2010 International Workshop on Nonlinear Circuits,Communication and Signal Processing
NCSP'10, Waikiki, Hawaii, March 3-5, 2010

Switch Coupling of Auto Gain Controlled Oscillators Containing Time Delay
Akiyoshi Ono, Hironori Kumeno and Yoshifumi Nishio
Dept. Electronic Eng.,
Tokushima University, Japan
Email: {akiyoshi, kumeno, nishio}@ee.tokushima-u.ac.jp

Abstract
In this study, we propose two chaotic oscillators coupled via
switch. The chaotic oscillators are auto gain controlled oscillators containing time delay. The oscillators have feedback systems which control the gain. The coupling switch
is placed between the feedback systems which control bothof the oscillators. The coupling switch connects alternately
with one subcircuit and the other with a fixed time interval.
We carry out computer calculations and circuit experiments
for two coupled auto gain controlled oscillators containing
time delay and investigate synchronization phenomena. In
the computer calculations, we observe coexistence phenomena of in-phase synchronization and opposite-phase synchro- Figure 1: Auto gain controlled oscillator containing time denization. In the circuit experiments, we observe coexistence lay.
phenomena of in-phase quasi-synchronization and oppositephase quasi-synchronization. Finally, to investigate the diﬀerence between the numerical results and the experimental results, we investigate eﬀects of parameter mismatch and noise.

1. Introduction
Synchronization and the related bifurcation of chaotic systems are good methods to describe various high-dimensional
nonlinear phenomena in the field of natural science. Studies on synchronization phenomena are extensively carried out
in various field [1]-[3]. The chaos phenomena are quite dependent on initial values and not periodical and predictable.
Moreover, we obtain the synchronization phenomena in coupled oscillators despite of behaviors of one oscillator are complex. Therefore, we are interested in the synchronization phenomena. However, a lot of synchronization phenomena of
coupled chaotic oscillators have not been solved yet and wehave to investigate them.
In this study, we propose switch coupling system for auto
gain controlled oscillators containing time delay. The coupling switch is placed between the feedback systems which
control both of the oscillators. The coupling switch connects alternately with one subcircuit and the other with a fixed
time interval. The oscillator used in this study is auto gain
controlled oscillator containing time delay, whose circuit isFigure 2: Chaotic attractors in auto gain controlled oscillator
containing time delay.

shown in Fig. 1. This is introduced in [4]. The oscillator exhibits bifurcation phenomena from one-periodic attractor tochaotic attractor. Figure 2 shows examples of experimental
chaotic attractor and corresponding attractor of the numericalsimulation. We investigate synchronization phenomena in thecoupled auto gain controlled oscillators containing time delay. We observe coexistence phenomena of in-phase synchronization and opposite-phase synchronization in the computer
calculations. Moreover, we observe coexistence phenomenaof in-phase quasi-synchronization and opposite-phase quasisynchronization in the circuit experiments. Finally, we investigate eﬀects of parameter mismatch and noise.

- 53 -2. Circuit model

Figure 3: Circuit model.

The circuit model used in this study is shown in Fig. 3. In
the circuit, two chaotic oscillators are coupled by periodicallychanging switch SW3. The circuit equations of the oscillator
are given as{dvC= gv − i,dt

di=v
dt

(1)

{dvdiC= −Gv − i, L = v.
dtdt

(2)

LThe periodically changing switch SW3 connects alternately
with one subcircuit and the other with a fixed time interval
Figure 4: Information V through periodically changing(the switching frequency is 2πγt ). The periodically changing
switch SW3.
switch can be controlled by the following sinusoidal function.
k = sin γt t.

(3)

Figure 4 shows operation of the SW3 and information
through the SW3. Two comparators of the subcircuits obtain
the information of the voltage from the subcircuit connected
via SW3 and compare the voltage of the subcircuit with agiven threshold Vth . Then, two comparators give the information to the delay systems of the subcircuits. The delay systems of the subcircuits delay the information whitch control
switches SW1 and SW2.
The periodically changing switch connects to the left subcircuit in Fig. 3 when k is positive, while to the right subcircuit in Fig. 3 when negative. The voltage of the subcircuit
connected via SW3 controls SW1 and 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 the voltage of the subcircuit rises above Vth , SW1and SW2 are connected to the positive resistor after the given
delay time during the time interval while the voltage of thesubcircuit is higher than Vth .

By using the following normalized variables and parameters,√ii = √ C/LVth xi , vi √= Vth yi ,(4)t=LCτ,
γ=γ/t√√ LC,g L/C = 2α, G L/C = 2β,
the normalized circuit equations are given as:{{ẋi = yiẏi = −xi + 2αyi

(5)ẋi = yiẏi = −xi − 2βyi .

(6)When SW1 and SW2 are connected to the negative resistor,
the circuit is governed by Eq. (5), while when SW1 and SW2
are connected to the positive resistor, the circuit is governedby Eq. (6).

