Switching Synchronization States ofa Ring of Coupled Chaotic Circuitswith One-Direction Delay Effects
Seiya Kita, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Eng., Tokushima University
2-1 Minami-Josanjima, Tokushima 770–8506, Japan
Email: {kita, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract—Synchronization states can be observed in coupledcircuits. Further, interesting synchronization states were confirmed in coupled chaotic circuits containing time delay. In thisstudy, we investigate the novel coupling methods and synchronization states observed in coupled chaotic circuits containing timedelay. The novel coupling methods is a ring of coupled chaotic
circuits with one-direction delay effects. Synchronization state
observed the novel coupling methods is switching synchronization
state. The switching synchronization state is changed by timedelay of each subcircuit. We focus on relationships between
switching synchronization state and the pattern of time delay.
Moreover, we investigate the cycle of switching synchronization
state.

I.

I NTRODUCTION

Studies on synchronization state are extensively carriedout in various fields [1]-[3]. Recently, in particular, synchronization states in chaotic oscillators are studied by manyresearchers. Behavior of chaotic oscillators is interesting. Then,chaos phenomena are quite dependent on initial values and not
periodical and predictable. Moreover the synchronization states
have caused very interesting phenomena. Synchronization andthe related bifurcation of chaotic systems are good methods todescribe various high-dimensional nonlinear phenomena in thefield of natural science. However, many synchronization statesof coupled chaotic oscillators have not been solved yet. The
synchronization phenomena in electric circuit make clear themechanism of the synchronization phenomena in our daily life.
There are many nonlinear systems containing time delay, suchas neural networks, control systems, meteorological systems,
biological systems and so on in the natural world. Namely, it
is considered that investigation of stability in such time-delaysystems is significant [4]. Generation of chaos in time delayed
system is reported self excited oscillation system containing
time delay [5]. The oscillators have feedback systems which
control gains in this study. This chaotic circuit can be easily
realized by using simple electric circuit element and analyzed
exactly. The coupling switch connects alternately with one
subcircuit and other with a fixed time interval. On the other
hand, there are examples of nonlinear phenomena, chaoticsynchronization and so on [6]. In particular, many studies
on synchronization of coupled chaotic circuits have beenreported [7].
In this study, we devise the novel coupling method thattakes advantage of features of the chaotic circuit containing978-1-5090-1570-2/16/$31.00 ©2016 IEEE

384

time delay. The novel coupled method is utilizing the characteristics of the circuit having time delayed feedback. Then,
we observe switching synchronization state. We carry out
computer calculations for three coupled auto gain controlled
oscillators containing time delay and investigate time delay of
subcircuits effects a change of synchronization state and thetime waveform.
II.

T IME D ELAYED C HAOTIC C IRCUIT

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 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.
VthDelayTd

TdinCCnC

vn L-gFig. 1.

SWG

Time delayed chaotic circuit.
v
Vthtime(b)(a) v< Vth(b) v> Vth(a)timeOPERATION
DELAY

timethe switch is connected to positive resistor

Fig. 2.

Switching operation.

APCCAS 2016

(A) In case of switch connected to −g,{ẋ = yẏ = 2αy − x,(B) In case of switch connected to G,{ẋ = yẏ = −2βy − x.

synchronization can not be observed or synchronization is lostin case of small coupling strength γ.
(1)
Td3

CC1
Td1

(2)
CC3

By changing the parameters and variable as follow:√√
Ci=Vth x, v = Vth y, t = LCτ ,√L√CC
g= 2α and G= 2β.
LL

Td2Fig. 3.

System including time delay in one direction.

IV.

The switching operation is shown in Fig. 2, it controls theamplitude of the oscillator. This switching operation is included time delay. Td denotes the time delay. First, the switch
is connected 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 remainedfor 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 the switch
is connected after Td . A set of switching operations control
the amplitude of v. By using mapping method to this circuit,we could derive the 1-dimensional Poincare map explicitlyfrom each circuit, and the Poincare map was proved to have apositive Liapunov number with computer assistances [3].
III.

CC2

S IMULATION RESULTS

The proposed model has three time delay Td . In this study,
we consider that the coupled circuits have symmetric delay
and asymmetric delay.
A. With symmetric delay effects
The simulation results are shown in the Figs. 4, 5 and6. These figures can be obtained by difference of time delay
Td1 , Td2 and Td3 . Figure 7 shows the magnified poincare
section of switching synchronization with symmetric delay
when Td1 , Td2 and Td3 change. Figure 8 shows the bifurcationand the cycle of switching synchronization with symmetricdelay. The amplitude of xn is switching sequentially in Figs. 4,5 and 6 (c). These synchronization states are switching synchronization state. The order of switching amplitude of theloop is subcircuit1 - subcircuit3- subcircuit2. The amplitude
is going divergence and convergence. Additionally the time ofS YSTEM I NCLUDING T IME D ELAY IN O NE
D IRECTION

The circuit in this study has characteristic time delays
methods. We have devised coupled systems as shown inFig. 3. This system is coupled by resistors R0 . It is called
coupled systems and “a ring of coupled chaotic circuits withone-direction delay effects” By changing the parameters and(a) Attractor

variables as follows:√√CVth xn , vn = Vth yn , t = LCτ ,in =L√√√CCCg= 2α, G= 2β and γ = R0.
LL
L
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 ),(B) In case of that switch is connected to G,{
ẋn = ynẏn = −xn − 2βyn + γ(yn−1 − 2yn + yn+1 ),(c) Time waveform(4)

where (n = 1, 2, 3) and x0 = x3 , x4 = x1 . In calculation result, in-phase synchronization state can be observed.
When the coupling strength γ is larger than 0.1, full inphase synchronization can be observed. However full in-phase978-1-5090-1570-2/16/$31.00 ©2016 IEEE(b) Lissajous figure385(d) Poincare section
Simulation results with symmetric delay of α = 0.015,
β = 0.5, γ ′ = 0.01 and Tdn = π.

