平成19年度電気関係学会四国支部連合大会

379

17-49

Research on Synchronization Phenomena in
Chaotic Oscillators and Noisy Oscillators
Ryo Imabayashi

Yoko Uwate(Tokushima University)

1 Introduction
Synchronization is one of typical nonlinear phenomena
observed in the ﬁeld of natural science. 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.
In this study, we investigate the breakdown of synchronization observed from four coupled chaotic oscillators. Inorder to understand the phenomenon, the model of coupled
modﬁed van der Pol oscillators with noise is considered.

Yoshifumi Nishio

where ξ is the parameter added to tune the period of thewaveform and σ(τ ) is the added noise.

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.

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

1.2

1.4

1.1

1.2

11

0.9


c

x1 0.8

0.8
0.6

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0.5

00.4

00.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14

0.1

0.2

(a)

c

The normalized circuit equations of the circuit are described as
4∑xj ẋk = β(xk + yk ) − zk − γ

1.4

1.2

1.2żk = xk + yk ,0.6



c0.4

0.5

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0.8

0.6

0.6

0.4

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0.2

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000

0.1

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0.3



(1)
(k = 1, 2, 3, 4)

where f (yk ) = 0.5(δ yk + 1 −| δ yk − 1|).
Next, we consider four coupled van der Pol oscillators.
In order to obtain the waveforms similar to those of thechaotic 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 ẋ = ξ[−y + ε{(1 + σ(τ ))xkkk− ν((1 + σ(τ ))xk )2 − ((1 + σ(τ ))xk )3 }]
4
∑
yj .
(k = 1, 2, 3, 4)ẏ=(1+σ(τ))x−γkk


(3)

j=1

While when we add the noise to the voltage period of
the modiﬁed van der Pol oscillator, the circuit equation of
the coupled oscillators are described as
23 ẋk = (1 + σ(τ4))ξ{−yk + ε(xk − νxk − xk )}∑yj .
(k = 1, 2, 3, 4) ẏk = xk − γ
j=1

0.5

10.8

0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14

j=1ẏk = α{β(xk + yk ) − zk − f (yk )}



0.4

(c)

1.4

1

Figure 1: Coupled chaotic oscillators.

0.3

(4)

(b)
(d)
Computer calculated results. (a) One-parameterbifurcation diagram of chaotic oscillator (α = 7.0, γ = 0.0 andδ = 100.0). (b) Breakdown of synchronization of coupled chaoticoscillators (α = 7.0 and δ = 100.0 ). (c) Breakdown of synchronization of modiﬁed van der Pol oscillators with noisy amplitude(ε = 0.5, ξ = 1.07 and ν = 0.1035). (d) Breakdown of synchronization of modiﬁed van der Pol oscillators with noisy period(ε = 0.5, ξ = 1.07 and ν = 0.1035).

Figure 2:

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

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. The simulation results suggested that chaotic systems might be synchronized more stably than simple periodic oscillators with noise.

References[1] Yoshifumi NISHIO and Akio USHIDA, “Chaotic Wandering and its Analysis in Simple Coupled Chaotic Circuits,” IEICE Transactions on Fundamentals, vol. E85-A,
no. 1, pp. 248-255, Jan. 2002.