IEEE Workshop on Nonlinear Circuit Networks
December 13-14, 2013

Synchronizations Phenomena
in Two Coupled Chaotic Circuits
Containing Time Delay Coupling
Naoto Kageyama

Yoko Uwate

Yoshifumi Nishio

Dept. of Electrical and Electronic Eng., Dept. of Electrical and Electronic Eng., Dept. of Electrical and Electronic Eng.,
Tokushima University,
Tokushima University,
Tokushima University,
2-1 Minami-Josanjima,
2-1 Minami-Josanjima,
2-1 Minami-Josanjima,
Tokushima, 770-8506 JAPANTokushima, 770-8506 JAPANTokushima, 770-8506 JAPAN
Email: n-kage@ee.tokushima-u.ac.jp
Email: uwate@ee.tokushima-u.ac.jp
Email: nishio@ee.tokushima-u.ac.jp

Abstract— Studies on chaos synchronization in coupled chaotic
circuits are extensively carried out in various fields. Moreover,interesting phenomena were confirmed in a system including atime delay. In this study, we investigate two Nishio-Inaba chaotic
circuits coupled via resistor containing time delay coupling.
Interesting synchronization phenomena can be confirmed bycircuit experiments and computer simulations.

I. I NTRODUCTION
Synchronization phenomena in complex systems are very
good models to describe various higher-dimensional nonlinear
phenomena in the field of natural science. Recently, manystudies have been investigated synchronization of chaoticcircuits. It is applied in the field of engineering, physics andbiology and so on [1]-[4]. Moreover,, interesting phenomena
were confirmed in a system including a time delay [5]. We
consider that it is very important to investigate the phenomena
related with chaos synchronization to realize future engineering application utilizing chaos.
In this study, we consider two Nishio-Inaba chaotic circuits[6]-[8] coupled via resistor containing time delay coupling.
Coupling this system is controlled by the switch. In additional, the switches connection is controlled by the amplitude.
Additionally, this switching operation is included delay, andvarious phenomena were occurred by varying connection time.
Therefore, we investigate the synchronization phenomena byusing the computer calculated and circuit experiments.

switch is not connected to the resistor immediately, whichis connected for Tc seconds after Td seconds. Namely, v1nreaches Vth , two chaotic circuits coupled via resistor R isconnected for Tc seconds after Td seconds. Figure 3 shows
the circuit realization. This circuit is consisted two Monostable
Multivibrators. Monostable Multivibrator 1 outputs a pulse atregular intervals in reaction to the rising edge of the input
pulse, and Monostable Multivibrator 2 outputs a pulse at regular intervals in reaction to the falling edge of the input pulse.
Pulse width of Monostable Multivibrator 1 is corresponding to
the delay time, and pulse width of Monostable Multivibrator
2 is corresponding to the connection time.

SWR
DELAY(Td)i 11i 12L1i 21v11

Figure 1 shows the circuit model. In the circuit, two NishioInaba circuits are coupled via resistor R and switch SW .
The circuit consists of a negative resistance, a nonlinear resistance consisting of two diodes, capacitor and two inductors.
Furthermore, SW is the switch controlled by the amplitude,
switch is connected to the resistor R in the case of amplitudeis larger than the threshold, and connect time is represented
by Tc . Also, this switching operation is included delay, anddelay time is represented by Td . Figure 2 shows the switching
operation. Voltage reaches the threshold at a certain time, the- 24 -L2Cvd-r

II. C IRCUIT M ODEL

L1L2CFig. 1.

i 22-rCircuit model.

By using the variables such thatv21vd√
√
C
L1 C

i=Vx,i=V yn , vn1 = V zn
 n1nn2
LL
12

√
√

C
L1L1 C
α=r, β=, δ = rd,
LLL

122
√


√


 γ = 1 L1 , t = L1 C2 τ, ” · ” = d .
R CdτMonostable
Multivibrator 2Ri 11n(L2Cvd-rv21vdyn +11− yn −δδ)
(3)time(a) v > Vth(b) v < Vth(b)

time

Td Tc

Figure 4 is obtained by cutting out a part of the state tovaries continuously.
Figure 4 shows circuit experiments results for the caseof varying the Tc and the parameters are fixed with L1 =500[mH], L2 = 200[mH], C = 0.0153[µF ], rd = 1.46[M Ω],
R = 43.2[kΩ] and Vth = 5.50[V ]. In Fig. 4(a), two circuits
are asynchronous including the in-phase synchronization andanti-phase synchronization. However, in Figs. 4(b) and 4(c),two circuits are synchronized in in-phase. As a result, we
confirmed two circuits are synchronized with the increase ofTc .
B. Computer Calculated Results

Vth(a)

Circuit realization.

