2011 International Workshop on Nonlinear Circuits,Communication and Signal Processing
NCSP'11, Tianjin, China, March 1-3, 2011

Anti-Phase Synchronization of Switching Phenomena in Globally Coupled Chaotic Circuits
without Parameter Mismatches
Takuya Nishimoto† , Yasuteru Hosokawa† and Yoshifumi Nishio‡†Shikoku University
Furukawa, Ohjin, Tokushima, Japan
Phone:+81-88-665-1300
FAX:+81-88-665-1318
Email: {s2137034, hosokawa}@edu.shikoku-u.ac.jp‡Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Phone:+81-88-656-7470
FAX:+81-88-656-7471
Email: nishio@ee.tokushima-u.ac.jp

Abstract
In this study, synchronizations of switching phenomena areinvestigated in globally coupled chaotic circuits without parameter mismatches. Especially, relationship between phenomena and the number of a chaotic circuits is investigated.

GG

G
G

1. Introduction
Many kinds of complex phenomena can be observed on thelarge-scale coupled chaotic circuits. Investigations of these
phenomena are very important works in order to declare nonlinear phenomena in the natural world. A electric circuit issuitable for models of large-scale coupled nonlinear systems
because of the following reasons. Getting electric parts iseasy and inexpensive, short experiment time, high repeatability of experiments and it’s a real physical system. Therefore,
there are many studies of large-scale coupled circuit systems.
In these studies, synchronization phenomena are attracted attentions.
On the other hand, some chaotic circuits have coexistingattractors. In these circuits, switching phenomena of attractors can be observed. Normally, in the case of a synchronization state, the switching of attractors is also synchronized.
And in the case of asynchronous states, the switching of attractors is also asynchronized. However, synchronization ofself-switching phenomena on full-coupled chaotic oscillators
is reported by [1]. Additionally, we could observe similar
phenomena in an other system [2][3].
Let us describe below the points of a synchronization ofswitching phenomena:

GG

Figure 1: System model (N = 4)
• Anti-phase synchronization of switching phenomena
can be observed whether a parameter has mismatches
or not.
In this study, a relationship between the number of chaotic
circuits and anti-phase synchronization of switching phenomena is investigated in globally coupled chaotic circuits without parameter mismatches.

2. System ModelIn this study, globally coupled chaotic circuits shown in
Fig. 1 is investigated. Elements of this system are coupled
by resistors. The element is a chaotic circuit [4] as shown in• The system keeps asynchronous state.
Fig. 2. This circuit consists of three memory elements, one• Each circuit switches in the almost same time.
linear negative resistor and bi-directionally-coupled diodes.
In this circuit, two attractors which are symmetrical about a
In previous study, we investigated in case that the number oforigin are observed by adjusting the value of the negative reelements is two. Following results were obtained.
sistor.
• In-phase and anti-phase synchronization of switching
In order to derive the system equation, bi-directionallyphenomena are observed.
coupled diodes are modeled as Fig. 3. Then, the v-i char-- 151 -

in1

in2
L2

L1
Cvnr

Figure 2: Circuit model

Figure 4: Bifurcation diagram of Fig. 2. Vertical axis: y.
Horizontal axis: α. β = 3.0 and γ = 470.0. Red: Initialvalues are x = 0.1, y = 0.1 and z = 0.1. Blue: Initial values
are x = −0.1, y = 0.1 and z = −0.1.

f (id)VThe normalized system equation is described as follows.
ẋn = zn+ αxn , 


1γ1

 ẏn = β zn −yn +− yn −,2γγ(4)!
N

X


 żn = −xn − yn − δ N zn −zk

rdid-V(a) Circuit schematic(b)v-i characteristics

k=1Figure 3: Coupled diodes model

where N is corresponding the number of the circuits and n iscorresponding the circuit number. Computer simulations arecarried out using this equation.

acteristic is described as follows.
vd =id +VV− id −rdrd

3. Computer Simulations

.

(1)

First of all, we describe how to distinguish two attractors.
Next, the computer simulation results are shown in the caseof changing the number of elements.

The others elements are modeled as linear elements. Eachcircuit number is defined as 1 ≤ n ≤ N . By using these 3.1. Definition of two attractors
models, the system equation is described as follows:
Figure 4 shows a bifurcation diagram by using two kindsofinitial values. These points are plotted when the solutiondin1
= vn + rin1 ,L1hitsz = 1 and ẏ < 0. From this diagram, two attractors
dt




aredistinguishedby y = 0.675. Coexisting of two attractors

din2rdVV
L2= vn −
in2 +− in2 −,are observed in α > 0.356. Therefore, we defined the statedt2rdrd !
R and B when the solution is y ≤ 0.675 and y > 0.675
N
Xdvn
respectively. Figure 5 shows a simulation result of the chaotic C= − (in1 + in2 ) − G N vn −vk .
dtcircuit as shown in Fig. 2. Coexisting attractors are observed.
k=1(2) This result is color-coded by the definition. In this paper, thisdefinition is applied in all of the simulations.
Changing parameters and variables as follows,
rCCt = L1 Cτ, in1 = V
xn , in2 = V
yn ,L1 r
L1Cd, α=r,vn = V zn , ”· ” =dτ
L1rr
L1C
L1β=, γ=rd and δ = G.
L2L1C√r

3.2. Case of N = 2(3)Figure 6 shows a computer simulation results in the case of
N = 2. In Fig. 6(a), horizontal axis shows the time. The vertical axes z1 and z2 are corresponding to voltages vn of thecircuit model as shown in Fig. 2. Fig. 6(b) shows the difference of two waveforms. The horizontal axis shows the time.

