IEEE Workshop on Nonlinear Circuit Networks
December 9-10, 2011

Investigation of Switching Phenomena in Coupled
Chaotic Circuits as a Ring Structure
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 double scroll chaotic circuits. In our previous study, a synchronization of switching phenomena in a full-coupled chaotic circuits
have been confirmed. It is very interested that the phenomenon
can be observed in an asynchronization state.
In this study, we investigate a coupled chaotic circuits as aring structure. By carrying out some computer simulations, amechanism of the phenomenon are revealed.

I. I NTRODUCTION
Synchronization can be observed in everywhere. For instance, there are synchronization of two pendulum clock
reported by Huygens, a relationship between rotation periodof the Earth and revolution period of moon, firefly synchronization and so on. Therefore, investigating synchronizationof coupled oscillatory systems is very important.
In coupled chaotic circuits, synchronization is also observed. Since this phenomenon reported by Pecora et al. [1],many researchers of coupled chaotic circuits pay attention tosynchronization.
On the other hand, some chaotic circuits have double scroll
attractors. In these circuits, switching phenomena of attractors
are observed by changing parameters. Self-switching phenomena in coupled double scroll circuits reported by Sekiya et al.
[2]. In this case, these circuits are not synchronized. Therefore,
we can say that this phenomenon includes synchronization
and asynchronization. We consider that this is very interesting
phenomenon.
In our previous study [3], we have observed similar phenomenon in another circuit and another coupling topology. The
phenomenon is also asynchronization state and anti-phase synchronization of switching of attractors are observed. However,
we could not explain the mechanism of this phenomenon.
In this study, we investigate the case of same chaotic circuit
and another coupling topology. This investigation reveals themechanism.
II. S YNCHRONIZATION OF S WITCHING P HENOMENA
A circuit model [4] using in this study is shown in Fig. 1.
Figure 2 shows its computer simulation result. Double scroll
attractors are observed and two attractors are symmetricalabout a origin. In this result, attractors are color-coded byfollowing rules. In the case of y ≥ 0.675, z = 1 and ẏ < 0,
the color is set as blue. The other case, namely, the case ofy < 0.675, z = 1 and ẏ < 0, the color is set as red. Wedefine the Poincaré section z = 1 and ẏ < 0. Colors aredistinguished by y = 0.675. This distinction is derived froma bifurcation diagram as shown in Figure 3 that is used twokinds of initial values. In this paper, this definition is appliedin all of the simulations. Double scroll attractors are observedin α > 0.356.
In our previous study, chaotic circuits coupled by resistors
as shown in Fig. 4 was investigated. The system equation isdescribed as follows:
din1


= vn + rin1 ,L1
dt

din2rdVV
L2= vn −in2 +− in2 −,dt
2rdrd !
NXdvn
=−(i+i)−GNv−vk .
Cn1n2ndt
k=1Changing parameters and variables as follows,rr√CCt = L1 Cτ, in1 = V
xn , in2 = V
yn ,L1 r
L1dC
vn = V zn , ”· ” =, α=r,dτ
L1 rr
L1CL1
β=, γ=rd and δ = G.
L2L1C

(1)

(2)

The normalized system equation is described as follows.
ẋn = zn+ αxn , 
γ11
 ẏn = β zn −yn +− yn −,2γγ(3)!
N

X

 żn = −xn − yn − δ N zn −zk
k=1

where N is corresponding the number of the circuits and n
is corresponding the circuit number. Figure 5 shows one ofits computer simulation results. Horizontal axes are time andvertical axes are z1 , z2 , z3 and z4 which are corresponding
to voltages of each circuit. Switching phenomena of attractors
are observed at the same time. However, all waves are notsynchronized each other. Additionaly, there are no parameter
errors. We call this phenomenon synchronization of switching
phenomena.

- 112 -in1

in2

C

vn

Ln2

Ln1

CCnf( in2)

-rFig. 1.

Circuit modelFig. 4.

Fig. 2. Computer simulation result of a chaotic circuit as shown in Fig. 1.
α = 0.40 and β = 3.0

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. Therefore, we consider that the phenomena is very
interesting phenomena. This phenomenon is reported by [2].
Differences between [2] and our study are follows.
In our system,
There are no parameter errors.
Two kinds of attractors are observed at the same time.
• Circuit topology is full-coupled.
••

In this study, we investigate switching phenomena in coupled chaotic circuits as a ring structure and reveal the mechanism of this phenomenon.