A. Circuit Experiments Results

V

OPERATIONtime

DELAY

The switch is conected to R
Fig. 2.

v11Fig. 3.

(n = 1, 2).

DELAY

L1

(2)

The nonlinear function f (yn ) corresponds to the I − Vcharacteristics of the nonlinear resistors consisting of thediodes and are assumed to be described as follows;

Td Tci 22n(n = 1, 2),

δ2i 21L2C-rz˙1 = −x1 − βyn − γ(z1 − z2 )z˙2 = −x2 − βyn + γ(z1 − z2 )f (yn ) =i 12

L1(1)That the equation (1) is normalized asx˙n = αxn + zn






 y˙ = z − f (y )nMonostable
Multivibrator 1

VthSwitching operation.

Figure 5 shows computer calculated results for the case ofvarying the Tc , and the parameters are fixed with α = 0.460,
β = 3.0, δ = 470, γ = 0.132 and Vth = 0.6. In Figs. 5(d),5(e) and 5(f), we confirmed the phase difference between twocircuits decrease with the increase of Tc . We can say that thisinteresting phenomenon can be observed from both computer
calculations and circuit experiments.
IV. C ONCLUSIONSIn this study, we have investigated the synchronization
phenomena observed from coupled chaotic circuits containing
time delay coupling. We confirmed the phase difference between two circuits decrease with the increase of connection
time and phase difference between two circuits switch inasynchronous and in-phase synchronization with the increaseof time delay
Investigating the detail of the states and statistical analysisof the observed phenomena are our important future work as
well as more detailed explanation of the mechanism of thegenerations.
R EFERENCES

III. S YNCHRONIZATION P HENOMENA
We investigate the phenomenon for the case of varying Tc
in the circuit of Fig. 1.

- 25 -[1] Y. Nishio and A. Ushida, “Chaotic Wandering and its Analysis in Simple
Coupled Chaotic Circuits,” IEICE Trans. Fundamentals, vol. E85-A, no.
1, pp. 248-255, 2002.(a) Td =100µs, Tc =100µs.

x2x2x2(b) Td =100µs, Tc =330µs.

x1x1x1(a) Td = 50, Tc = 100.

(b) Td = 50, Tc = 200.

(c) Td = 50, Tc = 300.

x1-x2τ(c) Td =100µs, Tc =560µs.
Fig. 4. Phase difference between two circuits for the case of varying
the Tc (circuit experiments).

(d) Td = 50, Tc = 100.

x1-x2[2] G. Abramson,V. M. Kenkre and A. R. Bishop, “Analytic Solutions for
Nonlinear Waves in Coupled Reacting Systems,” Physica A: StatisticalMecanics and its Applications, vol. 305, no. 3-4, pp. 427-436, 2002.
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3423-3433, 2005.
[4] M. Wada, Y. Nishio and A. Ushida, “Analysis of Bifurcation Phenomena in Two Chaotic Circuits Coupled by an Inductor,” IEICE Trans.
Fundamentals, vol. E80-A, no. 5, pp. 869-875, 1997.
[5] T. Maruyama, Y. Nishio, N. Inaba, S. Mori and S. Mori, ”Chaos in an
Auto Gain Controlled Oscillator Containing Time Delay” Transactionsof IEICE (in Japanese), vol. J72-A, no. 11, pp. 1814-1820, Nov. 1989.
[6] Y. Nishio, N. Inaba, S. Mori and T. Saito, “Rigorous Analyses ofWindows in a Symmetric Circuit,” IEEE Transactions on Circuits andSystems, vol. 37, no. 4, pp. 473-487, Apr. 1990.
[7] C. Bonatto and J. A. C. Gallas, “Periodicity Hub and Nested Spirals in
the Phase Diagram of a Simple Resistive Circuit,” Phys. Rev. Lett., 101,054101, Aug. 2008.
[8] R. Stoop, P. Benner and Y. Uwate, “Real-World Existence and Origins ofthe Spiral Organization of Shrimp-Shaped Domains,” Phys. Rev. Lett.,
105, 074102, Aug. 2010.

- 26 -τ(e) Td = 50, Tc = 200.

x1-x2
τ(f) Td = 50, Tc = 300.
Fig. 5. State variation for the case of varying the Tc . (a), (b) and (c)phase difference between two circuits (computer simulations). (d),(e) and (f) the value of x1 − x2 with varying time.