2012 International Workshop on Nonlinear Circuits,Communications and Signal Processing
NCSP'12, Honolulu, Hawaii, USA. March 4-6, 2012

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

in1Abstract

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 isa models suitable for large-scale coupled nonlinear systems.
The reason of why is follows. Getting electric parts is easy
and inexpensive, experiment time is very short, repeatabilityof experiments is high and an electric circuit is a real physicalsystem. Therefore, there are many studies of large-scale coupled circuit systems. In these studies, synchronization phenomena are attracted researcher’s 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].
In this study, synchronization of switching phenomena isinvestigated in a star-coupled chaotic circuits.
2. System Model
Figure 1 shows a circuit model used in this study. This
circuit consists of a linear negative resister, two inductors,
a capacitor and a bi-directionally coupled diodes. Figure 2

L2

L1

Some chaotic circuits have coexisting attractors. Switching
phenomena of coexisting attractors are observed in these circuits. However, there are a few of study that is focused on
switching phenomenon. In our past study, synchronization ofswitching phenomena are confirmed in coupled chaotic systems which have some kinds of topologies.
In this study, we investigate an existence of the phenomenain the case of a star-coupled system.

C-r

in2

CCnvnf( in2)

Figure 1: A circuit modelshows circuit experimental results. By increasing a value ofthe negative resistance, a periodic orbit (a), two types periodic orbits (b) and (c) which depending on initial values, twotypes chaotic orbits (d) and (e) which depending on initialvalues, chaotic orbit (f) which coexists of (d) and (e) are observed. Switching phenomena can be observed between twotypes chaotic orbits and the coexistence of two chaotic orbits.
In order to carry out the computer simulations, a bidirectionally coupled diodes models as follows. The i-v characteristic of a bi-directionally coupled diodes is described as:f (id ) = 2Vu(id ) − V and shown in Fig. 3.
By using this model, computer simulation results corresponding to Fig. 2 are obtained as shown in Fig. 4 The coexisting attractors are observed in Fig. 4 (c) which is corresponding to Fig. 2 (f). The attractors are classified by colors according to following definition. When the solution hitsy ≥ 0.675, z = 1 and ẏ < 0, then the color is set as blue. The
other case, namely, the case of y < 0.675, z = 1 and ẏ < 0,
the color is set as red. This definition is determined from ourprevious study [2]. This is applied to all simulation results ofthis study.
In our previous study, full-coupled, ladder and ring topology were applied to investigated system. In these system, synchronizations of switching phenomena are observed.
However, all of them are equivalent. In this study, starcoupled chaotic system is investigated. This system is oneto-many coupling topology. One center element is not equivalent to others. It is very interesting to observe the synchronization of switching phenomena.

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Figure 5 shows the star-coupled chaotic system. The system equation is described as follows:di11

= v1 + ri11 ,L1


dt



di

 L2 12 = v1 − V (2u(i12 ) − 1) ,dt
N∑
dv1(N − 1)v1 −
 .
(i)
=−+iC−Gv1112k
dtk=2din1L1dtdi L2 n2dt
dvnC
dt


(1)
=vn + rin1 ,=vn − V (2u(in2 ) − 1) ,=− (in1 + in2 ) − G (vn − v1 ) .
(n = 2, 3, · · · , N)

By changing parameters and variables as follows,√√
Ct = L1Cτ, in1 = Vxn , in2 = yn ,L1 √Cdvn = Vzn , ”· ” =, α=r,
dτL1
√√
L1 C
L1and δ = G,β=L2C
the system equation is described as follows:
A center circuitẋ1 = αx1 + z1