Fig. 4.

APCCAS 2016

(a) Attractor(a) Attractor(b) Lissajous figure(b) Lissajous figure(c) Time waveform(c) Time waveform(d) Poincare section(d) Poincare section

Simulation results with symmetric delay ofα = 0.015,
β = 0.5, γ ′ = 0.01 and Tdn = 0.5π.

Fig. 5.

divergence and convergence is different. Especially, the cycleof time waveforms is different in Figs. 4, 5 and 6 (c). Namely,
the number of cycle between the maximum of the amplitude
from next one is different. Figure 8 shows the bifurcation ofswitching synchronization with symmetric delay. When theattractor is high periodicity, the number of cycle is large.

Simulation results with symmetric delay of α = 0.015,
β = 0.5, γ ′ = 0.01 and Tdn = 1.5π.

Fig. 6.

(i) Td=0.5π(ii) Td=0.7π(iii) Td=π(iv) Td=1.3π(v) Td=1.5π

Magnified poincare section with asymmetric delay of
α = 0.015, β = 0.5 and γ ′ = 0.01.

Fig. 7.

B. With asymmetric delay effects
Next we change Td1 and fix Td2 = Td3 = 1.0π. Figures. 9and 12 show simulation results. We investigate cycle of switching synchronization with asymmetric delay. Figure 10 shows
the magnified poincare section of switching synchronization
with asymmetric delay. Figure 11 shows the bifurcation andthe cycle of switching synchronization with asymmetric delay.
This bifurcation is periodic. Figure 13 shows cycle of switching synchronization with symmetric or asymmetric delay. The
variation of cycle with asymmetric delay is less than symmetricdelay. The number of cycle is large when periodic bifurcation
is observed. Namely, the cycle of switching synchronization
state is stabilization when periodic solution can be observed.

V.

C ONCLUSION

In this study, we devised coupled systems that takes advantage of features of the time delayed chaotic circuit. We
investigated synchronization state of a ring of coupled chaotic
circuits with one-direction delay effects. As a result, somespecial synchronization states can be observed. We observed
in-phase switching synchronization state. When time delay ofsubcircuits changes, the cycle of switching synchronization
978-1-5090-1570-2/16/$31.00 ©2016 IEEE386

Fig. 8.

Bifurcation and cycle with symmetric delay.

state changes. The switching of the amplitude can be observedby difference of time delay. The variation of cycle withsymmtric delay is less than asymmtric delay. We consider thecycle of switching synchronization state is stabilization when
periodic solution can be observed.
R EFERENCES[1] G. Abramson, V. M. Kenkre and A. R. Bishop, “AnaAPCCAS 2016

(a) Attractor(a) Attractor(b) Lissajous figure(b) Lissajous figure(c) Time waveform(c) Time waveform(d) Poincare section(d) Poincare section

Simulation results with asymmetric delay of α = 0.015,
β = 0.5, γ ′ = 0.01, Td1 = 1.2π and Td2 = Td3 = π.
Fig. 9.

(i) Td1=0.5π(ii) Td1=0.7π(iii) Td1=π(iv) Td1=1.3π

Simulation results with asymmetric delay of α =0.015, β = 0.5, γ ′ = 0.01, Td1 = 1.4π and Td2 = Td3 = π.
Fig. 12.

(v) Td1=1.5π

Fig. 10. Magnified poincare section with asymmetric delay of
α = 0.015, β = 0.5, γ ′ = 0.01 and Td2 = Td3 = π.

Cycle of switching synchronization with symmetric or
asymmetric delay.

Fig. 13.

[4][5]Fig. 11.

Bifurcation and cycle with asymmetric delay.

lytic Solutions for Nonlinear Waves in Coupled Reactig
Systems,” Physical A: Statistial Mechanics and its Applications, vol. 305, no.3-4, pp. 427-436, 2002.
[2] C. M. Gray, “Synchronization Oscillators in Neural
Systems: Mechanisms and Functions,” J. Computational
Neuroscience, vol. 1, pp. 11-38, Feb. 1994.
[3] Y. Zhang, G. H. Du and J. J. Jiang, “Synchronization
Chaos by Impulsive Feedback Method,” International

978-1-5090-1570-2/16/$31.00 ©2016 IEEE387

[6]

[7]

Jounal of bifurcation and Chaos, vol. 11, no.8, pp. 22332243, 2001.
X. Liu, “Stability of impulsive control systems with timedelay. Mathematical and Computer Modelling,” Vol.39,pp.511-519, 2004.
T. Maruyama, N. Inaba, Y. Nishio and S. Mori, “Chaosin an Auto Gain Controlled Oscillator Containing TimeDelay,” Trans. IEICE, vol. J 72-A, pp. 1814-1820, Nov.
1989.
T. Maruyama, N. Inaba, Y. Nishio and S. Mori, “Chaosin Self Oscillator Circuit Containing Time Delay,” Proceedings of IEEE Midwest Symposium on Circuits andSystems (MWSCAS’90), vol. 2, pp. 1055-1058, Aug.
1990.
L. M. Pecora and T. L. Carroll, “Synchronization inChaotic Systems,” Physical Review Letters, vol. 64, no.8,
pp. 821-824, Feb. 1990.

APCCAS 2016