Synchronization State of Coupled Chaotic Circuits
Containing Time DelaySeiya Kita

Yoko Uwate

Yoshifumi Nishio

Dept. Electrical and Electronic Eng.
Tokushima University
Email: kita@ee.tokushima-u.ac.jp

Dept. Electrical and Electronic Eng.
Tokushima University
Email: uwate@ee.tokushima-u.ac.jp

Dept. Electrical and Electronic Eng.
Tokushima University
Email: nishio@ee.tokushima-u.ac.jp√Abstract— In this study, we investigate synchronization statesobserved in coupled chaotic circuits containing time delay. We
focus on relationships between synchronization state and chaotic
strength. Coexisting synchronization states depending on initialvalues can be observed in the proposed system. Moreover, weinvestigate the effect of chaotic behavior on the subcircuit tosynchronization states.

I. I NTRODUCTION
Generally, generation of chaos is reported self excitedoscillation system containing time delay. This chaotic circuit
can be easily realized by using simple electric circuit element and analyzed exactly [1]. Furthermore, a number ofstudies on synchronization of coupled chaotic circuits havebeen made [2]. In this study, we investigate synchronization
state observed in some chaotic circuits containing time delay
coupled by inductor. Especially, we focus on the transitionof synchronization state by changing the chaotic strength
in this system. By carrying out computer simulations, twotypes of synchronization state depending on initial values andparameters can be observed. Moreover, we investigate therelationship between parameters and synchronization state. In
addition we consider synchronization state of changes due tothe number of subcircuits.
II. C IRCUIT M ODELBy changing the parameters and variable as follow:√√Ci=Vth x, v = Vth y, t = LCτ ,LC= 2β.
LVthiv

LSW-gGFig. 1. Chaotic circuit containing time Fig. 2. Chaotic attractor obtained bycomputer simulation.
delay.

v
Vthtime(a) v < Vth(b) v > Vth(2)√Figure 2 is chaotic attractor observed from the circuit. Theswitching 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 remained
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. In the computer simulation, Td isC

Figure 1 shows the chaotic circuit containing time delay.
This circuit consists of one inductor L , one capacitor C , one
linear negative resistor −g and one linear positive resistor Rof which 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.

gC= 2α and GLTth(a)

(b)

(a)
time

OPERATIONtime
Td
the switch is connected to negative resistor

Fig. 3.

Switching operation.

fixed as π/(1 − α2 ). By increasing the parameter value of β,
the chaotic strength is qualitatively swelled since the solution
set of attractor is enlarged accordance with the increase theparameter value.
III. T WO C OUPLED C HAOTIC C IRCUITS
Figure 4 shows the coupled chaotic circuit. 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 = 2αyn − xn − γ(xn+1 − xn ).
(B) In{case of that switch is connected to G,
ẋn = yn(4)ẏn = −2βyn − xn − γ(xn+1 − xn ).
where (n = 1, 2) and x3 = x1 . By changing the parametersand variable as√follow:√C
Vth xn , vn = Vth yn , t = LCτ ,in =L√√CCLg= 2α, G= 2β and γ =.
LLL0Figure 5 shows example of simulation results. Figures 5(1) and (2) are in-phase synchronization and anti-phase synchronization states depending on the initial values. We investigate the basin of attraction regarding anti-phase and inphase synchronization. Figure 6 shows the example of thebasin. Red and blue colors denote the basin related to antiphase and in-phase synchronization state respectively. By
increasing the value of β, the proportion of basin regardinganti-phase synchronization state is increased. The relationof the parameter value of β and the population of antiphase synchronization states show Fig. 7. By increasing β,
the proportion of basin regarding anti-phase synchronization
state is increased. Namely, it is considered that anti-phasesynchronization state is induced by chaotic behavior of eachsubcircuits.
IV. A R ING OF THREE C OUPLED C HAOTIC C IRCUIT
Figure 8 shows schematic of a ring of N coupled chaotic
circuits containing time delay coupled by the inductor. The
normalized system equations are given as follows:(A) In case of switch connected to −g,{ẋn = yn(5)ẏn = 2αyn − xn − γ(2xn − xn+1 − xn−1 ).

(B) In case of switch connected to G,{ẋn = ynẏn = −2βyn − xn − γ(2xn − xn+1 − xn−1 ).
where (n = 1, 2, ..., N ) , x0 = xN and xN +1 = x1 .

(1) α = 0.015, β = 0.1, γ = 0.2.

(2) α = 0.015, β = 0.1, γ = 0.2.
Fig. 5. Simulation results of N = 2. (a) Attractor. (b) Lissajous figure.
(c) Timewaveform. Red: x1 , Blue: x2 .

(1) α = 0.015, β = 0.1, γ = 0.2(2) α = 0.015, β = 0.5, γ = 0.2(3) α = 0.015, β = 0.7, γ = 0.2(4) α = 0.015, β = 0.8, γ = 0.2

Fig. 6. Basin of attraction. Red and blue colors denote anti-phase synchronization state and in-phase synchronization state respectively. y1 = 0.1112,
y2 = −0.1113.

L0VthC

Vthv1

i1i2

LL

SW-gGCv2

SW-gG

Fig. 7.
Fig. 4.

Two coupled chaotic circuits containing time delay.

(6)

Proportion of anti-phase synchronization state.

