RISP

NCSP’16

International Workshop on
Nonlinear Circuits, Communications and Signal Processing

Honolulu, Hawaii, USA, March 6-9, 2016

Synchronization State of Chaotic Circuit Containing Time Delay in One Direction
Seiya Kita, Yoko Uwate and Yoshifumi Nishio
Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Phone/FAX:+81-88-656-7000/+81-88-656-7001
Email: {kita, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract
Synchronization state can be observed in coupled circuits.
Further, interesting synchronization state was confirmed incoupled time delayed chaotic circuits. In this study, we propose novel coupled systems and investigate synchronization
state in coupled time delayed chaotic circuits. The proposed
coupling methods of time delayed chaotic circuits depend on
attractor types. We focus on the relationships between synchronization state and the coupling methods. Moreover, weinvestigate the special coupling methods of time delayed circuit in this study.
1. Introduction
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 of stability in suchtime-delay systems is significant [1]. Generation of chaos ofthem all is reported self excited oscillation system containing
time delay [2]. This chaotic circuit can be easily realized byusing simple electric circuit element and analyzed exactly. On
the other hand, there are examples of nonlinear phenomena,
chaotic synchronization and so on [3]. In particular, manystudies on synchronization of coupled chaotic circuits havebeen reported [4].
In this study, we devise coupled systems that takes advantage of features of the time delayed chaotic circuit. The novel
coupled systems are utilizing the characteristics of the circuit having time delayed feedback. This circuit is auto gain
controlled chaotic oscillator containing time delay. The circuit has feedback systems which control the gain. We investigate synchronization state in coupled time delayed chaoticcircuits. By carrying out computer simulations, time delay of
subcircuits effects a change of synchronization state.

ear 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
are normalized as Eqs. (1) and (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:√√
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 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
VthDelay

Td

TdinCCnC

vn L-gSWG

2. Time Delayed Chaotic Circuit
Figure 1 shows the time delayed chaotic circuit. This circuit consists of one inductor L , one capacitor C , one lin-- 719 -Figure 1: Time delayed chaotic circuit.

v

TdVthCC1
time(a) v< Vth(b) v> VthL0Td(b)orR0TdCC3CC2(a)time

OPERATION

Figure 4: Ring coupled system by resistors.
DELAY

time

the switch is connected to positive resistor

Figure 2: Switching operation.

3.1 Coupled by resistors R0
Here is case of coupled system by resistors R0 . 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 ),Figure 3: Chaotic attractor obtained by computer simulation.

(B) In case of that switch is connected to G,{ẋn = ynẏn = −xn − 2βyn + γ(yn−1 − 2yn + yn+1 ),(4)

where (n = 1, 2, 3) and x0 = x3 , x4 = x1 . Figure 5 showssome of simulation results. In calculation result, in-phaseloop. Second, the system memorize the time as Tth while synchronization state can be observed. When the couplingv is exceeding the threshold voltage Vth and that state is re- strength γ is large, full in-phase synchronization can be obmained for Tth . In subsequent the instant of exceeding thresh- served. However full in-phase synchronization can not be obold Vth , the switch stays the state for Td . After that switch is served or synchronization is lost in case of small coupling
connected to positive resistor during Tth . The switch does strength γ.
not immediately connect in the positive resistor however theswitch is connected after Td . A set of switching operations 3.2 Coupled by resistors L0
control the amplitude of v. Figure 3 shows chaotic attractorIn case of time delayed chaotic circuits coupled by inducof time delayed chaotic circuit. By using mapping method
tors L0 , Fig 6 shows some of simulation results. By changto this circuit, we could derive the 1-dimensional Poincard
ing the parameters and variables when ring coupled system ismap explicitly from each circuit, and the Poincark map wasconnected by the inductor as follows:
proved to have a positive Liapunov number with computer√√assistances [3].
C
in =Vth xn , vn = Vth yn , t = LCτ ,L√√CCL= 2α, G= 2β and γ ′ =.
g
3. Ring Coupled Time Delayed Chaotic CircuitLLL0
The normalized circuit equations of the system are given as
In this section, we investigate synchronization state of thecoupling methods of three coupled time delayed chaotic cir- follows:(A) In case of that switch is connected to −g,cuits. Figure 4 shows the schematic of coupled three timedelayed chaotic circuits. Two cases of interest are consid- {ẋn = yn(5)ered: coupling elements are resistors R0 or inductors L0 . Byẏn = −xn + 2αyn + γ ′ (xn−1 − 2xn + xn+1 ),changing the parametersand variables as follows:√(B) In case of that switch is connected to G,√C
in =Vth xn , vn = Vth yn , t = LCτ ,{Lẋn = yn
(6)√√√ẏn = −xn − 2βyn + γ ′ (xn−1 − 2xn + xn+1 ),CCC
g= 2α, G= 2β and γ = R0.
L
LL
where (n = 1, 2, 3) and x0 = x3 , x4 = x1 . In-phase synchro-- 720 -(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 = π.