III. C OUPLED C HAOTIC C IRCUIT WITH A R ING
S TRUCTURE
We investigate coupled chaotic circuits which is coupledby resistors shown in Fig. 6. The circuit topology is a ringstructure.
Firstly, a system equation is derived. Bi-directionallycoupled diodes are modeled as Fig. 7. This model is describedas following function

System model in our previous study. N = 4.

Fig. 5.
Computer simulation result of coupled chaotic circuits as shownin Fig. 4. Horizontal axes are time and vertical axes are z1 , z2 , z3 and z4
which are corresponding to voltages of each circuits. α = 0.405, β = 3.0and δ = 0.20.

models, system equation is described as follows:din1
= vn + rn in1 ,L1dt
din2= vn − (2V u(id ) − V ) ,L2ndt
dvn
C= − (in1 + in2 ) − G (2vn − vn−1 − vn+1 ) .
dt(5)Changing parameters and variables as follows,r√Ct = Ln1 Cτ, in1 = V
xn , in2 = yn ,Ln1 rdC(6)vn = V zn , ”· ” =, α=r,
dτL1r
√
Ln1 C
Ln 1β=and δ = G.
L2nCThe normalized system equation is described as follows.
ẋn = αxn + zn ẏn = β {zn − (2u(yn ) − 1)}żn = −xn − yn(7)−δ(2z−z−z)nn−1n+1(n = 1, 2, · · · , N )wherez0 = zN ,vd = 2V u(id ) − V,(4)where V is a threashold voltage of a diode and u() is a stepfunction. The other elements are modeled as linear elements.
Each circuit number is defined as 1 ≤ n ≤ N . By using thesezN +1 = z1 .

(8)

IV. C OMPUTER S IMULATIONS
Figure 8 shows simulation results of the system as shownin Fig. 6. Horizontal axes show time and vertical axes show
voltages of each circuit. In this system, synchronization ofswitching phenomena is not observed except for the case of- 113 -Fig. 3. One parameter bifurcation diagram of result of a chaotic circuit as shown in Fig. 1. β = 3.0. Initial values are x = −0.1, y = −0.1, z = −0.1
and x = 0.1, y = 0.1, z = 0.1.

the number of circuit is 3. A ring and full-coupled topology
is same that in the case of N = 3. Figure 10 shows thephenomena in the case of the number of circuit is 3. In the caseof N > 3, when the number of circuits is even numbers, twoattractors are observed alternately like Fig. 8 (a). When thenumber of circuits is odd numbers, basically, two attractors
are observed alternately. However, there is a point where
neighboring attractors become a same type attractor. This point
keeps moving as shown in Figs. 8 (b) and 8 (c). These results
show that the neighboring circuit becomes a different attractor
in spite of coupled by resistors. We consider that the reason
as follows.
Figure 9 shows a simulation result in the case that thenumber of circuit is two. Horizontal axes are time. Verticalaxes are z1 , z2 , y1 and y2 which are corresponding to v1 ,v2 , i12 , i22 , respectively. Normally, in-phase synchronization
is observed in a system coupled by resistors. In this case,waveforms are similar each other. A current flows a coupling
resistor in order to decrease a voltage difference between twocircuits. Therefore, directions of two currents i12 and i22 arereverse each other. This flow makes directions of thresholdvoltages of bi-directionally-coupled diodes reverse each other.
This is the reason why these are not synchronized.

CC13nFig. 6.

[1] L. M . Pecora and T. L. Carroll, “Synchronization in Chaotic Systems,”
Phys. Rev. Lett., vol. 64, pp. 821–824, 1990.
[2] 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.

f( id)f( id )

Vid-V(a) Circuit schematic.
Fig. 7.

R EFERENCES

System model in this study.

idV. C ONCLUSIONSIn this study, switching phenomena in coupled chaoticcircuits as a ring structure is investigated. As a result, amechanism of synchronization of switching phenomena incoupled chaotic circuits are revealed.

2

N(b)i-v characteristics.

Bi-directionally-coupled diodes model.

[3] 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.
[4] 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.
[5] 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.

- 114 -50(a)

50(b)

250

Fig. 10.
Computer simulation results of the system model as shown inFig. 6. N = 2. α = 0.405, β = 3.0 and δ = 0.21.

50(c)Fig. 8. Computer simulation results of the system model as shown in Fig. 6.
α = 0.405, β = 3.0 and δ = 0.20. (a) N = 8. (b) N = 7. (c) N = 7.
Differential initial values are applied in (b) and (c).

50Fig. 9. Computer simulation results of the system model as shown in Fig. 6.
N = 2. α = 0.405, β = 3.0 and δ = 0.20.

- 115 -