2010 IEEE Workshop on
Nonlinear Circuit Networks, Tokushima, Dec 3-4, 2010

In-Phase and Anti-Phase Synchronization ofSwitching Phenomena in Coupled Chaotic Circuits
Takuya Nishimoto

Yasuteru Hosokawa

Yoshifumi Nishio

Dept. Information Business,
Shikoku University
Email: s2137034@edu.shikoku-u.ac.jp

Dept. Media and Information System,
Shikoku University
Email: hosokawa@keiei.shikoku-u.ac.jp

Dept. Electrical and Electronic Eng.,
Tokushima University
Email: nishio@ee.tokushima-u.ac.jp

Abstract— Switching phenomena of attractors can be observedin coupled chaotic circuits. In this study, we confirmed that inphase and anti-phase synchronizations of switching phenomena.
The phenomena can be observed in asynchronous states. Therefore, it is very interesting phenomena.

I. I NTRODUCTION
Many kinds of complex phenomena are observed on largescale coupled chaotic circuits. Investigations of these phenomena are very important works in order to declare nonlinear
phenomena in the natural world. Electric circuits are suitable
for models of large-scale coupled nonlinear systems. Because
getting electric parts is easily, the price is inexpensive, experiment time is very short, repeatability of experiments isvery high, an electric circuit is real physical system and soon. Therefore there are many studies of large-scale coupledcircuit systems. In these studies, synchronization phenomena
are attracted attentions.
On the other hand, some chaotic circuits have coexisting
attractors. In these circuits, switching phenomena of attractors
are observed by changing parameters. Normally, in the caseof synchronization states, switching of attractors are also synchronized. And in the case of asynchrization states, switchingof attractors are also asynchronized. However, there is thepossibility that switching of attractors are synchronized in thecase of asynchrization states.
In this study, we investigate coupled chaotic circuits andconfirm in-phase and anti-phase synchronizations of switching
phenomena. Especially, the case that the number of circuit is
two is investigated.
II. S YSTEM MODEL

in1
C

vnf (id)
Vrdid-V(a) Circuit schematicFig. 3.

Coupled diodes model

negative resistor and bi-directionally-coupled diodes. In thiscircuit, two attractors which are symmetrical about a origin
are observed by adjusting the value of the negative resistor.

CCnrn
vd =Fig. 1.

(b)v-i characteristics

In this study, we investigate coupled chaotic circuits whichis coupled by resistors shown in Fig. 2. Firstly, a system equation is derived. Bi-directionally-coupled diodes are modeled asFig. 3. Then, we can derive a following function

in2L2n

L1

System model (N = 4)

Fig. 2.

id +VV− id −rdrd
.

(1)

Circuit model

A circuit model [1] using in this study is shown in Fig. 1.
This circuit consists of three memory elements, one linear

The others elements are modeled as linear elements. Eachcircuit number is defined as 1 ≤ n ≤ N . System equation- 15 -

is described as follows:
din1


= vn + rn in1 , L1 dt






rddin2VV
L2n= vn −in2 +− in2 −,dt
2rdrd !

N

X

dvn



C=−(i+i)−GNv−vk .
n1n2n
dt
k=1

Changing parameters and variables as follows,rr√CCt = L1 Cτ, in1 = Vxn , in2 = Vyn ,L1L
1rdC
vn = V zn , ”· ” =, αn = rn,dτL1 rr
L1CL1βn =, γ=rd and δ = G.
L2nL1C(2)(3)