Figure 5: Simulation results of ring coupled system by resistors. Attractor. Figure 6: Simulation results of ring coupled system by inductors. AttracLissajous figure. Time waveform. Red, blue and green colors denote y1 , y2
and y3 respectively.

tor. Lissajous figure. Time waveform. Red, blue and green colors denote y1 ,y2 and y3 respectively.

nization and three-phase synchronization can be observed in
the ring coupled system by inductors L0 . When the couplingstrength γ ′ is equal to 0.01, synchronization is lost. Consequently, the certain level of coupling strength is required forsynchronization.

CC1

Td

Td
CC3

4. System Including Time Delay in One Direction

L0orR0CC2
Td

The circuit in this study have characteristic time delays
Figure 7: System including time delay in one direction.
methods. We have devised coupled systems as shown inFig. 7. This system is coupled by resistors R0 or inductors L0 .
It is called coupled systems and “system including time delay
be observed. When the coupling strength γ is bigger thenin one direction”
0.1, switching synchronization state is lost and full in-phasesynchronization state can be observed.
4.1 Coupled by resistors R
0Now, we use resistors R0 to coupling elements. The normalized circuit equations of this system are same to Eqs.(3)and (4). The result as shown in the Fig. 8 can be obtainedby difference of coupling strength γ. The time waveformof Fig. 8 (a) is in-phase synchronization and the amplitudeof yn is switching sequentially. However when the coupling strength γ is bigger then 0.1, switching synchronization state is lost and full in-phase synchronization state can4.2 Coupled by inductors L0
when we use inductors L0 to coupling elements , the result
as shown in the Fig. 9 can be obtained by difference of coupling strength γ ′ . Eqs.(5) and (6) are same to the normalized
circuit equations of coupled by inductors L0 . The time waveform of Fig. 9 (a) has the phase difference and the amplitudeof yn is switching sequentially. However when the coupling- 721 -(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 8: Simulation results of system including time delay in one direc-Figure 9: Simulation results of system that combines switching includingtion coupled by resistor. Attractor. Lissajous figure. Time waveform. Red,
blue and green colors denote y1 , y2 and y3 respectively.

ime delay in one direction coupled by inductor. Attractor. Lissajous figure.
Time waveform. Red, blue and green colors denote y1 , y2 and y3 respectively.

strength γ ′ is bigger then 0.1, switching synchronization stateis lost and synchronization can be observed by initial values.
Generally switching synchronization can be observed whensystem including time delay in one direction is coupled byresistors R0 or inductors L0 . The amplitude is going divergence and convergence. Additionally the time of divergenceand convergence is different.

special synchronization state can be observed. The switchingof the amplitude of voltage in addition to the in-phase synchronization state can be observed by difference of couplingstrength.
References[1] X. Liu, “Stability of impulsive control systems with time delay. Mathematical and Computer Modelling,” Vol.39, pp.511-519, 2004.

5. Conclusion
In this study, we investigated synchronization state of novel
coupled systems observed from some coupling methods ofring coupled by time delayed chaotic circuits. All the investigated coupling systems 4 types. In case of ring coupled by resistors, we observed in-phase synchronization state.
The other case of ring coupled by inductors, in-phase synchronization and three-phase synchronization state can be observed. We devised coupled systems that takes advantage offeatures of the time delayed chaotic circuit. As a result, some[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, Nov. 1989.
[3] T. Maruyama, N. Inaba, Y. Nishio and S. Mori, “Chaos in Self Oscillator Circuit Containing Time Delay,” Proceedings of IEEE Midwest Symposium on Circuits and Systems (MWSCAS’90), vol. 2, pp.
1055-1058, Aug. 1990.
[4] L. M. Pecora and T. L. Carroll, “Synchronization in Chaotic Systems,”- 722 -Physical Review Letters, vol. 64, no.8, pp. 821-824, Feb. 1990.