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

Figure 5: Coexisting attractors of a chaotic circuit as shownin Fig. 2. α = 0.35, β = 3.0, γ = 470.0
The vertical axis shows as follows. A black line shows z1 z2 . A green line shows the difference of two coexisting attractors. Namely, when both z1 and z2 are red or blue, theline becomes 0. When z1 is blue and z2 is red, the line becomes 1. When z1 is red and z2 is blue, the line becomes -1.
On the basis of above points, let us explain the phenomena
in Fig. 6. The black line of Fig. 6(b) shows that an asynchronous state is kept at all time. The color of the waveforms
as shown in Fig. 6 shows that two attractors are synchronizedwith in-phase. Thus, this phenomena is called as “in-phasesynchronization of switching phenomena”. Note that the circuits have parameter mismatches. Namely, this phenomena
cannot be observed in the case that parameter mismatches become zero.
Figure 7 shows the other computer simulation result in thecase of N = 2. In this case, an asynchronous state is also kept
at all time. A difference from Fig. 6 is a relationship between
colors of two waveforms. In this case, this phenomena iscalled as “anti-phase synchronization of switching phenomena”. This phenomena can be observed weather parameterhas mismatches or not.
From these results, we consider that to observe “in-phasesynchronization of switching phenomena” is difficult in thecase of increasing the number of elements because increasing
the number of elements means increasing a range of parameter mismatches. However, we consider that “anti-phase synchronization of switching phenomena” may be observed in
the case of increasing the number of elements and the result
is very interesting.

(a)

(b)
Figure 7: Anti-phase synchronization of switching phenomena case of N = 2. αn = 0.40, β = 3.0, γ = 470.0 andδ = 0.21.
two attractors becomes B : R = 1 : 2 or B : R = 1 : 2
constantly. Furthermore, a switching time is not so short asshown in part B of Fig. 8. This switching time is observed in
all the switching phenomena.
3.4. Case of N = 4Figure 9 shows a computer simulation results in the caseof N = 4. Switching phenomena are also observed. A proportion of two attractors becomes B : R = 2 : 2 constantly.
Furthermore, as mentioned above, a switching time can bealso observed. Whereas, there is another different point. The
switching time is longer than the case of N = 3. The systemneeds much time to determine the attractor of each circuit.
3.5. Cases of N = 5 to 8

The cases of N = 5 to 8 were also investigated. Figure 10shows one of computer simulation results. From results of investigations of N = 2 to 8, it is clear that the switching time
is longer and longer as increasing the number of N . Additionally,a proportion of two attractors becomes B : R = 1 : 1
3.3. Case of N = 3
in the case of N is equal to even number. And a proporFigure 8 shows a computer simulation results in the case tion of two attractors becomes B : R = N/2 : N/2 + 1 orof N = 3. In the center of Fig. 8, switching phenomena are B : R = N/2 + 1 : N/2 in the case of N is equal to odd
observed. Additionally, it is confirmed that a proportion of number.

- 153 -ACB

(a)

(a)
(b)
Figure 8: Anti-phase synchronization of switching phenomena case of N = 3. α = 0.40, β = 3.0, γ = 470.0 andδ = 0.151.
4. Conclusions(b)
In this study, a relationship between the number of chaoticcircuits and anti-phase synchronization of switching phenomena have been investigated in globally coupled chaotic cirFigure 9: Anti-phase synchronization of switching phenomcuits without parameter mismatches.
ena in the case of N = 4. α = 0.40, β = 3.0, γ = 470.0 andConsequently, we can be obtained following results. Atδ = 0.095.
least, anti-phase synchronization of switching phenomena
can be observed in the case of N = 2 to 8. The switchingtime is longer and longer as increase the number of elements.
There is a rule of proportion of two attractors.
References[1] H. Sekiya, S. Mori and I. Sasase, “Synchronization of
Self-Switching Phenomena on Full-Coupled Chaotic Oscillators,” IEICE Trans., vol. J83-A, no. 11, pp. 1264–1275, 2000.
[2] T. Nishimoto, Y. Hosokawa and Y. Nishio, “Anti-phase
Synchronization of Switching Phenomena in GloballyCoupled System of Chaotic Circuits,” IEICE TechnicalReport, no. NLP-48 & CAS2010-32, pp. 1–4, 2010.
[3] T. Nishimoto, Y. Hosokawa and Y. Nishio, “In-Phase andAnti-Phase Synchronization of Switching Phenomena in
Coupled Chaotic Circuits” Proc. of IEEE Workshop on
Nonlinear Circuit Networks (NCN’10) , pp. 15–18, 2010.
[4] Y. Nishio, N. Inaba, S. Mori and T. Saito, “RigorousAnalyses of Windows in a Symmetric Circuit,” IEEE Figure 10: Anti-phase synchronization of switching phenomTrans. Circuit and Systems, vol. 37, no. 4, pp. 473–487, ena in the case of N = 8. α = 0.40, β = 3.0, γ = 470.0 andδ = 0.043.
1990.

- 154 -