair of subcircuits because the number of circuit is odd. So self-switching phenomena of synchronization occurs. From these, we consider that interesting self-switching phenomena of synchronization occur when the number of subcircuit is odd (forexample the number of subcircuits is 5 or 7).
Second, we consider the case that there is phase difference of the rectangular wave. Fix ε = 1.50, α = 0.95,
ω = 1.50 and δ = 0.30, and phase shifts of the rectangular wave of the subcircuits to 2π/3 and 4π/3. In this case,self-switching phenomenon of synchronizations is not observed. Figure 11 shows one of three different types ofsynchronization 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 sequence of phase shift of therectangular wave.

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(a)Figure 11: 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 and 4π/3.
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.
Acknowledgments(b)
Figure 9: Sojourn time of self-swiching. Horizontal axis:
δ. Vertical axis: period. ε = 1.50, α = 0.95 and ω = 1.50.
(a) Mean value. (b) Standard deviation.

This work was partly supported by Yazaki Memorial
Foundation for Science and Technology.
References[1] I. Belykh, M. Hasler, M. Lauret and H. Nijmeijer,“Synchronization and graph topology,” Int. J. Bifurcation and Chaos, vol.15, no.11, pp.3423-3433, Nov.
2005.

(a)(b)
Figure 10: Time series. ε = 1.50, α = 0.95 and ω = 1.50.
(a) δ = 0.80. (b) δ = 20.0.
5. Conclusions
In this study, we investigated synchronization of parametrically excited van der Pol oscillators. By carrying out
computer calculations for two or three subcircuits case, weconfirmed that various kinds of synchronization phenomena of chaos were observed. In the case of two subcircuits,
we confirmed that synchronization phenomena are related
to phase difference of the functions that corresponding tothe parametric excitation. Then the two subcircuits are synchronized at the opposite-phase. In the case of three subcircuits, three coupling van der Pol oscillators which do notinclude parametric excitation are synchronized at the threephase. However, three coupling parametric excited van der
Pol oscillators generate self-switching phenomenon of synchronization states when there is not phase difference of the[2] J. Cosp, J. Madrenas, E. Alarcon, E. Vidal and G. Villar, “Synchronization of 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 of mapping 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 Bifurcationof the Almost Periodic Oscillation and the Generationof Chaos in a Parametric Excitation Circuit,” Trans. ofIEICE, vol. J68-A, no. 7, pp. 621-626, 1985.

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