2015 International Symposium on Nonlinear Theory and its Applications
NOLTA2015, Kowloon, Hong Kong, China, December 1-4, 2015

Synchronization State of Coupled Changing Time Delayed Chaotic Circuits
Seiya Kita, Yoko Uwate and Yoshifumi Nishio
Department of Electrical and Electronic Engineering, Tokushima University
2–1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: {kita,uwate,nishio}@ee.tokushima-u.ac.jp
Abstract—In this study, we investigate synchronization
states observed in coupled time delayed chaotic circuits byinductor. Time delay of chaotic circuit depends on attractor
types. We focus on relationships between synchronization
state and time delay. Single type synchronization state canbe observed by changing time delay of subcircuits. Coexisting synchronization states depending on initial values
can be observed in specific time delay values. Moreover,
we investigate the eﬀect of various time delay values on
the subcircuit to changing synchronization states.

Moreover, synchronization state can be classified by thenumber of coupled chaotic circuit whether the number iseven or odd.
In this study, we investigate synchronization states bychanging time delay of subcircuits in the proposed systems.
The proposed systems denote coupled time delayed chaotic
circuits by inductor as a bridge or a ring. By carrying out
computer simulations, time delay of subcircuits eﬀects achange synchronization state.
2. Circuit Model

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 naturalworld. Namely, it is considered that investigation of stability in such time-delay systems is important [1]. Generation of chaos of them all is reported self excited oscillation
system containing time delay. This chaotic circuit can beeasily realized by using simple electric circuit element andanalyzed 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
previous study, we have investigated synchronization state
observed in coupled time delayed chaotic circuits by inductor [5]. Coupled nonlinear circuits by inductor can beobserved two types of synchronization states [6][7]. As aresult, two types of synchronization state is caused by increasing the chaotic strength of subcircuit. In roughly divided, two types of synchronization states depending on
initial values and number of subcircuits can be observed.

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
R of which amplitude is controlled by the switch containing
time delay. The current flowing through the inductor L isi, and the voltage between the capacitor C is v. The circuit
equations are normalized as Eqs. (1) (2) by changing thevariables 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.
v
VthtimeVth(a) v < Vth(b) v > VthTd1

Tth(a)

(b)

(a)

i1
Cv1

Figure 1:circuit.

L(2)timeSW-g

OPERATION
time
Td
the switch is connected to negative resistorGTime delayed chaotic Figure 2: Chaotic attractor obtained by computer simulation.

- 926 -Figure 3: Switching operation.

By changing the parameters and variable as follow:

3. Two Coupled Chaotic Circuits√√
CVth x, v = Vth y, t = LCτ,√L
√CCg
= 2α and G= 2β.
LL

i=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. T d denotes the time delay. First, theswitch is connected to a negative resistor. In state of that,the voltage v is amplified up to while v is oscillating, theamplitude exceeds the threshold voltage Vth which is thethreshold control loop. Second, the system memorize thetime as T th while v is exceeding the threshold voltage Vthand that state is remained for T th . In subsequent the instantof exceeding threshold Vth , the switch stays the state forT d . After that switch is connected to positive resistor during T th . The switch does not immediately connect in thepositive resistor however the switch is connected after T d .
A set of switching operations control the amplitude of v.
Figure 4 is bifurcation of time delayed chaotic circuit. By
increasing the parameter value of time delay T d , width of
dot area changes. Time delayed chaotic circuit is changed
time delay from π/2 to 3π/2 in previous study [5]. In thisstudy, we consider range of T d is changed from 5π/8 toπ. We investigate synchronization state when coupled ofperiodic attractor and chaotic attractor.

Figure 5 shows the coupled chaotic circuit by inductor.
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 = ynẏn = −2βyn − xn + γ(xn+1 − xn ).

(4)

where (n = 1, 2) and x3 = x1 . By changing the parametersand variable as follow:√√
Cin =Vth xn , vn = Vth yn , t = LCτ,L√√CCL
g= 2α, G= 2β and γ = .
LLL0We investigate the synchronization when time delay
T d1 and T d2 of chaos circuits are changed from 1.96 to
3.14. Figure 7 shows some of simulation results. Table 1 shows the synchronization states each time delay.
When (T d1 , T d2 ) = (1.96, 3.14), coexisting in-phasesynchronization and anti-phase synchronization state canbe observed depending on initial values. By increasing(T d1 , T d2 ) = (2.60, 3.14) or (T d1 , T d2 ) = (3.14, 3.14), synchronization state is fixed anti-phase synchronization. Inphase synchronization is readily induced when T d is small.
4. A Ring of Three Coupled Chaotic Circuit
Figure 6 shows the schematic of a ring of three coupledtime delayed chaotic circuits 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 ).