 ẏ1 = β {z1 − (2u(y1 ) − 1)}




N

∑






(N − 1)z1 −ż=−x−y−δz


111k

k=2
Other circuits
ẋnẏ nżn
(2)=

αxn + zn=

β {zn − (2u(yn ) − 1)}=−xn − yn − δ (zn − z1 )(3)and blue waves are corresponding to the attractor states asshown in Fig. 4(a) and (b). Green lines show diﬀerences ofattractor state. The synchronization of switching phenomena
is observed in a black frame. Note that switching phenomenameans transition of attractors in this paper. This system keeps
asynchronous state at all times by reason that an amplitude ofz1 − z2 is observed. Though asynchronous state is kept at thissystem, switching phenomenon is generated in both circuit at
the same time. This phenomenon is very interesting at thepoint of the simultaneity in asynchronous behavior. We also
confirmed this phenomena in circuit experiment [2].
We guess a mechanism of the phenomena qualitatively asfollows. Basically, two circuits want to be synchronized witheach other. Since the coupling elements are resistors. In order to be synchronized, a current flows through coupling elements. However, the current also flows diodes as shown inFig. 6. Because both circuits have two symmetrical equilibrium points which are caused by i − v characteristic of diodes,
two attractors are attracted to reverse equilibrium points eachother. Therefore, two attractors can not be synchronized.
3.2. Case of N ≥ 3

Figure 7 shows the simulation result case of N = 3. The
horizontals show the time, the vertical show the zn , yn and z1 −zn . The synchronization of switching phenomena is observedin a black frame. It seems that the switching phenomenon ofCC2 and CC3 have occurred in accordance with CC1 of it. We
can also confirm the reverse currents with y1 and y2 , y1 andy3 respectively. However there is a lag with the switching ineach circuit.
We also investigate case of N = 4, 5, 6. Observation of(4)synchronization of switching phenomena is a bit diﬃcult inthese case. One of the simulation results is shown in Fig. 8.
The horizontal axes show the time, the vertical axes show thezn , yn and z1 − zn . Switching of CC1 can be clearly observed.
However, Switching of CC2 , CC3 and CC4 are not clear. We
consider that the reason is as follows. The eﬀect of the current
from CC1 to CCn is decreased by increasing the number ofthe circuits because currents from CC1 to each circuits are(5) divided into N − 1. In the case of N ≥ 7, synchronization ofswitching phenomena could not be observed.

(n = 2, 3, · · · , N)

4. Conclusions

where N is a number of circuits, u() is a step function and xn ,yn and zn are corresponding to in1 , in2 and vn , respectively.

In this study, we have investigated synchronization ofswitching phenomena in star-coupled system. As results,

3.1. Case of N = 2• Synchronization of switching phenomena can be observed in the cases of the number of circuits are fromtwo to six.

Figure 6 shows simulation result of N = 2. The horizontals
show the time, the vertical show the zn , yn and z1 − zn . Red• The phenomena can be observed clearly in the case of
two and three.

3. Computer Simulations- 77 -• The observation of synchronization of switching phenomena becomes diﬃcult by increasing the number ofthe circuits.
References[1] H. Sekiya, S. Mori and I. Sasase, “Synchronization ofSelf-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.

(a) r = 3.52 [kΩ][3] T. Nishimoto, Y. Hosokawa and Y. Nishio, “In-Phase andAnti-Phase Synchronization of Switching Phenomena inCoupled Chaotic Circuits” Proc. of IEEE Workshop onNonlinear 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
Trans. Circuit and Systems, vol. 37, no. 4, pp. 473–487,
1990.

(b) r = 3.96 [kΩ](c) r = 3.96 [kΩ](d) r = 4.99 [kΩ](e) r = 4.99 [kΩ](f) r = 5.29 [kΩ]

Figure 2: Circuit experimental results. Vertical axes: v1[V/div]. Horizontal axes: y1 [0.1mA/div]. L1 = 642.8[mH],L2 = 211.1[mH] and C = 7.06[nF].

- 78 -f( in2)
Vin2-V

Figure 3: i-v characteristic(a) α = 0.35

Figure 6: A computer simulation result. N = 2, α = 0.41,
β = 3.0 and δ = 0.26.

(b) α = 0.35

Figure 7: A computer simulation result. N = 3, α = 0.41,
β = 3.0 and δ = 0.26.

(c) α = 0.36

Figure 4: Computer simulation results. β = 3.0.

CC3

CC2

CC1

CCN

CCn

Figure 5: System model

Figure 8: A computer simulation result. N = 4, α = 0.41,
β = 3.0 and δ = 0.24.

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