IEEE Workshop on Nonlinear Circuit Networks
December 12-13, 2014

Propagation of Switching Phenomena observed on a
Ring of Coupled Chaotic Circuits in State of
N-Phase Synchronization
Takuya Nishimoto∗ , Yoko Uwate∗ , Yasuteru Hosokawa† , Yoshifumi Nishio∗ and Daniéle Fournier-Prunaret‡∗Dept. of Electrical and Electronic Eng., Tokushima University,
2-1 Minami-Josanjima, Tokushima, 770-8506 Japan
Email: {nishimoto, uwate, nishio}@ee.tokushima-u.ac.jp†
Dept. of Media and Information Systems, Shikoku University,
Furukawa, Ohjin, Tokushima, 771-1192, Japan
Email: hosokawa@keiei.shikoku-u.ac.jp‡
LAAS-CNRS, 7 avenue du colonel Roche, F-31077 Toulouse Cedex 4, France
Email: daniele.fournier@insa-toulouse.fr
Abstract—2 types of synchronization phenomena, which arein-phase and N -phase synchronization, can be observed on a ringof coupled chaotic circuits. In state of N -phase synchronization,
the system generates propagation of switching phenomena. In
this study, we focus on the relationships between the propagationof switching phenomena and parameters. Various propagation
patterns depending on the number of circuits and propagation
velocity are investigated.

I.

This paper is organized as follows. In Section II, we
propose a ring of coupled chaotic system. In Section III, 2type chaos synchronization are shown. Be based on Section III,
propagation of switching phenomena and its patterns areshown in Section IV. Finally, Section V concludes this paper.
II.

S YSTEM MODELI NTRODUCTION

Chaos synchronization observed on large-scale coupledchaotic systems have been attracted attention in various sciencefields because it can be regarded as models of a lot of real
physical systems. For instance, chaos phenomena have beenreported in engineering, biology, economics, astronomy and soon. Therefore the investigation of synchronization phenomena
on high-order chaotic systems are very important to grasp
essentials of the phenomena observed in natural system.
On the other hand, some chaotic circuit have coexisting attractors [1]. Multi-scroll attractor observed on coupledchaotic system causes some interesting phenomena [2], [3].
In our past study, we investigated coexisting synchronization
phenomena on a ring of coupled chaotic system [4]. The
phenomena are interesting since the system generates N -phasesynchronization phenomena although the coupling element is
simple positive resister. Moreover, we analyzed the phenomena
by carrying out averaging method on a ring of coupled van der
Pol circuit and clarified the stable parameter region of N -phasesynchronization.
In this study, we investigate propagation of switching phenomena observed on the ring of coupled chaotic system in stateof N -phase synchronization. In the past study, asynchronous
double-mode oscillations and wave propagations have reported[5], [6]. However, observation of propagation phenomena ona coupled chaotic circuit coupled by simple positive resister
have not been reported before. Thus, propagation of switching
phenomena in state of N -phase synchronization on coupledchaotic circuit coupled by simple positive resister is the firstobservation. Various propagation patterns and relationships
between propagation velocity and parameters are investigatedin proposed system model. By increasing the number ofcircuits, propagation patterns are increased and it means a lotof coexisting state can be observed.

R

R
R

R
i N1 i N2
L1

L2
vN

-r

L1
f(i N2)-r

Fig. 1.

i 21 i 22

i 11 i 12
L2
v1

L1
-r
f(i 12)L2
v2
f(i 22)

System model.

The system model is shown in Fig. 1. Each subcircuit iscoupled by resistor. Subcircuit is chaotic circuit which consistsof three memory elements, one linear negative resistor, and bidirectionally-coupled diodes proposed by Nishio et al [8]. Bychanging the parameters and variables,
√
√
√
C
C
xn , in2 = V
yn ,t = L1 Cτ, in1 = V
L1 √
L1
d
C
L1
(1)
vn = V zn , ”· ” =,α = r
,β =
dτ
L
L2
1
√
√
1 L1
C
and δ =
γ = rdL1
R C
the normalized circuit equation is described as following 3×N dimensional ordinary differential equation:ẋn = αx{n + zn ,

}



 ẏn = β zn − fˆ(yn ) ,(2)żn = −xn − yn



+δ
(z
−
2z
+
z
)
,
n−1
n
n+1


(n = 1, 2, · · · , N )- 109 -

wherethe propagation phenomena, we treat the case of the numberof the circuit is even number in this paper.

