Journal of Signal Processing, Vol.20, No.4, pp.125-128, July 2016

SELECTED PAPER AT NCSP'16

Synchronization State of Chaotic Circuit Containing Time Delay in One Direction
Seiya Kita, Yoko Uwate and Yoshifumi Nishio
Department of Electrical and Electronic Engineering, Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan
E-mail: {kita, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract
A synchronization state can be observed in coupled circuits.
Furthermore, an interesting synchronization state was confirmed in coupled time-delayed chaotic circuits. In this study,
we propose novel coupled systems and investigate the synchronization state in coupled time-delayed chaotic circuits.
The proposed coupling methods of time-delayed chaotic circuits depend on the attractor type. We focus on the relationships between the synchronization state and the coupling
method. Moreover, we investigate the special coupling methods of time-delayed circuits in this study.
1. Introduction
There are many nonlinear systems containing a time delay,
such as neural networks, control systems, meteorological systems, biological systems and so forth, in the natural world.
Thus, the investigation of the stability in such time-delayed
systems is important [1]. The generation of chaos has beenreported in all self-excited oscillation systems containing atime delay [2]. Such a chaotic circuit can be easily realized
by using a simple electric circuit element and can be analyzed
exactly. On the other hand, there are examples of nonlinear
phenomena, such as chaotic synchronization and so forth [3].
In particular, many studies on the synchronization of coupledchaotic circuits have been reported [4].
In this study, we devise coupled systems that take advantage of the features of time-delayed chaotic circuits. The
novel coupled systems utilize the characteristics of circuits
having time-delayed feedback. These circuits also containgain-controlled chaotic oscillators with a time delay and a
feedback system for controlling the gain. We investigate
the synchronization state in coupled time-delayed chaotic circuits. By carrying out computer simulations, it is shownthat the time delay of subcircuits changes the synchronization
state.

tive resistor −g and one linear positive resistor G, whose amplitude is controlled by a switch containing a time delay. Thecurrent flowing through the inductor L is i, and the voltage
across the capacitor C is v. The circuit equations are normalized to Eqs. (1) and (2) by changing the variables as below.
(A) In the case of a switch connected to the negative resistor,{ẋ = y(1)ẏ = 2αy − x(B) In the case of a switch connected to the positive resistor,{ẋ = y(2)ẏ = −2βy − x
We change the parameters and variables as follows.
√√CVth x, v = Vth y, t = LCτ
i=L√√CC= 2α and G= 2βgLLThe switching operation used to control the amplitude of theoscillator is shown in Fig.2. This switching operation includes a time delay denoted by Td . First, the switch is connected to the negative resistor. Then, the voltage v is amplified while it is oscillated until v exceeds the threshold voltage
Vth . Second, the system memorizes the time during whichv exceeds the threshold voltage Vth as Tth . After time Td ,switch is connected to the positive resistor for time Tth . AVthDelayTdTdinCCnCvn L-gSWG2. Time-Delayed Chaotic Circuit
Figure 1 shows a time-delayed chaotic circuit. This circuit
consists of one inductor L, one capacitor C, one linear negaJournal of Signal Processing, Vol. 20, No. 4, July 2016

Figure 1: Time-delayed chaotic circuit

125

v
Vthtime(a)time

OPERATION
time

DELAY

switch connected to positive resistor

Figure 2: Switching operation

Td
CC1
L0

Td

or

R0

Td
CC3

CC2

Figure 3: Ring system coupled by resistors

set of switching operations is used to control the amplitude ofv. By applying a mapping method to this circuit, we can derive the one-dimensionale Poincare map explicitly from eachcircuit, and the Poincare map was proved to have a positive
Lyapunov number by computer simulation [3].
3. Coupled Time-Delayed Chaotic Ring Circuit==yn−xn + 2αyn + γ(yn−1 − 2yn + yn+1 )(3)(B) In the case that the switch is connected to the positiveresistor{ẋn = yn(4)ẏn = −xn − 2βyn + γ(yn−1 − 2yn + yn+1 )(b)(a) v< Vth(b) v> Vthresistor{ẋnẏnwhere n = 1, 2, 3 and x0 = x3 , x4 = x1 . Figure 4 showssome simulation results, in which the in-phase synchronization state can be observed. When the coupling strength γ
is large, full in-phase synchronization can be observed. However, full in-phase synchronization cannot be observed or synchronization is lost in the case of a small γ.
3.2 Coupled by inductors L0Figure 5 shows some simulation results in the case of timedelayed chaotic circuits coupled by inductors L0 . We change
the parameters and variables when the coupled ring system isconnected by inductors as follows.
√√CVth xn , vn = Vth yn , t = LCτ
in =L√√
LCC= 2α, G= 2β and γ ′ =gL
LL0The normalized circuit equations of the system are given as
follows:(A) In the case that the switch is connected to the negative
resistor{ẋn = yn(5)ẏn = −xn + 2αyn + γ ′ (xn−1 − 2xn + xn+1 )In this section, we investigate the synchronization state for(B) In the case that the switch is connected to the positive
three time-delayed chaotic circuits coupled by different meth- resistor
ods. Figure 3 shows a schematic of the three coupled time- {ẋn = yn
delayed chaotic circuits. Two cases of interest are considered:(6)ẏn = −xn − 2βyn + γ ′ (xn−1 − 2xn + xn+1 )the coupling elements are resistors R0 and inductors L0 . Wechange the parametersand variables as follows.
√√C
in =Vth xn , vn = Vth yn , t = LCτ
L
√
√
√
CC
C= 2α, G= 2β and γ = R0gLL
L
3.1 Coupled by resistors R0

