2013 International Workshop on Nonlinear Circuits,Communications and Signal Processing
NCSP'13, Isrand of Hawaii, Hawaii, USA. March 4-7, 2013

Synchronization Phenomena in Coupled
van der Pol Oscillators with Nonlinearity Errors
Kentaro Kaishima, Takahiro Nagai, Yoko Uwate and Yoshifumi Nishio
Department of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima-shi, Tokushima, 770-8506, Japan
E-mail: {kaishima, nagataka, uwate, nishio}@ee.tokushima-u.ac.jp

pending on the error parameters. By using computer simulations, we focus on the phase difference between the coupled
In this study, we investigate synchronization phenomena ob- oscillators in each group. We observe interesting synchroserved in completely coupled ten van der Pol oscillators. Ten nization phenomena when the nonlinearity error is changed.
oscillators are divided into three groups. Each group consistsof two, three and five oscillators. When the nonlinearity ofeach oscillator has different values, we observe the interest- 2. Circuit model
ing synchronization phenomena in the circuit system.
Figure 1 shows the conceptual circuit model of our investigation. Ten van der Pol oscillators are completely coupled
1. Introduction
and divided into three groups. Group1 is composed of 1stSynchronization phenomena can be observed everywhere and 2nd oscillators. Group2 is composed of 3rd, 4th and 5thin natures such as firefly luminescences, cry of frogs, car- oscillators. Other oscillators belong to group3.
Abstract

diac heartbeat and so on. Coupled oscillators are often used
to investigate the synchronization phenomenon. Also studies on synchronization phenomena of coupled oscillators areextensively carried out in various fields, physics, biologies,engineerings and so on [1], [2]. Therefore, many types of
the coupled circuit models using van der Pol oscillators areproposed to understand synchronization phenomena. Endo etal. have reported the details of a theoretical analysis and thecorresponding circuit experiments on electrical oscillators arranged in a ladder, a ring and a two-dimensional array topologies [3]-[5]. Moreover, coupled oscillatory systems can also
produce interesting phase patterns, including wave propagation, clustering and complex patterns. Setou et al. haveobserved interesting synchronization phenomena (oscllation
death, independent oscillation and double mode oscillation)
when van der Pol oscillators with different frequencies arecoupled by means of a resistor in a star topology [6]. We assume that any frustrations such as different frequencies haveone possibility to produce interesting synchronization phenomena in oscillatory systems. However, there are not many
discussions the coupled oscillatory systems with nonlinearity
errors.
In this study, we investigate synchronization phenomena
observed in completely coupled ten van der Pol oscillators,
when ten van der Pol oscillators are divided into three groups.
Each oscillator has different values of the nonlinearity de-Figure 1: Conceptual circuit model.

The electrical circuit model of van der Pol oscillator isshown in Fig. 2(a). Figure 2 shows the circuit model whenten van der Pol oscillators are completely coupled by resistors.
The vk − iRk characteristics of the nonlinear resistor areapproximated by the following equation.

- 137 -iRk = −g1 vk + g3 vk3(k = 1, 2, ..., 10),(1)

The parameter of the standard nonlinearity is fixed with ε =1.00, and the coupling strength is set to γ = 0.01. In thissection, we investigate synchronization state when the error
is added to the nonlinearity of each oscillator. We define theerror rate of each oscillator as follows:
ε1 = ε
ε2 = ε + δε3 = ε + δ + g





ε4 = ε + 2δ + g ε5 = ε + 3δ + g(a) van der Pol oscillator.

(3)

ε6 = ε + 4δ + 2g






ε7 = ε + 5δ + 2g ε8 = ε + 6δ + 2g
ε9 = ε + 7δ + 2g
 ε10 = ε + 8δ + 2g,(b) Completely coupled ten van der Pol oscillators.
Figure 2: Circuit model.

By using the variables and the parameters,t=√√LCτ, vk =g1
xk , ik =g3

√

g1
g3

√

CykL

√√
L1 L
ε = g1, γ=,CR C
The normalized circuit equations of ten coupled van derPol oscillators are given as follows:10∑dxk= ε(1 − x2k )xk − yk − γxk
dτ
k=1
 dyk = x .
kdτ

where, δ denotes the small error and g denotes the large error of the nonlinearity. We set the range of the parameters
δ=[0.00:0.030], g=[0.00:0.30].
First, we investigate synchronization phenomena in ten
coupled oscillators when the nonlinearity errors are fixed with
δ = 0.01, g = 0.00. In this case, we confirm that all oscillators
are synchronized at in-phase state.
Next, we investigate the synchronization phenomena whenthe nonlinearity errors are fixed with δ = 0.01, g = 0.22. Figure 3 shows the phase difference between two oscillators ineach group. We can see that the phase difference between 1stand 2nd oscillators in group1 is changing and has amount of
the phase shift (Fig. 3 (a)). While, 3rd/4th and 6th/7th oscillators in group2 and group3 are synchronized at in-phase asin Figs. 3 (a),(b).
Figure 4 shows the phase difference between 1st and 2ndoscillators with iteration time. From this figure, we observe
the switching phenomenon of the phase difference. The phasedifference changes from 5 to 70 degrees periodically.

