IEEE Workshop on Nonlinear Circuit Networks
December 12-13, 2014

Synchronization Pattern of Chaotic System
Containing Time Delay
Seiya Kita, Takuya Nishimoto, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University,
2-1 Minamijosanjima, Tokushima, 770-8506 Japan
Email: {kita, nishimoto, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract—In this study, we investigate synchronization phenomena observed in coupled chaotic circuits containing timedelay. Various synchronization states can be observed . Especially, we focus on relationships between synchronization stateand parameters. Coexisting synchronization states depending
on initial values can be observed in the proposed system bycarrying out computer calculation. Moreover, we investigate theeffect of chaotic behaviour on subcircuit to synchronization
states. Chaotic strength of subcircuit possibly induces anti-phasesynchronization.

I. I NTRODUCTION
Recently, researches on chaos phenomena have been attracting attention. The chaos phenomenon shows a complex
unpredictable behavior. It can be observed in various sciencefields, such as biology, economics, physics and astronomy.
A number of studies on synchronization of coupled chaotic
circuits have been made [1]. The synchronization is one of
the typical phenomena observed in nature. There are flashingof fireflies, rhythm of the heart cells and laser oscillation in
examples of the synchronization. In addition, synchronization
phenomenon, there is an advantage that a large energy isobtained by synchronizing the small energy. By utilizing thisproperty, it is possible to realize a pacemaker that can control
the synchronization of the cardiomyocytes of the heart, whichis actually used as a medical device. On the other hand,electric circuit observed the chaotic phenomena. There area lot of merits using electric circuit to investigate chaoticbehaviour. For instance, inexpensive element, short experimenttime, high repeatability of the experiment and so on. There arevarious types of chaotic circuits. We focus on the chaos circuit
including a also time delay in that [2]-[4].
In this study, we investigate two coupled chaotic circuits
containing time delay. Especially we focus on the synchronization phenomena of our proposed circuit. By carrying outcomputer simulations, two types of synchronization phenomena depending on initial values can be observed. Moreover,
we investigate the relationships between parameters and synchronization phenomena.
II. C IRCUIT M ODELThe chaotic circuit used in this study is shown in Fig. 1. The
circuit is composed of a −gLC oscillator and the amplitudecontrol loop. −g is a negative resistance, the current flowing

through the inductor L is i, the voltage between the capacitor
C is v. Switching operation is shown in Fig. 2, it controls
the amplitude of the oscillator. The switching controls theamplitude of v by changing from the negative resistance tothe positive resistance when the amplitude is larger than thethreshold value. This switching operation is included timedelay. Td denotes the time delay. First, the switch is connected
to a negative resistance. v is amplified while oscillating,the amplitude is bigger a certain threshold voltage Vth . Theswitch does not immediately connect in the positive resistancehowever the switch is connected after Td seconds. In thecomputer simulation, Td is set to π/(1 − α2 ).

VthiCvFig. 1.

LSW-gGCircuit model.

The circuit equations of the system are given as follow:(A) In case of switch connected to -g, L di = v
dt
(1)
dv

 C= gv − i.
dt(B) In case of switch connected to G, L di = v
dt
(2)
dv

 C= −Gv − i.
dt
By changing the parameters and variable as follows:√√C
i=Vth x, v = Vth y, t = LCτ ,L- 52 -v

L0VthVthVthtimei1C(b)(a) v< Vth(b) v> VthDELAYDELAYSW-gGCv2

LSW-g

G

timeFig. 3.

time

the switch is connected to negative resistor

Fig. 2.

√L(a)OPERATION

gv1i2Switching operation.

√CC= 2α and G= 2β.
LL

The normalized circuit equations of the system are given as
follow:(A) In case of switch connected to −g,{ẋ = y(3)ẏ = 2αy − x.
(B) In case of switch connected to G,{ẋ = yẏ = −2βy − x.

A coupled 2 chaotic circuits containing time delay

anti-phase synchronization states, respectively. By increasing
the parameter β, anti-phase synchronization phenomena areinduced and in-phase synchronization gradually become unstable. The parameter β qualitatively denotes the chaotic strength.
Namely, it is considered that anti-phase synchronization stateis induced by chaotic behavior of each subcricuit as shown in
Fig. 4 (3).

(4)(1) α = 0.015, β = 0.1, γ = 0.2.

III. T WO COUPLED CHAOTIC CIRCUITS
Figure 3 shows the schematic of two chaotic circuits containing time delay coupled by a inductor. By changing theparameters and variables
as follows:√√Cin =Vth xn , vn = Vth yn , t = LCτ ,L√√CCLg= 2α, G= 2β and γ =;LL
L0The normalized circuit equations of the system are given as
follows:(A) In case of the switch is connected to −g,{ẋn = yn(5)ẏn = 2αyn − xn − γ(xn+1 − xn ).
(B) In case of switch connected to G,{ẋn = ynẏn = −2βyn − xn − γ(xn+1 − xn ).

(2) α = 0.015, β = 0.1, γ = 0.2.

(3) α = 0.015, β = 0.13, γ = 0.2.
Fig. 4. Simulation results of N = 2. (a) Attractor. (b) Lissajous figure.
(c) Timewaveform. Red and blue colors denote x1 and x2 respectively.

(6)

where
(n = 1, 2),x3 = x1 .

Figure 4 shows simulation results in case of two coupledchaotic circuits. Figure 4 (1) and (2) are two types of synchronization phenomena depending on the initial values which
are corresponding to almost in-phase synchronization and- 53 -IV. C ONCLUSION
In this study, we have investigated synchronization phenomena observed by coupled chaotic circuits containing time delay.
As a result, induction of anti-phase synchronization causedby chaotic strength of subcircuit have clarified. In our futurework, we will investigate the parameter region of coexisting
synchronization and the mechanism of induction of anti-phasesynchronization phenomena.
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Controlled Oscillator Containing Time Delay,” Trans. IEICE, vol. J 72-A,
pp. 1814-1820, Nov. 1989.
[3] T. Maruyama, N. Inaba, Y. Nishio and S. Mori, “Chaos in Self Oscillator
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Aug. 1990.
[4] T. Kousaka, Y. Umakoshi, T, Ueta and H. Kawakami, “Bifurcation ofnonlinear circuits with periodically operating switch,” IEEE, vol. 84, Issue
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