where n = 1, 2, 3 and x0 = x3 , x4 = x1 . In-phase synchronization and three-phase synchronization can be observed in
the ring system coupled by inductors L0 . When the couplingstrength γ ′ is equal to 0.01, synchronization is lost. Consequently, a certain coupling strength is required for synchronization.
4. System Including Time Delay in One Direction

The circuits in this study employ characteristic time deHere we discuss case of a system coupled by resistors R0 .
lay methods. We have devised the coupled system shown in
The normalized circuit equations of the system are given asFig.6. This system is coupled by resistors R0 or inductors L0 .
follows:
This system includes a time delay in one direction.
(A) In the case that the switch is connected to the negative126

Journal of Signal Processing, Vol. 20, No. 4, July 2016

(a) α = 0.015, β = 0.5, γ = 0.01 and Td = π(a) α = 0.015, β = 0.5, γ ′ = 0.1 and Td = π(b) α = 0.015, β = 0.5, γ = 0.1 and Td = π(b) α = 0.015, β = 0.5, γ ′ = 0.2 and Td = πFigure 4: Simulation results of ring system coupled by resis- Figure 5: Simulation results of ring system coupled by inductors
tors
4.1 Coupled by resistors R0First, we use resistors R0 as coupling elements. The normalized circuit equations of this system are the same asEqs.(3) and (4). The result shown in Fig.7 can be obtained
from the difference in the coupling strength γ. The time
waveform in Fig.7(a) shows in-phase synchronization and theamplitude of yn is switched sequentially. However, when γ
is larger than 0.1, the switching synchronization state is lostand a full in-phase synchronization state can be observed.

CC1TdTdCC3L0orR0

CC2
Td

Figure 6: System including time delay in one direction

4.2 Coupled by inductors L0
When we use inductors L0 as coupling elements, the result
shown in Fig.8 can be obtained from the difference in the coupling strength γ ′ , where the normalized circuit equations areEqs.(5) and (6). The time waveform in Fig.8(a) shows a phasedifference and the amplitude of yn is switched sequentially.
However, when γ ′ is larger than 0.1, the switching synchronization state is lost. Generally, switching synchronization
can be observed when the system including a time delay in
Journal of Signal Processing, Vol. 20, No. 4, July 2016

one direction is coupled by resistors R0 or inductors L0 . The
amplitude alternately diverges and converges with different
divergence and convergence times.
5. Conclusions
In this study, we investigated the synchronization state ofnovel coupled systems of time-delayed chaotic ring circuits
127

(a) α = 0.015, β = 0.5, γ = 0.01 and Td = π(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(b) α = 0.015, β = 0.5, γ ′ = 0.1 and Td = π

Figure 7: Simulation results of system coupled by resistors Figure 8: Simulation results of system coupled by inductorsthat combines switching including time delay in one direction
including time delay in one direction

coupled by various methods. Four types of coupling systemwere investigated. In the case of a ring circuit coupled by resistors, we observed an in-phase synchronization state. In theother cases of the ring circuit coupled by inductors, in-phasesynchronization and three-phase synchronization states wereobserved. We devised coupled systems that take advantage ofthe features of the time-delayed chaotic circuits. As a result,some special synchronization states were observed. Switching of the amplitude of the voltage in addition to the in-phasesynchronization and three-phase synchronization state wereobserved from the difference of coupling strength and coupling elements.

Acknowledgment

References[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: Chaos in
an auto gain controlled oscillator containing time delay,
Trans. IEICE, Vol. J 72-A, pp. 1814-1820, 1989.
[3] T. Maruyama, N. Inaba, Y. Nishio and S. Mori: Chaos inself oscillator circuit containing time delay, Proceedingsof IEEE Midwest Symposium on Circuits and Systems(MWSCAS’90), Vol. 2, pp. 1055-1058, 1990.
[4] L. M. Pecora and T. L. Carroll: Synchronization inchaotic systems, Physical Review Letters, Vol. 64, No.
8, pp. 821-824, 1990.

This work was partly supported by JSPS Grant-in-Aid forChallenging Exploratory Research 26540127.
128

Journal of Signal Processing, Vol. 20, No. 4, July 2016