IEEE Workshop on Nonlinear Circuit Networks
December 11-12, 2015

Synchronization State of System
Including Time Delay in One Direction
Seiya Kita

Yoko Uwate

Yoshifumi Nishio

Dept. of Electrical and Electronic Eng., Dept. of Electrical and Electronic Eng., Dept. of Electrical and Electronic Eng.,
Tokushima University
Tokushima University
Tokushima University
2-1 Minami-Josanjima,
2-1 Minami-Josanjima,
2-1 Minami-Josanjima,
Tokushima 770–8506, Japan
Tokushima 770–8506, Japan
Tokushima 770–8506, Japan
Email: kita@ee.tokushima-u.ac.jp
Email: uwate@ee.tokushima-u.ac.jp
Email: nishio@ee.tokushima-u.ac.jp

Abstract—Synchronization states can be observed in coupledcircuits. Further, interesting synchronization state were confirmed in coupled time delayed chaotic circuits. In this study,
we investigate the new coupled system and synchronization
states observed in coupled time delayed chaotic circuits. The
pattern of coupled system by time delayed chaotic circuits
depends on attractor types. We focus on relationships between
synchronization state and the pattern of coupling. Moreover, weinvestigate the specific use of time delay of time delayed circuitin this study.

I. I NTRODUCTION
There are many nonlinear systems containing time delay,
such as neural networks, control systems, meteorologicalsystems, biological systems and so on in the natural world.
Namely, it is considered that investigation of stability in suchtime-delay systems is important [1]. Generation of chaos ofthem all is reported self excited oscillation system containing
time delay. This chaotic circuit can be easily realized byusing simple electric circuit element and analyzed exactly [2].
There are examples of nonlinear phenomena, chaotic synchronization, clustering phenomenon and so on [3]. In particular,a number of studies on synchronization of coupled chaotic
circuits have been made [4].
In this study, we investigate synchronization states by a
new coupled system. The new coupled system is utilizing
the characteristics of the circuit to be used in this study. By
carrying out computer simulations, time delay of subcircuits
effects a change synchronization state.

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) (2) by changing the variables as below.
(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:√√
Ci=Vth x, v = Vth y, t = LCτ ,L√√CCg= 2α and G= 2β.
LLv
Vthtime

VthDelayTdTd(a) v< Vth(b) v> VthinCCnCvn L-g(a)timeSWGFig. 1. Time delayed chaotic circuit.

(b)

OPERATION
Fig. 2. Chaotic attractor obtained by computer simulation.

DELAY

the switch is connected to positive resistor
Fig. 3. Switching operation.

-8-timeFigure 1 is chaotic attractor observed from the circuit. The
switching operation is shown in Fig. 3, it controls the amplitude of the oscillator. This switching operation is included
time delay. Td denotes the time delay. First, the switch isconnected 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 remainedfor 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. Figure 2 shows chaotic attractorof time delayed chaotic circuit.

IV. S YSTEM INCLUDING TIME DELAY IN ONE DIRECTION
The circuit in this study have characteristic time delays
methods. We have devised a coupled system as shown inFig. 6. It will be called the coupled system and ”system
including time delay in one direction”

III. R ING COUPLED TIME DELAYED CHAOTIC CIRCUITIn this study, we investigate the pattern of coupled threetime delayed chaotic circuits. Figure 4 shows the schematicof coupled three time delayed chaotic circuits by resistors R.
By changing the parameters and variables as follows:√√
Cin =Vth xn , vn = Vth yn , t = LCτ ,L
√
√
√
CCC
g= 2α, G= 2β and γ = R.
LL
L
In case of coupled system by resistors R, the normalized
circuit equations of the system are given as follows:(A) In case of that switch is connected to −g,{ẋn = yn(3)ẏn = −xn + 2αyn + γ(yn−1 − 2yn + yn+1 ),(B) In case of that switch is connected to G,{ẋn = ynẏn = −xn − 2βyn + γ(yn−1 − 2yn + yn+1 ).

(a) α = 0.015, β = 0.5, γ = 0.01 and Td = π.

(4)where (n = 1, 2, 3) and x0 = x3 , x4 = x1 . Figure 5 showssome of simulation results. In calculation result, in-phasesynchronization state can be observed. When the couplingstrength γ is large, full in-phase synchronization can beobserved. However full in-phase synchronization can not beobserved or synchronization is lost in case of small couplingstrength γ.

(b) α = 0.015, β = 0.5, γ = 0.1 and Td = π.
Fig. 5. Simulation results of ring coupled system by resistors. Attractor.
Lissajous figure. Time waveform. Red, blue and green colors denote y1 , y2
and y3 respectively.

Td

CC1

Td
R

CC1

R
Td

R

CC3
Td

R

RR

CC3

CC2

CC2
Td

Td
Fig. 4. Ring coupled system by resistors.
Fig. 6. System including time delay in one direction.

-9-In case of the ring coupled by resistors R, such a result asshown in the Fig. 7 was obtained by difference of couplingstrength γ. when the coupling strength γ is bigger then
0.1, switching synchronization state is lost and full in-phasesynchronization state can be observed.
Generally switching synchronization can be observed whensystem including time delay in one direction is coupled byresistors R.
V. C ONCLUSION
In this study, we have investigated synchronization state
observed some pattern of ring coupled by time delayed chaotic
circuits. In case of ring coupled by resistors, we observed
in-phase synchronization state. We devised a coupled systemthat takes advantage of features of the time delayed chaoticcircuit. As a result, some special synchronization state canbe observed. The switching of the amplitude of voltage inaddition to the in-phase synchronization state can be observedby difference of coupling strength.
R EFERENCES[1] X. Liu, “Stability of impulsive control systems with timedelay. Mathematical and Computer Modelling,” Vol.39,
pp.511-519, 2004.
[2] T. Maruyama, N. Inaba, Y. Nishio and S. Mori, “Chaosin an Auto Gain Controlled Oscillator Containing TimeDelay,” Trans. IEICE, vol. J 72-A, pp. 1814-1820, Nov.
1989.
[3] T. Maruyama, N. Inaba, Y. Nishio and S. Mori, “Chaosin Self Oscillator Circuit Containing Time Delay,” Proceedings of IEEE Midwest Symposium on Circuits andSystems (MWSCAS’90), vol. 2, pp. 1055-1058, Aug.
1990.
[4] L. M. Pecora and T. L. Carroll, “Synchronization inChaotic Systems,” Physical Review Letters, vol. 64, no.8,
pp. 821-824, Feb. 1990.

(a) α = 0.015, β = 0.5, γ = 0.01 and Td = π.

(b) α = 0.015, β = 0.5, γ = 0.1 and Td = π.
Fig. 7. Simulation results of system including time delay in one direction
coupled by resistor. Attractor. Lissajous figure. Time waveform. Red, blue andgreen colors denote y1 , y2 and y3 respectively.

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