2013 International Workshop on Nonlinear Circuits,Communications and Signal Processing
NCSP'13, Isrand of Hawaii, Hawaii, USA. March 4-7, 2013

Synchronization of Switching Phenomena in a Coupled Chaotic System with Inaba Circuit
Takuya Nishimoto† , Yasuteru Hosokawa† and Yoshifumi Nishio‡†Shikoku University
Furukawa, Ohjin, Tokushima, Japan
Email: s2137034@edu.shikoku-u.ac.jp‡Tokushima University
2-1 Minami-Josanjima, Tokushima, Japan
Email: nishio@ee.tokushima-u.ac.jp

shows i-v characteristic of the diodes that are idealized model.
Figure 3 shows a coupled chaotic system with Inaba circuit.

Abstract
Synchronization of switching phenomena can be observed ina few coupled chaotic systems. In our past study, we have observed synchronization of switching phenomena in coupledchaotic circuits with Nishio circuit. However, these phenomena have reported only two cases. In this study, we investigate
the phenomena in other chaotic system with Inaba circuit.
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. An 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 is 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 coexisting
attractors. 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 have observed similar phenomena in some chaotic systems with Nishio circuit[2][3]. We investigated full-coupled, ladder-coupled, starcoupled and ring-coupled chaotic system with Nishio circuit
is investigated.
In this study, we investigate these phenomena in fullcoupled chaotic systems with Inaba circuit.

Figure 1: A circuit modelf( i n2 )αi n2-αFigure 2: System model

CC 1CC 2CC 4CC 32. System Model
Figure 1 shows a circuit model used in this study. Thiscircuit consists of a linear negative resister, two inductors,
a capacitor and a bi-directionally coupled diodes. Figure 2- 133 -Figure 3: System model(N = 4)

Full coupled chaotic system is investigated in this study. Thesystem equation is described as follows:()
N∑dvnC= gvn − (in1 + in2 ) − G N vn −vk ,dtk=1
din1= vn ,L1dt
din2
L2= vn − f (in2 ).
dt
(n = 1, 2, · · · , N )(1)
By changing parameters and variables as follows,√√CC
vn = xn , in1 =yn , in2 =zn ,L1
L1√√
L1d(2), b=g,
t = L1 Cτ, ”· ” =dτC√
L2L1n=and δ = G.
L1Cwhere z = 0.7 and ẏ < 0. From this bifurcation diagram, theattractors is classified by colors according to following definition. When the solution hits the poincaré map and y ≤ 0.32,
the color is set as blue. The other case, namely, When thesolution hits the poincaré map and y > 0.32, the color is set
as red. This definition is important to investigate switching of
attractors and as such, is being applyed to the simulations of
this study.

the normalized system equation is described as follows:when zn > 0,()
N∑

zk ,
 ẋn = bxn − yn − zn − δ N zn −Figure 4: Coexisting attractors. α = 0.7, b = 1.49, n =0.0495 and V0 = 1.

k=1ẏn żn=xn ,=α1xn − .
nnwhen zn = 0, ẋn = ẏn(3)

1.5
0.7-3y3-1.5

3.2 Case of N = 2
Figure 6 shows simulation result of N = 2. These figures(bxn − yn − zn − δN zn −)N∑

zk(a),30(4)k=1=xxn .S

τwhen zn < 0, ẋn =
ẏn żn(bxn − yn − zn − δN zn −zk
k=1

=xn ,

=α
1xn + .
nn)

N
∑

,(5)

(b)

30

τ

where N is a number of circuits. xn , yn and zn are corresponding to vn , in1 and in2 , respectively.
3. Computer Simulations

(c)

30

This section shows numerical experiments of the systemmodel.

τ(1)3.1 Coexisting Attractors(2)

Figure 6: Computer simulation results. N = 2, (a) δ =Figure 5 shows a bifurcation diagram. The points are plot- 0.30. (b) δ = 0.15. (c) δ = 0.09. (1)Time waveforms of xi
ted when a solution hits the poincaré map that is defined and difference. (2) z1 vs z2 .