The normalized system equation is described as follows.
ẋn = zn + αn xn , 



11γ

 ẏn = βn zn −− yn −yn +,2γγ(4)!
N

X



zk .
 żn = −xn − yn − δ N zn −k=1

In the next section, by using this equation, computer simulations are carried out.
III. C OMPUTER SIMULATIONS
A. All phenomena observed in this study
Firstly, all phenomena observed in this study are shown.
Figure 4 shows a coexisting attractors of a chaotic circuit asshown in Fig. 1. The orbit is color-coded according to two
attractors. This color coding is defined by an investigation ofa bifurcation diagram [2]. In following all simulation results,
this definition is applied.
Figure 5 shows a computer simulation result in the caseof an anti-phase synchronization of switching phenomena.
Figure 5 (a) shows waveforms of two circuits. Horizontal
axis show time. Vertical axes show z1 and z2 . These arecorresponding to voltage vn of the circuit model as shownin Fig. 1. Figure 5 (b) shows a difference of two waveforms.
Horizontal axes show time. Vertical axis shows as follows. A
black line is z1 - z2 . A green line shows a difference of two
attractors. Namely, when both of z1 and z2 is red or blue,
the line becomes 0. When z1 is blue and z2 is red, the line
becomes 2. When z1 is red and z2 is blue, the line becomes-2. This rule is applied to following simulation results. In thiscase, z1 and z2 is not synchronized and switching phenomenaof two attractors are synchronized with anti-phase. Figure 6
shows a computer simulation result in the case of in-phasesynchronization of switching phenomena. In this case, z1 andz2 are not synchronized and two attractors are synchronizedwith in-phase.
Above two kinds of phenomena are observed in a steadystate. We consider that these are interesting phenomena.

Figure 7 shows a computer simulation result in the caseof an asynchronization of switching phenomena. Figure 8
shows a computer simulation result in the case of switching
phenomena on one circuit only. In this case, if circuits are
not coupled, z2 keeps red in a steady state. Figure 9 shows acomputer simulation result in the case of holding one attractor
state in each circuit. In this case, by changing initial values,
an opposite pattern of the Fig. 9 is also observed. Figure 10
shows a computer simulation result in the case of boundary
area between Fig. 5 and Fig. 7. Same phenomena with Fig. 7
are observed in the left side area and the right side area.
Same phenomena with Fig. 5 are observed in the center area.
Figure 11 shows a computer simulation result in the case ofboundary area between Fig. 6 and Fig. 7. This case looks
like a high frequency state of Fig. 6. However, switching
timing is not in time. It is also shown in green line ofFig 11 (b). Figure 12 shows a computer simulation result in thecase of an anti-phase synchronization of switching phenomena
with synchronization phenomena. In this case, synchronization
phenomena are observed between anti-phase synchronizationsof switching phenomena. This synchronization time is corresponding to parameter δ mainly.
B. Relationship between parameters and phenomena
Next, we investigate the relationship between parametersand phenomena which is mentioned above. Figure 13 showsa relationship among parameter β2 , δ and phenomena. Parameters β2 and δ are corresponding to the inductor L2 andthe resistor G which is coupling strength. The relationshipbetween colors and observed phenomena is shown in Table I.
Following points could be mentioned from Fig. 13.
• In-phase and anti-phase synchronizations of switching
phenomena are observed between anti-phase synchronization states as shown in Fig. 7 and the state of holding one
kinds of attractors as shown in Fig. 9.
• Anti-phase synchronizations of switching phenomena areobserved in the case of β1 = β2
• In-phase synchronizations of switching phenomena areobserved in the case that β2 is differ from β1 .
• In the case that β2 is nearly equal to 12, in-phase andanti-phase synchronizations of switching phenomena areobserved by changing δ only.
Additionally, we confirmed that by increasing a value of δ,sojourn time of until switching becomes long in the case of inphase and anti-phase synchronization of switching phenomena.
We consider that in the case of increasing the number of thecircuit, anti-phase synchronizations of switching phenomena
are observed and in-phase synchronizations of switching phenomena are not observed. Because anti-phase can be observedin the case of β1 = β2 and in-phase cannot be observed in thecase of β1 = β2 or β1 ' β2 . Namely, the difference of βn is
not ignore in the case of increasing the number of the circuit.
Figure 14 shows a relationship among parameters α2 , δand phenomena. Parameters α2 is corresponding to negative
resistor r2 . The relationship between colors and observed
phenomena is shown in Table I. Anti-phase synchronizations- 16 -TABLE I
C OLOR DEFINITION OF F IG . 13 AND F IG . 14.
Fig.
Fig. 5
Fig. 6Fig. 7
Fig. 8Fig. 9Fig. 10
Fig. 11
Fig. 12