- 54 -

3. Synchronization phenomenasynchronization.

In this section, synchronization phenomena generated in
the coupled auto gain controlled oscillators containing timedelay are investigated by computer calculation and circuit experiment.
3.1. Computer calculations
We carry out computer calculations for the coupled auto
gain controlled oscillators containing time delay. We observe coexistence phenomenon of in-phase synchronizationand opposite-phase synchronization. Figure. 5 shows examples of the observed attractors and phase diﬀerences at theparameters α = 0.0186, β = 0.87 and γ = 2510. The two
subcircuits are synchronized at in-phase as shown in Fig. 5(a),while the two subcircuits are synchronized at opposite-phaseas shown in Fig. 5(b). These states are observed for the same
parameters, although initial values are diﬀerent. Thus, the inFigure 6: Attractors and phase diﬀerences in experimental
phase synchronization and the opposite-phase synchronizaresults. (a) In-phase quasi-synchronization of periodic attraction coexist.
tors. (b) Opposite-phase quasi-synchronization of periodicattractors. (c) Quasi-synchronization of chaotic attractors.

4. Eﬀect of noise and parameter mismatch

Figure 5: Attractors and phase diﬀerences in numerical results for α = 0.0186, β = 0.87 and γ = 2510.

3.2. Circuit experiments
We carry out circuit experiments for the coupled oscillators. We observe coexistence phenomenon of in-phase quasisynchronization and opposite-phase quasi-synchronization
as shown in Fig. 6. Two periodic attractors are quasisynchronized at in-phase in Fig. 6(a). Two attractors are
quasi-synchronized at opposite-phase in Fig. 6(b). Twochaotic attractors are quasi-synchronized at in-phase inFig. 6(c). These states are observed for the same parameters, although initial values are diﬀerent. These results isdiﬀerent from the computer calculations. Figure 6 show thephase diﬀerence and the attractor on unstable quasi-in-phase

The synchronization phenomena observed in the experimental results are slightly diﬀerent from the synchronization phenomena observed in the computer calculation, although coexistence phenomena of in-phase synchronizationand opposite-phase synchronization. Synchronization in thepractical experiment will be aﬀected by two factor: noiseand parameter mismatch between two oscillators. On one
hand it is impossible to make the parameters of one subcircuit to be completely consistent with the other. On the other
hand, noise inevitably exists in the any realistic physical system. These perturbations may kick the motion oﬀ the synchronization manifold. We consider the influence from thenoise and the diﬀerence of parameter mismatch cause the difference between the numerical results and the experimentalresults. Then, carry out computer calculation for the coupledcircuits containing the noise and changing the parameter mismatch. Normalized circuit equations containing the noise andthe parameter mismatch are described as:{x˙1 = y1(7)y˙1 = −x1 + 2αy1 + ξ1- 55 -{{x˙1 = y1y˙1 = −x1 − 2βy1 + ξ2 ,(8)x˙2 = σ1 y2y˙2 = −x2 + 2(1 + σ2 )αy2 + ξ1(9){x˙2 = σ3 y2y˙2 = −x2 − 2(1 + σ4 )βy2 + ξ2 ,(10)In this study, we have proposed the switch coupling system for auto gain controlled oscillators containing time delay, and investigated the synchronization phenomena generated in the coupled oscillators. couped auto gain controlled
oscillators containing time delay. We observed coexistence
phenomena of in-phase synchronization and opposite-phasesynchronization on the computer calculations for this circuit.
On the other hands, we observed coexistence phenomenaof in-phase quasi-synchronization and opposite-phase quasisynchronization on the experiments for the circuit.
Our future research is to make clear the mechanism of thegeneration of the coexistence phenomenon and the cause ofthat we observed quasi-synchronization.

where σ correspond to the parameter mismatch, and ξ isthe noise. Figure 7 shows simulation results for influencesof the parameter mismatch and the noise. Criterial attractors and phase diﬀerences without the parameter mismatchand the noise are shown in Fig. 7(a). Attractors and phase
diﬀerences containing only the noise are shown in Fig. 7(b).
Attractors and phase diﬀerences containing only the parameter mismatch are shown in Fig. 7(c). Attractors and phase
diﬀerences containing both of the noise and the parametermismatch in Fig. 7(d).
The parameter mismatch and the noise perturb the synchronization. From the results, it seem that the parameter mis- Acknowledgment
match largely aﬀect the synchronization to be perturbed in
the experimentation.
This work was partly supported by CASIO Science Promotion Foundation.
References[1] G. Abramson, V. M. Kenkre and A. R. Bishop, “Analytic solutions for nonlinear waves in coupled reactingsystems,” Physica A: Statistial Mechanics and its Applications, vol. 305, no. 3–4, pp. 427–436, 2002.
[2] C. M. Gray, “Synchronous oscillators in neural systems:mechanisms and functions,” J. Computational Neuroscience, vol. 1, pp. 11–38, 1994.
[3] Y. Zhang, G. H. Du and J. J. Jiang, “Synchronizing Chaos
by Impulsive Feedback Method,” International Jounal ofbifurcation and Chaos, vol. 11, no. 8, pp. 2233–2243,
2001.
[4] T. Maruyama, N. Inaba, Y. Nishio and S. Mori, “Chaosin Auto Gain Controlled Oscillator Containing Time Delay,” Transactions of IEICE, vol. J72-A, no. 11, pp. 1814–1820, 1989.

Figure 7: Attractors and phase diﬀerences in numerical results including noise and parameter mismatch. (a) Without the noise and the parameter mismatch. (b) Containing
only the noise. (c) Containing only the parameter mismatch.
(d) Containing both of the noise and the parameter mismatch.

5. Conclusions- 56 -