- 134 -ybFigure 5: Bifurcation diagram. α = 0.7, V0 = 1 and n = 0.0495.
show time waveforms(1) and lissajous figure(2). In the time point that there is behavior which has simultaneity in asynchronous behavior. Fig. 6 (b) and (c) show simulation results
which can be observed when decrease the coupling strengthfrom Fig. 6 (a). In this case, synchronization of switching
phenomena cannot be observed and lissajous figure gradually changed. This phenomenon is collapsing according to400
decrease the parameter value of δ.
In our past study, similar phenomena is observed withNishio circuit. However, there is a point of difference between Nishio circuit and Inaba circuit. The synchronization
τof switching phenomena in Nishio circuit is anti-phase attractor switching as shown in Fig. 7. In contrast, the phenomena
Figure 7: Synchronization of switching phenomena in Nishio in Inaba circuit is in-phase attractor switching. It is a interesting that though the structure of both circuit have a similarcircuit.
structure, the switching of these circuits are different.
waveforms, there are red and blue waves show that the attractor states. Green lines also show a difference of attractor 3.3 Case of N = 3state. Namely, if each circuit takes different attractor state,
the amplitude of this green line is observed. Synchronization
Figure 8 shows the simulation result case of N = 3. Weof switching phenomena occured in Fig. 6 (a). However this can confirm a synchronization of switching phenomena. Thesystem keeps asynchronous state at all because the wave of behavior of N = 3 is nearly synchronous state compare withx1 −x2 is observed. Though asynchronous state is kept at this case of N = 2. Because the amplitude of difference wave
system, switching phenomena are occured in both circuit at become smaller than N = 2. Moreover, frequency of thealmost same time. This phenomenon is very interesting at the switching is decreased. This tendency also can be observed in- 135 -the system with Nishio circuit. The behavior of the phenom- can be around N = 3, the phenomena in Inaba circuit canbe observed around N = 5. In the case of N > 5, the system becomes almost synchronous state and synchronizationof switching phenomena cannot be observed.
4. Conclusions30

τFigure 8: A computer simulation result for N = 3. α =0.7, V0 = 1, b = 1.56, n = 0.0495 and δ = 0.27.

In this study, synchronization of switching phenomena isinvestigated with Inaba circuit. As a result, In-phase synchronization of switching phenomena is observed. This result isdifferent from the synchronization of switching phenomenain the system with Nishio circuit. The switching of the system with Inaba circuit is in-phase and the phenomena can beobserved around N = 5. Moreover, there are tendencies asfollows. In case of the number of the circuit is increased, frequency of switching is decreased and gradually become synchronous state. The waveform of difference becomes bigger
before occur the switching.
References

ena have characteristics, such as the waveform of difference[1] H. Sekiya, S. Mori and I. Sasase, “Synchronization ofbecomes larger before occur the switching.
Self-Switching Phenomena on Full-Coupled Chaotic Oscillators,” IEICE Trans., vol. J83-A, no. 11, pp. 1264–
3.4 Case of N > 31275, 2000.
We also investigate case of N > 3. One of the simula- [2] T. Nishimoto, Y. Hosokawa and Y. Nishio, “Anti-phasetion results is shown as Fig. 9. Frequency of switching isSynchronization of Switching Phenomena in Globallydecreased than case of N < 4. Although observation of
Coupled 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,” IEEETrans. Circuit and Systems, vol. 37, no. 4, pp. 473–487,
1990.

30

τ
Figure 9: A computer simulation result for N = 4. α =0.7, V0 = 1, b = 1.54, n = 0.0495 and δ = 0.21.
the phenomena in Nishio circuit is remarkably difficult at
N > 3, in Inaba circuit, still can be observed. In the caseof increased the number of the circuit from Fig. 9, the system becomes almost synchronous state. Though, observationof synchronization of switching phenomena in Nishio circuit- 136 -