A. In case of N = 3
Figure 9 shows the simulation results in case of N = 3. We
observed the two types of synchronization state same with
N = 2. In-phase synchronization and three-phase synchronization states depending on initial values can be observed
as shown in Fig. 9 (1) and (2). Also, by increasing theparameter value β, in-phase synchronization state is graduallychanged and finally the system generates only three-phasesynchronization as shown in Fig 9 (3). This behavior implies
the basin of attraction regarding to enlargement of three-phasesynchronization for increase of the parameter. In state of threephase synchronization, the phase difference is almost 120◦with respect to neighboring subcircuit.

L0

L0

L0L0VthVthVthiNi2i1
CFig. 8.

vNLSW-gGCv1

LSW-gC

Gv2

LSW-gG

A ring of coupled chaotic circuits containing time delay.

B. In case of N = 4
Figure 10 shows the simulation results in case of N = 4.
Red, blue green and black colors denote x1 , x2 x3 and x4
respectively. In this case, similar behavior with the case ofN = 2 can be observed. For instance, in-phase synchronization and anti-phase synchronizations state can be observed incase of N = 4. It seems that anti-phase synchronization tend
to cause the stable state for the system inasmuch as the orbit
is exactly periodic. Moreover, from the results of N = 2and N = 4, the system generates in-phase and anti-phasesynchronization in case of that the number of coupled chaotic
circuit is even number in any number of coupled circuit.

(1) α = 0.015, β = 0.1, γ = 0.2.

C. In case of N = 5
Similar synchronization state in N = 3 can be observedin N = 5 as shown in Fig. 11. Red, blue and green colors
denote x1 , x2 and x3 respectively. Except for x1 , x2 , x3 , time
waveforms are described with black color. In the parameter of
weak chaotic region, the system generates coexisting synchronization state. And only five-phase synchronization is inducedin certain chaotic. The two types Lissajous figures depending
initial value can be observed the parameter value of β as shownin Fig. 11 (3) and (4). In state of five-phase synchronization,
there is phase difference of 72◦ with respect to neighboring
subcircuit. We compare the value of β when coupled circuits
are attracted to N -phase synchronization state. β in case ofN = 5 is smaller than N = 3.

(2) α = 0.015, β = 0.1, γ = 0.2.

(3) α = 0.015, β = 0.8, γ = 0.2.

V. C ONSIDERATION
Table I summarizes results from this study. The synchronization state can be classified by the number of coupledchaotic circuit whether the number is even or odd. When thenumber of subcircuits is large, synchronization state is easily
attracted anti-phase synchronization or N -phase synchronization.
VI. C ONCLUSIONSIn this study, we have investigated synchronization state
observed in two coupled chaotic circuits containing timedelay. As a result, induction of anti-phase synchronization
or N -phase synchronization state caused by increasing theFig. 9. Simulation results of N = 3. (a) Attractor. (b) Lissajous figure.
(c) Timewaveform. Red, blue and green colors denote x1 , x2 and x3
respectively.

chaotic strength of subcircuit has confirmed. Furthermore, wehave investigated synchronization state observed on a ring ofcoupled chaotic circuits containing time delay. As a result,induction of anti-phase or N -phase synchronization causedby chaotic strength of subcircuit have clarified. Moreover,synchronization state can be classified by the number ofcoupled chaotic circuit whether the number is even or odd.
In future work, We will investigate the parameter region ofcoexisting synchronization and the mechanism of induction of

TABLE IS YNCHRONIZATION STATE .
Coupled circuits
N =2
in-phaseθd ≃ 0[◦ ]anti-phaseθd ≃ 180[◦ ]N =3...

in-phaseθd ≃ 0[◦ ]3-phaseθd ≃ 120[◦ ]A ring of coupled circuits
N =5N =nodd
in-phasein-phase...
in-phase◦◦θd ≃ 0[ ]θd ≃ 0[ ]θd ≃ 0 [rad]anti-phase
5-phaseN-phase[rad]θd ≃ 180[◦ ]θd ≃ 72[◦ ]θd ≃ 2πnN =4N =kevenin-phaseθd ≃ 0 [rad]anti-phaseθd ≃ π [rad](1) α = 0.015, β = 0.1, γ = 0.2.

(1) α = 0.015, β = 0.1, γ = 0.2.

(2) α = 0.015, β = 0.1, γ = 0.2.

(2) α = 0.015, β = 0.1, γ = 0.2.

(3) α = 0.015, β = 0.8, γ = 0.2. type1(3) α = 0.015, β = 0.8, γ = 0.2.
(4) α = 0.015, β = 0.8, γ = 0.2. type2
Fig. 10.
Simulation results of N = 4. (a) Attractor. (b) Lissajous figure.
(c) Timewaveform. Red, blue green and black colors denote x1 , x2 x3 and Fig. 11.
Simulation results of N = 5. (a) Attractor. (b) Lissajous figure.
x4 respectively.
(c) Timewaveform. Red, blue and green colors denote x1 , x2 and x3 respectively. Black color shows the xn (n > 3).

anti-phase or N -phase synchronization state.

R EFERENCES[1] T. Maruyama, N. Inaba, Y. Nishio and S. Mori, “Chaos in an Auto Gain
Controlled Oscillator Containing Time Delay,” Trans. IEICE, vol. J 72-A,
pp. 1814-1820, Nov. 1989.

[2] L. M. Pecora and T. L. Carroll, “Synchronization in Chaotic Systems,”
Physical Review Letters, vol. 64, no.8, pp. 821-824, Feb. 1990.