z0 = zN , zN +1 = z1 andIV.

fˆ(yn ) = 12 (|γyn + 1| − |γyn − 1|).
Figure 2 shows an example of asymmetric chaos attractor
depending on initial value obtained from one subcircuit. For
the computer simulation, we fixed the parameters as β = 3.0
and γ = 470.0. Asymmetric attractors are distinguished with
2 colors in simulation result. In this study, we distinguish theattractors as follows:(1) A Poincaré section is defined at zn = 1.0 andẏn < 0.
(2) When the solution hits the Poincaré section, andyn > 0.675, the color is set as red and such stateis defined state {M }. In the case of yn ≤ 0.675,
the color is set as blue and defined state {P }.
The switching phenomenon is behavior that of attractor transition. This definition is important to grasp the domain of
attractor in each subcircuit and as such, is being applied tofollowing simulation results in this paper.
III.

2- TYPE CHAOS SYNCHRONIZATION

2-type synchronization phenomena depending on initialvalues can be observed in the system as shown in Fig. 1.
Figure 3 shows the examples of synchronization phenomena.
N-phase chaos synchronization is shown in Fig. 3 (a), and insynchronization in Fig. 3 (b). Red and blue colors show thestate {M } and {P } respectively. Both synchronization states
are not synchronized rigorously since the system generates
chaotic signal. However, those behaviour shown in Fig. 3 (a)and (b) is almost synchronized. Therefore these phenomena arecalled N-phase synchronization and synchronization respectively in this paper. It should be noted that when the subcircuits(n is odd number) take state {P }, while the others (n is evennumber) take state {M } in Fig. 3. Namely, the subcircuits take
opposite attractor state with respect to the neighbor subcircuitsin this case. Figure 4 is simulation results in state of N-phasechaos synchronization. The phase difference between z1 andzn is enlarged from z1 vs z2 to z1 vs z6 . Moreover, z6 , which
are located diagonal position of z1 , is almost synchronizedwith anti-phase. This behaviour also can be observed in circuit
experiment as shown in Fig. 5. Figure 6 shows time waveforms
and Lissajous figure of z1 and z6 observed in simulation andcircuit experiment. While z1 takes state {P }, z6 takes state{M } and almost synchronized with anti-phase as shown in
Fig. 6.

P ROPAGATION OF SWITCHING PHENOMENA

Propagation pattern in case of N = 6 are shown in
Fig. 7 (b). The attractors of the circuit located on diagonal
position take state P , while the other take state M . These attractor states are propagated with clock rotation. The direction
is determined by initial conditions. Figure 8 shows propagation
velocity, which is expressed by v = 1/T . T is period of
propagation derived by computer calculation. By increasing
the coupling strength, the propagation velocity is also gradually
increased and finally extinguished. Similarly, the propagation
damping are induced by decreasing the coupling strength. Oneof the stable state of this system is shown in Fig. 7 (a). The
subcircuits take opposite attractor with respect to neighboring
subcircuits. These attractor states set shown in Fig. 7 (b) isstable N -phase synchronization state and propagation phenomena cannot be observed in any number of coupled chaotic
circuit. Figure 9 shows propagation patterns in case of N = 8.
Various stable propagation states are coexisting. Basically, thepropagation velocity is almost constant for any propagation
pattern. Figure 9 (b) is stable state which corresponds to thestate of Fig 7 (a). Figure 9 (e) seems the propagation isextinguished, however, the system produces propagation withexact N -phase synchronization. All propagation patterns for
N = 6 and N = 8 are shown in Table I. These patterns
are depending on initial conditions. Moreover, in the case of
uncoupled subcircuit, the circuit generates periodic orbit for
the fixed parameter sets of α = 0.285, β = 3.0 and γ = 470.
It implies that the system in state of producing propagation is
the states of torus or periodic orbit. With the increase in thenumber of the coupled chaotic circuits, propagation patternincreases exponentially. These propagation phenomena can beobserved in large-scale case. Figure 10 shows an example ofpropagation phenomena in case of N = 50. The propagation
phenomena can be observed clearly in case of large-scale case.
V.

