平成18年度電気関係学会四国支部連合大会

311

Breakdown of Synchronizationin Three Coupled Oscillators

17 - 15

Ryo Imabayashi

Yoko Uwate(Tokushima University)

1 Introduction

Yoshifumi Nishio

The normalized equations are described as follows.

Synchronization in nonlinear systems is one of typicalnonlinear phenomena observed in the field of natural science. Breakdown of synchronization is a kind of cooperative phenomenon for dissipated assembly oscillators and isimportant to clarify its mechanism for better understanding of higher-dimensional complicated phenomena. In thisstudy, we investigate the breakdown of synchronization
observed from three coupled oscillators consisting of twotypes of oscillators; the first one is chaotic oscillator andthe second one is van der Pol oscillators with noise. We
compare the breakdown of the synchronization observed
from two types of coupled oscillators.

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

½¾((1 + σ(τ ))xk )3dxk
= −yk + ε (1 + σ(τ ))xk −3 dτ3Xdyk= (1 + ∆ωk )(1 + σ(τ ))xk − γyjdτj=1(k = 1, 2, 3)

where σ(τ ) is the added noise.

(3)

3 Computer Calculated Results
Computer simulated results are shown in Fig 2. The
horizontal axis shows the strength of the added noise andthe vertical axis shows the coupling parameter at which
the synchronization breaks down. The circuit parameters
are fixed as α = 10.0, β = 0.06, δ = 100.0 and ∆ωk =0.002(k − 1), (k=1,2,3) for computer calculations.

Figure 2: Coupling parameter at which synchronization
breaks down (computer calculated results). ε = 0.1 and∆ωk = 0.002(k − 1), (k=1,2,3).

Figure 1: Circuit model.
The normalized circuit equations of the circuit are described as
3X
dxk


=β(x+y)−z−γxj

kkk

dτ

j=1



dyk
= α{β(xk + yk ) − zk − f (yk )}
 dτ



 dzk = (1 + ∆ωk )(xk + yk ) dτ(1)For van der Pol oscillators without noise, the breakdown of the synchronization is observed for γ ≤ 0.0021.
While, for chaotic oscillators, the breakdown appear forγ ≤ 0.0026. This means that the coupled van der Pol oscillators are more stable than the chaotic system. When weadd noise to van der Pol oscillators, the parameter for thebreakdown increases as shown in Fig. 2. Around σ=0.35,
the coupled van der Pol oscillators become to show similar
breakdown to the chaotic system.

4 Conclusions
In this study, we investigated the breakdown of the synchronization observed from both three chaotic circuits andthree van der Pol oscillators with noise.

(k = 1, 2, 3)

Referenceswheref (yk ) = 0.5(δyk + 1 − |δyk − 1|),(2)and ∆ωk corresponds to the difference between the naturaloscillating frequency of the reference oscillator and thoseof other oscillators.
Next, we consider three coupled van der Pol oscillatorswith noise. In this study we assume that noise with uniform distribution is added to each voltage of the oscillator.

[1] Seiichiro MORO, Yoshifumi NISHIO and Shinsaku
MORI, “Synchronization Phenomena in Oscillators Coupled by One Resistor,” IEICE Transactions on Fundamentals, vol. E78-A, no. 2, pp. 244-253, Feb. 1995.
[2] 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.