Breakdown of Synchronizationin Chaotic Oscillators and Noisy Oscillators
Ryo Imabayashi, Yoko Uwate and Yoshifumi Nishio
Department of Electrical and Electronic Engineering, Tokushima University,
2-1 Minami-Josanjima, Tokushima 770-8506, JAPAN
Telephone: +81-88-656-7470, Fax: +81-88-656-7471
Email:{bayashi, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract— In this study, the breakdown of synchronization
observed from four coupled chaotic oscillators is investigated.
In order to understand the phenomenon, the model of coupledmodified van der Pol oscillators with noise is considered. The
comparison of the coupled chaotic oscillators with the coupledmodified van der Pol oscillators with noise gives us someinteresting results.

I. I NTRODUCTION
Synchronization is one of typical nonlinear phenomena
observed in the field of natural science. There have been many
investigations on mutual synchronization of oscillators. Breakdown of synchronization is a kind of cooperative phenomenon
for dissipated assembly oscillators and is important to clarify
its mechanism for better understanding of higher-dimensional
complicated phenomena [1]-[6]. In the case of coupled chaoticoscillators, chaotic fluctuations of their waveforms are supposed to play a role to break the synchronization and thebreakdown sometimes causes chaotic wandering phenomenon.
On the other hand, there is certainly noise in actual physicalsystems. In these years, investigations on systems including
noise attract many researchers’ attentions. We are interested in
the difference between chaotic fluctuation and noise, because
sometimes chaos exhibits better performance than random
noise in information processing tasks [7].
In this study, we investigate the breakdown of synchronization observed from four coupled chaotic oscillators. In order tounderstand the phenomenon, we consider the model of coupledmodified van der Pol oscillators with noise. The comparisonof the coupled chaotic oscillators with the coupled modified
van der Pol oscillators with noise gives us some interesting
results.
II. C IRCUIT M ODEL
A. Chaotic Oscillators
Figure 1 shows the circuit model. In this circuit, four
identical chaotic oscillators are coupled by one resistor. Bychanging the variables and the parameters,CC
Exk , ik =Eyk , vk = Ezk ,Ik =L1L1 
CL1(1)
t = L1 Cτ, α =, β=r,LL
21CCγ=R, δ = rd,L1
L11-4244-1342-7/07/$25.00 ©2007 IEEEFig. 1.

Coupled chaotic oscillators.

the normalized circuit equations of the circuit are described as
4
dxk



= β(xk + yk ) − zk − γxj

 dτ
j=1
 dyk
= α{β(xk + yk ) − zk − f (yk )}
(2)
 dτ
dzk= xk + yk(k = 1, 2, 3, 4)
 dτ
wheref (yk ) = 0.5(δyk + 1 − |δyk − 1|).

(3)

Figure 2 shows an example of the observed four-phasequasi-synchronization of chaos. In the figures the phase differences of x2 , x3 and x4 with respect to x1 are almost 90◦ ,180◦ and 270◦ , respectively.
Because of symmetry of the coupling structure, six different
combinations of phase states coexist;
S1 : (0◦ , 90◦ , 180◦ , 270◦ ),S2 : (0◦ , 90◦ , 270◦ , 180◦ ),S3 : (0◦ , 180◦ , 90◦ , 270◦ ),S4 : (0◦ , 180◦ , 270◦ , 90◦ ),S5 : (0◦ , 270◦ , 90◦ , 180◦ ),S6 : (0◦ , 270◦ , 180◦ , 90◦ ).

(4)

As the coupling resistance R (γ) decreases, we can observe that the synchronization states become unstable andthat the self-switching phenomenon of the states occurs [6].
The breakdown of the synchronization means the start of theself-switching, hence it is important to investigate how thesynchronization breaks down.

922

Figure 4 shows an example of the observed four-phasesynchronization of the modified van der Pol oscillators.

(a)

(b)

(c)(a)

(d)

(c)

(e)

Fig. 2. Four-phase quasi-synchronization of chaos (computer calculated result). α = 7.0, β = 0.10, γ = 0.10 and δ = 100.0. (a) x1 vs.
x2 . (b) x1 vs. x3 . (c) x1 vs. x4 . (d) x1 vs. z1 . (e) Time waveforms.

B. Modified van der Pol Oscillators
Next, we consider four coupled van der Pol oscillators.
In order to obtain the waveforms similar to those of theFig. 3.

(b)Coupled van der Pol oscillators.

chaotic oscillator, we modify the van der Pol oscillator with
the nonlinear resistor whose v − i characteristics are describedby the following asymmetric functionir (vk ) = −g1 vk + g2 vk2 + g3 vk3 (g1 , g2 , g3 > 0).
By changing the variables and the parameters,

√
g1
Cg1
vk =xk , ik =yk , t = LCτ,
g3Lg

 3CL
g2, ε = g1,
ν=√, γ=Rg1 g3LC
the normalized circuit equations are given as dxk
= ξ −yk + ε(xk − νx2k − x3k )dτ
4dyk= xk − γyj dτj=1
(k = 1, 2, 3, 4)

(5)

(6)

(d)

(e)

Fig. 4. Four-phase synchronization of modified van der Pol oscillators(computer calculated result). ε = 0.50, γ = 0.60, ξ = 1.07 andν = 0.1035. (a) x1 vs. x2 . (b) x1 vs. x3 . (c) x1 vs. x4 . (d) x1 vs.
y1 . (e) Time waveforms.