(2)

where ε is the nonlinearity and γ is the coupling strength.
(a) 1st-2nd

3. Simulation results
For the computer simulations, we calculate Eq. (2) using
a fourth-order Runge-Kutta method with step size h = 0.005.

(b) 3rd-4th(c) 6th-7th

Figure 3: Phase difference between two oscillators in eachgroup (δ = 0.01, g = 0.22).

- 138 -

(a) 1st-2nd

Figure 4: Phase difference of 1st and 2nd oscillators. (δ =0.01, g = 0.22).

(b) 3rd-4th(c) 6th-7th

Figure 6: Phase difference between two oscillators in group(δ = 0.01, g = 0.30).

Figure 5 shows the phase difference between the groups forδ = 0.01, g = 0.22. The synchronization state of oscillators
between group1 and group2 or group1 and group3 are asynchronous (Figs. 5 (a), (b)). The oscillators between group2and group3 are synchronized with amount of phase shift asshown in Fig. 5 (c). From Fig. 5, we can see that the synchronization states between first and second oscillators in group1are affected by group2 and group3.

Figure 7: Phase difference of 1st and 2nd oscillators. (δ =0.01, g = 0.30).

(a) group1-group2(b) group1-group3(c) group2-group3

Figure 5: Phase difference between groups (δ = 0.01, g =0.22).
(a) group1-group2

By increasing the value of g, we observe the different typeof synchronization phenomena. In Fig. 6, the results of phasedifferences between two oscillators in each group for δ =0.01, g = 0.30 are shown. The two oscillators in each groupare almost synchronized at in-phase state. Figure 7 shows thephase difference between first and second oscillators with iteration time. We can see that the aperiodic oscillation of thephase difference from 5 to 20 degrees.
Figure 8 shows the phase difference between the groups forδ = 0.01, g = 0.30. In this case, we confirm that the synchronization state of oscillators between groups are asynchronous
for all cases.
Next, we focus on synchronization phenomena dependingon the parameter dependency. Figure 9 shows the phase difference between 1st and 2nd oscillators when the parameters(b) group1-group3(c) group2-group3

Figure 8: Phase difference between groups (δ = 0.01, g =0.30).
of the nonlinearity error δ and g are changed. By increasing
the value of δ, the oscillation frequency of the phase difference becomes slow and the range of the phase shift becomeslarge (see Fig. 4 and Fig. 9 (a)). In comparison between Fig. 7and Fig. 9 (b), we can see that the range of the phase difference becomes large.
Finally, the sojourn time of in-phase state and one-period ismeasured by changing the nonlinearity error g. In this simulation, in-phase state is defined when the value of the phase difference smaller than 15 degrees. One-period means the sim-- 139 -(a) δ = 0.01.

(a) δ = 0.02, g = 0.19.

(b) δ = 0.02.

(b) δ = 0.02, g = 0.30.
Figure 9: Phase difference of first and second oscillators.

Figure 10: Sojourn time of the in-phase state.

References
ulation time between the peaks of the phase difference. The
simulation results for δ=0.01 and 0.02 are shown in Fig. 10. [1] G. Abramson, V.M. Kenkre and A.R. Bishop, “AnalyticThe horizontal axis is the nonlinearity error g and the verticalsolutions for nonlinear waves in coupled reacting sysaxis is the sojourn time (τp ). From these results, we confirm
tems,” Physica A: Statistical Mechanics and its Applicathat the sojourn time of in-phase and one-period decreases
tions, vol.305, nos.3-4, pp.427-436, Mar. 2002.
with the nonlinearity error g.
[2] I. Belykh, M. Hasler, M. Lauret and H. Nijmeijer, “Synchronization and graph topology,” Int. J. Bifurcation and4. ConclusionsChaos, vol.15, no.11, pp.3423-3433, Nov. 2005.
In this study, we have investigated synchronization phe- [3] T. Endo and S. Mori, “Mode analysis of a multimodeladder oscillator,” IEEE Trans. Circuits Syst., vol.23,
nomena observed in completely coupled ten van der Pol ospp.100-113, Feb. 1976.
cillators, when ten van der Pol oscillators are divided into
three groups. Each oscillator has different values of the non[4] T. Endo and S. Mori, “Mode analysis of two-dimensional
linearity depending on the error parameters. By using comlow-pass multimode oscillator,” IEEE Trans. Circuits
puter simulations, we have observed interesting synchronizaSyst., vol.23, pp.517-530, Sep. 1976.
tion phenomena when the nonlinearity error is changed.
For the future work, we would like to investigate the ob- [5] T. Endo and S. Mori, “Mode analysis of a ring of atained synchronization phenomena in detail and apply thislarge number of mutually coupled van der Pol oscillaproposed circuit system to general large-scale networks.
tors,” IEEE Trans. Circuits Syst., vol.25, no.1, pp.7-18,
Jan. 1978.
Acknowledgment[6] Y. Setou, Y. Nishio and A. Ushida, Synchronization Phenomena in Resistively Coupled Oscillators with Different
This work was partly supported by JSPS Grant-in-Aid forFrequencies IEICE Trans. Fundamentals, vol. E79-A, no.
Scientific Research 22500203.
10, pp. 1575-1580, 1996.

- 140 -