C ONCLUSION

In this study, we have investigated propagation of switching
phenomena on a ring of coupled chaotic circuits. As a result,

Zn

From this result, we can confirm that subcircuits which arelocated in diagonal position are synchronized with anti-phasein state of N -phase synchronization. Namely, sum of thesetime waveforms become almost 0, in case of that the numberof the circuit is even number. Therefore, for observation of

(a)

Zn

(b)
Fig. 2. Coexisting chaotic attractor observed in a subcircuit. Horizontal: y1 .
Vertical: z1 . α = 0.35, β = 3.0 and γ = 470.0.

Fig. 3. 2-type synchronization phenomena depending on initial values. (a)N-phase chaos synchronization. (b) Synchronization. Vertical: zn . Horizontal:
τ . N = 10, α = 0.425, β = 3.0, γ = 470.0 and δ = 0.20.

- 110 -TABLE I.

State:SN n
S61
S62
S63
State
S81
S82
S83
S84
S85
S86

CC1
P or M
P(M)
P(M)
P(M)
P(M)
P(M)

CC1
P or M
P(M)
P(M)
CC2
P or M
M(P)
M(P)
M(P)
M(P)
M(P)

CC2
P or M
M(P)
M(P)
CC3
P or M
P(M)
M(P)
P(M)
M(P)
P(M)

CC3
P or M
P(M)
M(P)
CC4
P or M
M(P)
M(P)
M(P)
P(M)
M(P)

S TATE TABLE

CC4
P or M
M(P)
P(M)
CC5
P or M
P(M)
P(M)
P(M)
M(P)
M(P)

CC5
P or M
P(M)
M(P)
CC6
P or M
M(P)
M(P)
M(P)
M(P)
M(P)

CC6
P or M
M(P)
M(P)
CC7
P or M
P(M)
M(P)
M(P)
M(P)
M(P)

Propagation
×
×
⃝
CC8
P or M
M(P)
M(P)
M(P)
M(P)
M(P)

Propagation
×
×
⃝
⃝
⃝
⃝

(a) z1 vs z2(b) z1 vs z3(c) z1 vs z4(a) v1 vs v2(b) v1 vs v3(c) v1 vs v4(d) z1 vs z5(e) z1 vs z6(f) z1 vs z7(d) v1 vs v5(e) v1 vs v6(f) v1 vs v7(g) z1 vs z8(h) z1 vs z9(i) z1 vs z10(g) v1 vs v8(h) v1 vs v9(i) v1 vs v10

Fig. 4. Lissajous figures in state of N-phase chaos synchronization (computer
simulations). z1 vs zn . α = 0.425, β = 3.0, γ = 470.0 and δ = 0.2.

various patterns of propagation and relationships between
propagation velocity and parameters have clarified. Moreover,with the increase in the number of coupled chaotic circuits,
propagation pattern also increases exponentially. This result is
extremely interesting and would give the effective suggestion
to make clear the nonlinear phenomena in large-scale highdimensional system. In our future work, we will investigate
the mechanism of propagation of switching phenomena andregularity of the propagation patterns in proposed system.

Fig. 5. Lissajous figures in state of N-phase chaos synchronization (circuit
experiments). v1 vs vn [10V/div]. L1 = 625[mH] ± 1%, L2 = 209[mH] ±1%, C = 1.49[nF] ± 4% and R = 100.0[kΩ] ± 1%.

[5]

[6]

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- 111 -

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Z1

(a)
Z6

(a) State:S82 .
Z1

(b)
Z6

(1)

(2)

Fig. 6. Almost anti-phase synchronization state for N = 10. (a) Simulation
result. (1) Time waveforms of z1 and z6 . (2) z1 vs z6 . (parameter values are
the same as those in Fig. 4). (b) Circuit experiment. (1) Time waveforms ofv1 and v6 . (2) v1 vs v6 . (parameter values are the same as those in Fig. 5).

(b) State:S83 .

(c) State:S84 .

(a) State:S62 .
(d) State:S85 .

(b) State:S63 .
Fig. 7. Propagation of switching phenomena for N = 6. α = 0.285, β = 3.0
and δ = 0.068.

(e) State:S86 .
Fig. 9. Propagation of switching phenomena for N = 8. α = 0.285, β = 3.0
and δ = 0.1.

Fig. 8. Propagation velocity for N = 6. Vertical axis: v. Horizontal axis: δ.

Fig. 10. An example of propagation of switching phenomena for N = 50.
α = 0.285, β = 3.0 and δ = 0.4.

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