IEEE Workshop on Nonlinear Circuit Networks
December 9-10, 2016

Switching Synchronization States ofRing System of Chaotic Circuit
Including Time Delay in One-Direction
Seiya Kita, Yoko Uwate and Yoshifumi Nishio
Dept. Electrical and Electronic Eng.,
Tokushima University
Email: {kita, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract—In this study, we investigate the novel coupling modeland synchronization states observed in coupled time-delayed
chaotic circuits. Moreover, we compare the standard coupling
model and the novel coupling model and investigate the transitionof chaos. The novel coupling model is a ring of coupled chaoticcircuits with one-direction delay effects. The synchronization
state of the novel coupling model is switching synchronization
state. The switching synchronization state is changed by timedelay of each subcircuit. We focus on relationships between
switching synchronization state and the pattern of time delay.
Finally, we investigate the cycle of switching synchronization
state.

I. I NTRODUCTION
Studies on synchronization state are extensively carried out
in various fields [1]-[3]. Recently, in particular, synchronization states in chaotic oscillators have been studied by many
researchers. The behavior of chaotic oscillators is interesting.
Then, chaos phenomena are quite dependent on initial values
and not periodical and predictable. Moreover the synchronization states have caused very interesting phenomena. Synchronization and the related bifurcation of chaotic systems aregood methods to describe various high-dimensional nonlinear
phenomena in the field of natural science. However, many
synchronization states of coupled chaotic oscillators have notbeen solved yet. The synchronization phenomena in electric
circuit make clear the mechanism of the synchronization
phenomena in our daily life. There are many nonlinear systems
containing time delay, such as neural networks, control systems, meteorological systems, biological systems and so on in
the natural world. Namely, it is considered that investigation ofstability in such time-delay systems is significant [4]. Generation of chaos in time delayed system is reported self excitedoscillation system containing time delay [5]. The oscillators
have feedback systems which control gains in this study. Thischaotic circuit can be easily realized by using simple electric
circuit element and analyzed exactly. The coupling switch
connects alternately with one subcircuit and the other witha fixed time interval. On the other hand, there are examples ofnonlinear phenomena, chaotic synchronization and so on [6].
In particular, many studies on synchronization of coupledchaotic circuits have been reported [7].

In this study, we devise the novel coupling model that takes
advantage of features of the time-delayed chaotic circuit. The
novel coupled model is utilizing the characteristics of the circuit having time-delayed feedback. Then, we observe switching synchronization state. We carry out computer calculations
for three coupled auto gain controlled oscillators containing
time delay and investigate time delay of subcircuits effects achange of synchronization state and the time waveform.
II. T IME -D ELAYED C HAOTIC C IRCUIT
Figure 1 shows the time-delayed chaotic circuit. This circuit
consists of one inductor L, one capacitor C, one linear
negative resistor −g and one linear positive resistor G ofwhich amplitude is controlled by the switch containing timedelay. The current flowing through the inductor L is i, and thevoltage between the capacitor C is v. The circuit equations arenormalized as Eqs. (1) and (2) by changing the variables asbelow.
(A) In case of switch connected to −g,{ẋ = y(1)ẏ = 2αy − x,(B) In case of switch connected to G,{ẋ = yẏ = −2βy − x.

(2)

By changing the parameters and variable as follow:√√
C
Vth x, v = Vth y, t = LCτ ,i=√L√CCg= 2α and G= 2β.
LLThe switching operation is shown in Fig. 2, it controls theamplitude of the oscillator. This switching operation is included time delay. Td denotes the time delay. First, the switch
is connected to a negative resistor. In state of that, the voltage v
is amplified up to while v is oscillating, the amplitude exceeds
the threshold voltage Vth which is the threshold control loop.
Second, the system memorize the time as Tth while v isexceeding the threshold voltage Vth and that state is remained- 67 -

for Tth . In subsequent the instant of exceeding threshold
Vth , the switch stays the state for Td . After that switch isconnected to positive resistor during Tth . The switch does
not immediately connect in the positive resistor however theswitch is connected after Td . A set of switching operations
control the amplitude of v. By using mapping method tothis circuit, we could derive the one-dimensional Poincare
map explicitly from each circuit, and the Poincare map wasproved to have a positive Liapunov number with computerassistances [3].

delayed chaotic circuits. Two cases of interest are considered:
the coupling elements are resistors R and inductors L0 . Wechange√the parameters and variables as follows.
√√CCin =Vth xn , vn = Vth yn , t = LCτ , g= 2α,LL√√LCCG= 2β, γR = Rand γL0 =.
LLL0The normalized circuit equations of the system are given as
follows:(A) In the case that the switch is connected to the negative
III. R ING C OUPLED T IME -D ELAYED C HAOTIC C IRCUIT
resistor
In this section, we investigate the synchronization state for { ẋn = yn
(3)three time-delayed chaotic circuits coupled by different eleẏn = −xn + 2αyn + γm (xn−1 − 2xn + xn+1 )ments. Figure 3 shows a schematic of the three coupled time(B) In the case that the switch is connected to the positiveresistor{ẋn = ynVth(4)Delayẏn = −xn − 2βyn + γm (xn−1 − 2xn + xn+1 )Td

Td

where n = 1, 2, 3, m = R, L0 and x0 = x3 , x4 = x1 .

inCCnCvn L-gFig. 1.