In this study, we consider the case that the noise is added
to the modified van der Pol oscillator in order to simulate
the breakdown of the synchronization in the coupled chaoticoscillators caused by chaotic fluctuations of their waveforms.
When we add the noise to the voltage amplitude of themodified van der Pol oscillator, the circuit equation of thecoupled oscillators are described as dxk= ξ[−yk + ε{(1 + σ(τ ))xkdτ− ν((1 + σ(τ ))xk )2 − ((1 + σ(τ ))xk )3 }]
4(8)
dyk


= (1 + σ(τ ))xk − γyj

dτ

j=1
(k = 1, 2, 3, 4)
where σ(τ ) is the added noise.
While when we add the noise to the voltage period of themodified van der Pol oscillator, the circuit equation of thecoupled oscillators are described as dxk= (1 + σ(τ ))ξ{−yk + ε(xk − νx2k − x3k )}

dτ

4

dyk
(9)
= xk − γyj

dτ


j=1

(k = 1, 2, 3, 4).

(7)

In this study we concentrate on the case that the added noiseis the additive white Gaussian noise (AWGN) with the average
0 and the variance σ 2 .

where ξ is the parameter added to tune the period of thewaveform.

When the coupling parameter γ is relatively large, boththe coupled chaotic oscillators and the modified van der Pol
oscillators with noise exhibit four phase synchronizations.

III. B REAKDOWN OF S YNCHRONIZATION

923

While for relatively smaller γ, the synchronizations break
down and we observe the switchings of phase states. Thismeans that a critical value of the coupling parameter exists
which divides the parameter space into the synchronization
region and the self-switching region. We define this critical
coupling parameter as γc and investigate how γc changes when
the strength of chaos or noise increases.
First, we define the breakdown of synchronization as atleast one switching during 50,000 periods. Next, in order
to compare the coupled chaotic oscillators with the modified
van der Pol oscillators, we set the parameters as follows. We
investigate the critical coupling parameter γc for the coupledchaotic oscillators when the chaotic oscillator exhibits the first
period-doubling bifurcation (β=0.0425). This value γc =1.189
is set as the standard value. We tune the parameter of themodified van der Pol oscillators without noise in order to thesynchronization breaks down at this standard value of γc . By
this setting of the parameter, the chaotic oscillators without
chaotic fluctuation and the modified van der Pol oscillators
without noise are equalized in the sense of the stability of thesynchronization.
Figures 5 and 6 show one-parameter bifurcation diagram of
the chaotic oscillator and the critical coupling parameter γc ,respectively. In Fig. 6, the region over the curve corresponds
to the synchronization region and the region under the curve
corresponds to the self-switching region. We can observe thatγc increases as β increases. This means that the synchronization becomes easier to be broken down for larger β, namely
larger chaotic fluctuation.

Figure 7 shows the critical coupling parameter γc of thecoupled modified van der Pol oscillators when the noise isadded to their voltage amplitude. The horizontal axis shows
the standard deviation of the added noise σ. For σ = 0.20,
γc takes the minimum value. When σ becomes larger beyond
0.20, γc increases.
Figure 8 shows the critical coupling parameter γc of thecoupled modified van der Pol oscillators when the noise isadded to their voltage period. Similar to the previous case, forγc takes the minimum value σ = 0.20 and γc increases for
σ > 0.20.
1.4
1.2
1

γc0.8
0.6
0.4
0.2
000.1

0.2

0.3

0.4

0.5

0.6

σFig. 7. Breakdown of synchronization of modified van der Pol oscillatorswith noisy amplitude (ε = 0.5, ξ = 1.07 and ν = 0.1035).

1.4
1.2
1

γc

1.3

0.8
0.6

1.2

0.4

1.1

0.2

1x1

00.9

00.8

0.1

0.2

0.3

0.4

0.50.6

σ0.7

Fig. 8. Breakdown of synchronization of modified van der Pol oscillatorswith noisy period (ε = 0.5, ξ = 1.07 and ν = 0.1035).

0.6
0.5
0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14

βFig. 5. One-parameter bifurcation diagram of chaotic oscillator (α = 7.0,
γ = 0.0 and δ = 100.0).
1.4
1.2

We noticed that the values of γc for the coupled modified
van der Pol oscillators are much larger than those for thecoupled chaotic oscillators. Although we have to investigate
more precisely before concluding, chaotic systems may be
synchronized more stably than simple periodic oscillators withnoise.

