平成20年度電気関係学会四国支部連合大会

377

17-8

Breakdown of Chaos Synchronization
and Noise Eﬀect on Simple Oscillators
Ryo Imabayashi

Yoko Uwate(Tokushima University)

1 Introduction
In our previous research, we have investigated thebreakdown of synchronization observed from four coupled chaotic oscillators. In order to understand the phenomenon, the model of coupled modiﬁed van der Pol oscillators with the additive white Gaussian noise (AWGN) wasproposed. By computer simulations, we have conﬁrmedthat chaotic systems were synchronized more stably thanmodiﬁed van der Pol oscillators with AWGN.
In this study, we propose other two types of noise foradding the van der Pol oscillators to conﬁrm the diﬀerent between chaotic systems and van der Pol systems withAWGN. First noise has two band characteristic, and second noise is a scaled AWGN. By adding the scaled AWGN,
modiﬁed van der Pol oscillators are close to coupled chaoticsystems is conﬁrmed.

2 Circuit Model
Fig. 1 shows the circuit model. In the circuit, four identical chaotic circuits are coupled by one resistor.

Yoshifumi Nishio

While when we add the noise to the voltage period of
the modiﬁed van der Pol oscillator, the circuit equation ofthecoupled oscillators are described asdxk
23 dτ = (1 + ρk nk (τ ))ξ{−yk + ε(xk − νxk − xk )}
4
∑dyk

=x−
γ
yj ,(k = 1, 2, 3, 4)
k
 dτ
j=1

(4)
where nk (τ ) is the added noise and ρk is constant to tune
the amplitude of the noise. The noise nk (τ ) is the additive
white Gaussian noise (AWGN) with the average 0 and thevariance σ 2 .

3 Computer Calculated Results
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. While for relatively smaller γ, the synchronizations
break down and we observe the switchings of phase states.
We deﬁne this critical coupling parameter as γc and investigate how γc changes when the strength of chaos or noise
increases.
1.4

1.4

Non-scaled  AWGN


AWGN
Noise with two band characteristics

1.210.8

0.8

cc
0.6

0.6

0.40.4

0.2

0.2

0

Figure 1: Coupled chaotic oscillators.

(1)j=1żk = xk + yk ,(k = 1, 2, 3, 4)

where f (yk ) = 0.5(δ yk + 1 −| δ yk − 1|).
Next, we consider four coupled modiﬁed van der Pol oscillators. In order to obtain the waveforms similar to thoseof the chaotic oscillator, we modify the van der Pol oscillator with the nonlinear resistor whose v − i characteristics
are described by the following asymmetric functionir (vk ) = −g1 vk + g2 vk2 + g3 vk3(g1 , g2 , g3 > 0).

(2)

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 + ρk nk (τ ))xk dτ
2− ν((1 + ρk nk (τ ))xk ) − ((1 + ρk nk (τ ))xk )3 }]dyk dτ = (1 + ρk nk (τ ))xk − γ4
∑

yj ,

0.1

0.2

0.3

0.4



4∑xj ẋk = β(xk + yk ) − zk − γẏk = α{β(xk + yk ) − zk − f (yk )}



00

The normalized circuit equations of the circuit are described as(k = 1, 2, 3, 4)

j=1(3)

               



1.2

1(a)

0.5

0.6

0

0.1

0.2

0.3



0.4

0.5

0.6

(b)

Figure 2: Breakdown of synchronization of modiﬁed van der
Pol oscillators with noisy amplitude (² = 0.5, ξ = 1.07, ν = 0.3,
µ = 2.795 and σ = 1.0). (a) Noise with two band characteristics(ρk = 1.0). (b) Scaled AWGN.

Fig. 2 (a) show synchronization of modiﬁed van der Pol
oscillators with noise with two band characteristics. Fromthese results, the breakdown of synchronization is lightly
aﬀected by the distribution of noise.
It is seen that non-scaled AWGN has a smaller area of
stable synchronization in Fig. 2 (b). From these results,
we can say that by adding the scaled AWGN the modiﬁed
van der Pol oscillators behave more similar to the coupledchaotic systems.

4 Conclusions
In this study, the breakdown of synchronization observed
from four coupled chaotic oscillators has been investigated.
In order to understand the phenomenon, the model of coupled modiﬁed van der Pol oscillators with noise was considered. By adding the scaled AWGN, the modiﬁed van der
Pol oscillators are conﬁrmed to be closer to the coupledchaotic circuits.

References[1] R. Imabayashi, Y. Uwate and Y. Nishio, “Breakdown of Synchronization in Chaotic Oscillators and Noisy Oscillators,” Proceedings of European Conference on Circuit Theory and Design(ECCTD’07), pp. 922-925, 2007.