A. Coupled by resistors RFigure 5 shows some simulation results, in which thequasi-in-phase synchronization state can be observed. Whenthe coupling strength γR is larger than 0.1, full in-phasesynchronization can be observed. However, full in-phase synchronization cannot be observed or synchronization is lost in
the case of a small γR . The Poincare section in Fig.5(d) shows
the presence of chaos.

SWG

Time-delayed chaotic circuit.
v

B. Coupled by inductors L0
VthFigure 6 shows some simulation results in the case of timedelayed chaotic circuits coupled by inductors L0 . In-phasesynchronization state and three-phase synchronization state
can be observed when γL0 is smaller than 0.1. When γL0 sets
0.01, three-phase and in-phase are switched. When γL0 sets0.5, only three-phase synchronization state can be observed.
The Poincare section in Fig.6(d) shows the presence of chaos.

time(b)(a) v< Vth(b) v> VthTth(a)timeOPERATIONtime

Td
Time Delaythe switch is connected to positive resistor

Fig. 2.

Switching operation.
Td3

CC1

Td1
Td1

CC1
L0or

CC3

R

Td2

Td3
CC3

R

CC2
Td2

CC2
Fig. 4.
Fig. 3.

L0 or

Ring coupling model.

- 68 -System including time delay in one direction.

(a) Attractor(a) Attractor(b) Lissajous figure(b) Lissajous figure(c) Time waveform(c) Time waveform(d) Poincare section(d) Poincare section

Fig. 5. Simulation results of system coupled by resistors R. of

α = 0.015, β = 0.5, γR = 0.01 and Tdn = π.

Simulation results of system coupled by inductors L0 .
α = 0.015, β = 0.5, γL0 = 0.01 and Tdn = π.
Fig. 6.

IV. S YSTEM I NCLUDING T IME D ELAY IN O NE D IRECTION

B. Coupled by inductors L0

The circuit in this study has characteristic time delays
methods. We have devised coupled systems as shown in Fig. 4.
It is called coupled systems and “a ring of coupled chaotic
circuits with one-direction delay effects” The normalized
circuit equations of this system are the same as Eqs.(3) and(4). It is different pnly how to use a time-delayed signal. The
novel coupling model has three time delays Tdn .

When we use inductors L0 as coupling elements, thesimulation results are shown in the Fig. 8 can be observed.
The time waveform in Fig.8(a) shows a phase difference andthe amplitude of yn is switched sequentially. The switching
synchronization state with phase difference can be observed.
The amplitude alternately diverges and converges with different divergence and convergence times. However, when γL0
is larger than 0.1, the switching synchronization state is lost.
The presence of small chaos is shown Fig. 8(d). The chaos ofFig. 8(d) is changing compared to Fig. 6(d) by one-direction
delay effects. Changing the time delays Tdn change the thecycle of the time waveform. Figure 9 shows cycle of switching
synchronization with symmetric or asymmetric delay. Thevariation of cycle in asymmetric delay is less than symmetricdelay. Moreover, when compared with the coupling element,
the number of cycles is larger for system coupled by resistorsR.

A. Coupled by resitors RWe use resistors R as coupling elements. The simulation
results are shown in the Fig. 7. The time waveform in Fig.7(a)shows in-phase synchronization and the amplitude of yn isswitched sequentially. This phenomenon is defined as theswitching synchronization state. There are three definitionsof switching synchronization state. First, maintain a constantphase difference. Second, amplitude oscillations in the order.
Third, patterns of amplitude oscillation. However, when γRis larger than 0.1, the switching synchronization state is lostand a full in-phase synchronization state can be observed.
Figure 7(d) shows the presence of small chaos. The chaos ofFig. 7(d) is changing compared to Fig. 5(d) by one-direction
delay effects. Changing the time delays Tdn change the cycleof the time waveform. Figure 9 shows the cycle of switching
synchronization with symmetric or asymmetric delay. The
variation of cycle in asymmetric delay is less than symmetricdelay.

V. C ONCLUSION
In this study, we devised coupled systems that take advantage of features of the time-delayed chaotic circuit. We
investigated the synchronization state of a ring of coupledchaotic circuits with one-direction delay effects. As a result,some special synchronization states can be observed. This is
the switching synchronization state. The switching synchronization state. There are existence or non-existence of phasedifference in the switching synchronization state. Moreover,- 69 -

(a) Attractor(a) Attractor(b) Lissajous figure(b) Lissajous figure(c) Time waveform(c) Time waveform(d) Poincare section(d) Poincare section

Fig. 7. Simulation results of system coupled by resistors R.
α = 0.015, β = 0.5, γR = 0.01 and Tdn = π.

Simulation results of system coupled by inductors L0 .
α = 0.015, β = 0.5, γL0 = 0.01, Tdn = π.
Fig. 8.

when time delay of subcircuits changes, the cycle of switching
synchronization state changes. The switching of the amplitude
can be observed by difference of time delay. The variation ofcycle in asymmetric delay is less than symmetric delay.
ACKNOWLEDGMENT
This work was partly supported by JSPS Grant-in-Aid for
Challenging Exploratory Research 26540127.
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Fig. 9. Cycle of switching synchronization coupled by R or L0

with symmetric or asymmetric delay.

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