平成21年度電気関係学会四国支部連合大会

341

18-17
Synchronization Phenomena in Coupled Auto Gain
Controlled Oscillators Containing Time Delay
Akiyoshi ONO

Hironori KUMENO(Tokushima University)

1. Introduction

Yoshifumi NISHIO{Recently, the synchronization phenomena in coupled chaotic oscillators are studied by many researchers. Because the synchronization phenomena
have caused very interesting phenomena. However, alot of synchronization phenomena of coupled chaoticoscillators have not been solved yet and we have toinvestigate them.
In this study, we propose chaotic oscillators coupled
via switch. We investigate synchronization phenomena in coupled auto gain controlled oscillators containing time delay. We observe several interesting
synchronization phenomena.

{ẋi = yiy˙i = −xi + 2αyi

(3)ẋi = yiy˙i = −xi − 2βyi .

(4)When SW1 and SW2 are connected to the negativeresistor, the circuit is governed by Eq. (3), while when
SW1 and SW2 are connected to the positive resistor,
the circuit is governed by Eq. (4).

3. Synchronization phenomena
We carry out computer calculations for coupled
auto gain controlled oscillators containing time delay. We observe coexistence phenomena of in-phasesynchronization and opposite-phase synchronization.
Figure. 2 shows examples of the observed attractors
and phase diﬀerences at the parameters α = 0.015,
β = 0.89 and γ = 0.02911. The two subcircuits
are synchronized at in-phase in Fig. 2(a), while thetwo subcircuits are synchronized at opposite-phase inFig. 2(b). These states are observed for the same parameters, however initial values are diﬀerent. Thus,
the in-phase synchronization and the opposite-phasesynchronization coexist.

2. Circuit modelFigure 1: Circuit model.
The circuit model used in this study is shown inFig. 1. In the circuit, two chaotic oscillators which areintroduced in [1] are coupled by periodically changing
switch SW3. The periodically changing switch connects alternately with one subcircuit and the otherwith a ﬁxed time interval (the switching frequency
is 2πγt ). Two comparators of the subcircuits obtain
information of the voltage from the subcircuit connected via SW3 and compare the voltage of the subcircuit with a given threshold Vth . The periodicallychanging switch can be controlled by the following
sinusoidal function.
k = sin γt t.

(1)

The periodically changing switch connects to the left
subcircuit in Fig. 1 when k is positive, while to theright subcircuit in Fig. 2 when negative. The voltage of the subcircuit connected via SW3 controls SW1and SW2 at the same time. If the voltage of the subcircuit is lower than the threshold Vth , SW1 and SW2
are connected to the negative resistors, while if thevoltage of the subcircuit rises above Vth , SW1 andSW2 are connected to the positive resistor after thegiven delay time during the time interval while thevoltage of the subcircuit is higher than Vth .
By using the following normalized variables and parameters,√√, vi = Vth yi , t =ii = C/LV√ LCτ,√ th xi√γt = γ/ LC, g L/C = 2α, G L/C = 2β,(2)the normalized circuit equations are given as:

Figure 2: Attractors and phase diﬀerences α = 0.015,
β = 0.89 and γ = 0.02911.

4. Conclusions
In this study, we have proposed couped auto gain
controlled oscillators containing time delay. In thiscircuit, we observed coexistence phenomena of inphase synchronization and opposite-phase synchronization. Our future research is to make clear themechanism of the generation of the coexistense andto carry out circuit experiments.
References[1] T. Maruyama, N. Inaba, Y. Nishio and S. Mori,“Chaos in Auto Gain Controlled Oscillator Containing Time Delay,” Transactions of IEICE, vol. J72-A,
no. 11, pp. 1814-1820, Nov. 1989.