平成 28 年度電気関係学会四国支部連合大会 講演論文集 (2016 徳島大学)
2016 SHIKOKU-SECTION JOINT CONVENTION RECORD OF THE INSTITUTES OF ELECTRICAL AND RELATED ENGINEERS (TOKUSHIMA)

1-7
Switching Synchronization State of System
Including Time Delay in One Direction
Seiya KITA, Yoko UWATE and Yoshifumi NISHIO(Tokushima University)

1. Introduction
The generation of chaos has been reported in all selfexcited oscillation systems containing a time delay [1].
In this study, we devise coupled systems that take advantage of the features of time-delayed chaotic circuits.
The novel coupled systems utilize the characteristics ofcircuits having time-delayed feedback. We investigate thesynchronization state in coupled time-delayed chaotic circuits. By carrying out computer simulations, it is shownthat the time delay of subcircuits changes the synchronization state.

VthCC1

Td3
R

Td1
RDelay

Td

TdinR

CC3

CCnCC2C

vn L-gSW
G

Td2

Figure 1: System including time delay in one direction.

2. Circuit Model(a) γ = 0.05 and Tdn = π.

The circuits in this study employ characteristic time delay methods. We have devised the coupled system shownin Fig. 1. This system is coupled by resistors R. This system includes a time delay in one direction. The normalized circuit equations of the system are given as follows:(A) In the case that the switch is connected to the negative resistor{ẋnẏn==yn−xn + 2αyn + γ(yn−1 − 2yn + yn+1 )(b) γ = 0.01 and Tdn = π.

(1)(c) γ = 0.01 and Tdn = 0.5π.

(B) In the case that the switch is connected to the positive resistor{ẋnẏn==yn−xn − 2βyn + γ(yn−1 − 2yn + yn+1 )(2)where n = 1, 2, 3 and y0 = y3 , y4 = y1 . Generally, switching synchronization can be observed when the system including a time delay in one direction is coupled by resistors R. The amplitude alternately diverges and converges
with diﬀerent divergence and convergence times.

(d) γ = 0.01 , Td1 = 0.5π and Td2 = Td2 = π.
Figure 2: Time waveform α = 0.015, β = 0.5.

3. Simulation Results
The result shown in Fig. 2 can be obtained from thediﬀerence in the coupling strength γ and the time delayTdn . The time waveform in Fig. 2(a) shows in-phase synchronization and the amplitude of xn is switched sequentially. However, when γ is larger than 0.05, the switching
synchronization state is lost and a full in-phase synchronization state can be observed. Furthermore, the synchronization state is changed by time delay. The cycleof synchronization state shows in Fig. 3. When the timedelay Tdn are asymmetric, the cycle approaches stable
regardless of the time delay.

4. Conclusions
In this study, we investigated the synchronization stateof novel coupled systems of time-delayed chaotic ring circuits coupled by various methods. As a result, some special synchronization states were observed. In the case of aring circuit coupled by resistors, we observed an in-phasesynchronization state. Switching of the amplitude of the7Figure 3: Cycle of switching synchronization.
voltage in addition to the in-phase synchronization state
was observed from the diﬀerence of coupling strength andtime delay. Furthermore, the cycle of switching synchronization state is changed by combination of the time delay.
Reference[1] T. Maruyama, N. Inaba, Y. Nishio and S. Mori, “Chaos inan Auto Gain Controlled Oscillator Containing Time Delay,”
Trans. IEICE, vol. J 72-A, pp. 1814-1820, Nov. 1989.