StateRedGreenBlue
Light blue
GrayPink
Light green
Yellow(a)(b)of switching phenomena are only observed. The area is around0.55 ≤ δ ≤ 0.63 and 0.46 ≤ α2 ≤ 0.48. Namely, anti-phasesynchronization is not observed in the case of α1 ' α2 ora small value of coupling factor δ. In yellow areas as shownin Fig. 14, by decreasing α2 , synchronization time becomeslong.

Fig. 5.
Anti-phase synchronization of switching phenomena. αn = 0.40,
β1 = 3.0, β2 = 4.5, γ = 470.0 and δ = 0.31.

(a)

IV. C ONCLUSIONSIn this study, we have investigated coupled chaotic circuits
and have observed in-phase and anti-phase synchronizations ofswitching phenomena. In the case that the number of circuit
is two, the relationship between parameters and phenomena isshown.

(b)Fig. 6.
In-phase synchronization of switching phenomena. αn = 0.40,
β1 = 3.0, β2 = 15.0, γ = 470.0 and δ = 0.33.

R EFERENCES[1] Y. Nishio, N. Inaba, S. Mori and T. Saito, “Rigorous Analyses of Windows
in a Symmetric Circuit,” IEEE Trans. Circuit and Systems., vol. 37, no. 4,
pp. 473–487, 1990.
[2] T. Nishimoto, Y. Hosokawa and Y.Nishio, “Anti-phase Synchronization ofSwitching Phenomena in Globally Coupled System of Chaotic Circuits,”
IEICE Technical Report, pp. 1–4, 2010.
[3] H. Sekiya, S. Mori and I. Sasase, “Synchronization of Self-Switching
Phenomena on Full-Coupled Chaotic Oscillators,” IEICE Trans., vol. J83A, no. 11, pp. 1264–1275, 2000.
[4] T. Yamada, Y. Hosokawa and Y. Nishio, “Quasi Synchronization Phenomena in Coupled Chaotic Systems as a Ladder,” Proc. of IEEE International
Workshop on Nonlinear Circuit Networks, pp. 68-71, 2008.

(a)(b)
Fig. 7. Asynchronization of switching phenomena. αn = 0.40, β1 = 3.0,β2 = 9.0, γ = 470.0 and δ = 0.10.

Fig. 4. Coexisting attractors of a chaotic circuit as shown in Fig. 1. α1 =0.40, β1 = 3.0 and γ = 470.0.

(a)

(b)
Fig. 8. Switching phenomena on one circuit only. αn = 0.40, β1 = 3.0,
β2 = 12.0, γ = 470.0 and δ = 0.10.

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δ
0.4(a)
0.3(b)
Fig. 9. Holding one attractor state in each circuit. αn = 0.40, β1 = 3.0,
β2 = 7.5, γ = 470.0 and δ = 0.25.

0.20.1(a)03(b)
Fig. 10. Boundary area between Fig. 5 and Fig. 7. αn = 0.40, β1 = 3.0,β2 = 6.0, γ = 470.0 and δ = 0.22.

6912151821β２Fig. 13.
Relationship among parameter β2 , δ and observed phenomena.
αn = 0.40, β1 = 3.0 and γ = 470.0.

δ0.7(a)0.6(b)
Fig. 11. Boundary area between Fig. 6 and Fig. 7. αn = 0.40, β1 = 3.0,β2 = 18.0, γ = 470.0 and δ = 0.21.
0.50.4(a)α２0.30.4(b)Fig. 12. Anti-phase synchronization of switching phenomena with synchronization phenomena. α1 = 0.40, α2 = 0.46, βn = 3.0, γ = 470.0 andδ = 0.52.

0.42

0.44

0.46

0.48

Fig. 14.
Relationship among parameter α2 , δ and observed phenomena.
α1 = 0.40, βn = 3.0 and γ = 470.0.

- 18 -