1γcIV. D ISTRIBUTION OF A MPLITUDE AND P ERIOD

0.80.60.4
0.2
00.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14

βFig. 6. Breakdown of synchronization of coupled chaotic oscillators (α = 7.0and δ = 100.0 ).

In order to evaluate the effect of the noise added to thecoupled modified van der Pol oscillators, we compare the statistical characteristics of amplitudes and periods of the chaoticoscillator with the noisy modified van der Pol oscillators.
The distributions of the voltage amplitude are shown inFigs. 9-11. The distributions of the voltage period are shownin Figs. 12-14.

924

1600

1600

1400

1400

1200

1200

1000

1000

n 800

n 800

600

600

400

400

200

200

00.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.301140

1.4

Fig. 9. Distribution of voltage amplitude of chaotic oscillators (α = 7.0,
β = 0.1, γ = 0.065 and δ = 100.0 ).

1600

1400

1400

1200

1200

1200

1220

1240

1260

1280

1000

1000

n 800

n 800
600

600

400

400

200

200

0.6

0.70.80.911.1

1.2

1.301140

1.41160

1180

1200

1220

1240

1260

1280

Tx1
Fig. 10. Distribution of voltage amplitude of modified van der Pol oscillatorswith noisy amplitude (γ = 0.942, ε = 0.5, ξ = 1.07, ν = 0.1035 andσ = 0.5).

Fig. 13. Distribution of voltage period of modified van der Pol oscillatorswith noisy amplitude (γ = 0.942, ε = 0.5, ξ = 1.07, ν = 0.1035 andσ = 0.5).

1600

1600

1400

1400

1200

1200
1000

1000

n 800

n 800

600

600

400

400

200

200
00.5

1180

Fig. 12. Distribution of voltage period of chaotic oscillators (α = 7.0,β = 0.1, γ = 0.065 and δ = 100.0 ).

1600

00.5

1160

Tx1

0.6

0.7

0.8

0.9

1

1.1

1.2

1.301140

1.4

1160

1180

1200

1220

1240

1260

1280

Tx1
Fig. 11. Distribution of voltage amplitude of modified van der Pol oscillatorswith noisy period (γ = 0.973, ε = 0.5, ξ = 1.07, ν = 0.1035 and σ = 0.5).

Fig. 14. Distribution of voltage period of modified van der Pol oscillatorswith noisy period (γ = 0.973, ε = 0.5, ξ = 1.07, ν = 0.1035 and σ = 0.5).

The distributions show the difference between the chaotic
oscillator and the modified van der Pol oscillator with noise.
Namely, the chaotic oscillator shows a kind of two-bandcharacteristics, while the modified van der Pol oscillatorshows the normal distribution. We consider that this may be
the main reason of the difference on the breakdown of thesynchronization.

ACKOWLEDGMENTS
This work was partly supported by Yazaki Memorial Foundation for Science and Technology.

V. C ONCLUSION
In this study, the breakdown of synchronization observed
from four coupled chaotic oscillators has been investigated. Inorder to understand the phenomenon, the model of coupledmodified van der Pol oscillators with noise was considered.
The simulation results suggested that chaotic systems might
be synchronized more stably than simple periodic oscillatorswith noise. We also compared the statistical characteristics ofthe amplitudes and the periods of the chaotic oscillator with
the noisy modified van der Pol oscillators and considered that
their difference might be the reason of the difference on thebreakdown of the synchronization.

925

R EFERENCES[1] N. Platt, E.A. Spiegel and C. Tresser, “On-Off Intermittency: A Mechanism for Bursting,” Phys. Rev. Lett., vol. 70, no. 3, pp. 279-282, 1993.
[2] E. Ott and J.C. Sommerer, “Blowout Bifurcations: the Occurrence ofRiddled Basins and On-Off Intermittency,” Phys. Lett., vol. A188, no. 3,pp. 39-47, 1994.
[3] P. Ashwin, J. Buescu and I. Stewart, “Bubbling of Attractors andSynchronization of Chaotic Oscillators,” Phys. Rev. Lett., vol. A193,
no. 3, pp. 126-139, 1994.
[4] T. Kapitaniak and L.O. Chua, “Locally-Intermingled Basins of Attractionin Coupled Chua’s Circuits,” Int. J. Bifurcation and Chaos, vol. 6, no. 2,pp. 357-366, 1996.
[5] M. Wada, Y. Nishio and A. Ushida, “Analysis of Bifurcation Phenomena on Two Chaotic Circuits Coupled by an Inductor,” IEICE Trans.
Fundamentals, vol. E80-A, no. 5, pp. 869-875, 1997.
[6] Y. Nishio and A. Ushida, “Chaotic Wandering and its Analysis in Simple
Coupled Chaotic Circuits,” IEICE Trans. on Fundamentals, vol. E85-A,
no. 1, pp. 248-255, 2002.
[7] Y. Hayakawa and Y. Sawada, “Effects of the Chaotic Noise on thePerformance of a Neural Network Model for Optimization Problems,”
Phys. Rev. E, vol. 51, no. 4, pp. 2693-2696, 1995.