Figure 4: Bifurcation. T d is variable. α = 0.015 and β = 0.8.

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

L0(6)where (n = 1, 2, 3) , x0 = x3 and x4 = x1 .
VthVthTd1

Td2

i1

Cv1

Li2
SW-gGCv2

LL0

L0L0VthSW-gVthVthG

Td1

i1

Td2

Td3

i3

i2
Cv1

LSW
-gG

Cv2

L

SW
-gC

Gv3

L

SW
-g

Figure 5: Coupled time delayed chaotic circuits.
Figure 6: A ring of three coupled time delayed chaotic circuits.

- 927 -G

Table 1: Synchronization state N = 2.
T imedelay
Synchronization stateT d1
T d2 In -phase Anti-phase
1.96
◦2.20
◦2.35
◦
1.96 2.40
◦
2.52
◦
2.60
◦
3.14
◦
◦
2.20
◦
2.35
◦
2.20 2.40
◦
2.52
◦
2.60
◦
3.14
◦
2.35
◦
2.40
◦
2.35 2.52
◦
2.60
◦
3.14
◦
2.40
◦
2.40 2.52
◦
2.60
◦
3.14
◦
2.52
◦
2.52 2.60◦
3.14◦
2.60 2.60◦
3.14◦
3.14 3.14◦(1) In-phase synchronization state T d1 = 3.14 and T d2 = 1.96.

(2) Anti-phase synchronization state T d1 = 3.14 and T d2 = 1.96.

(3) Anti-phase synchronization state T d1 = 3.14 and T d2 = 3.14.

Figure 7: Simulation results of bridge chaotic circuits. Attractor. Timewaveform. Red and blue colors denote y1 and y2 respectively.

We investigate the synchronization when three chaos circuits are changed time delay range from 1.96 to 3.14. Table 2 shows synchronization states. We observed the two
types of synchronization state. Figure 8 shows some ofsimulation results. Switching synchronization states can beobserved as shown in time waveform of Fig. 8 (1). Switching synchronization state is constituted anti-phase synchronization and three-phase synchronization state. However
only three-phase synchronization state can be observed asshown in time waveform of Fig. 8 (2). When anti-phasesynchronization state is observed, single voltage amplitudeof subcircuits1 or 2 becomes about zero for only a moment.
5. Conclusions
In this study, we have investigated synchronization state
observed in two coupled variable time delayed chaotic circuits by inductor. As a result, when time delay is small,in-phase synchronization state can be observed. However induction of anti-phase synchronization state caused
by increasing time delay to 3.14 of subcircuit has confirmed. Furthermore, we have investigated synchronization
state observed on a ring of three coupled variable time delayed chaotic circuits by inductor. As a result, inductionof three-phase synchronization caused increasing time de-Table 2: Synchronization state N = 3.
T imedelay
Synchronization state
T d1
T d2
T d3 Three-phase Switching
1.96
◦
2.20
◦
2.35
◦
3.14 3.14 2.40
◦
2.52
◦
2.60
◦
3.14
◦
lay to 3.14 of subcircuit has confirmed. However switching synchronization state can be observed when single timedelay of subcircuits is too small. Moreover, switching synchronization state is constituted anti-phase synchronizationand three-phase synchronization states when a ring of three
coupled time delayed chaotic circuits.
References[1] X. Liu, “Stability of impulsive control systems withtime delay. Mathematical and Computer Modelling,”- 928 -

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1999.

(1)Switching synchronization state T d1 = 3.14, T d2 = 3.14 andT d3 = 1.96.

(2) Anti-phase synchronization state T d1 = 3.14, T d2 = 3.14 andT d3 = 3.14.

Figure 8: Simulation results of a ring coupled three chaotic circuits.
Attractor. Time waveform. Red, blue and green colors denote y1 , y2 andy3 respectively.

Vol.39, pp.511-519, 2004.
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1989.
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1990.
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