Synchronization Patterns Generated ina Ring of Cross-Coupled Chaotic Circuits
Yumiko Uchitani and Yoshifumi Nishio
Abstract— Studies on chaos synchronization in coupledchaotic circuits are extensively carried out in various ﬁelds. In
this study, synchronization patterns generated in a ring of crosscoupled chaotic circuits are investigated. Computer simulations
show that this coupled system produces several phase patterns.

I. I NTRODUCTION
ANY people have been trying to develop some applications to information processing by exploiting oscillatory phenomena in neural networks. Such neural networks
can produce some kinds of phase patterns, and they may beutilized for associative memory or image processing [1]-[4].
On the other hand, since synchronization phenomena incoupled oscillatory systems are good models to describe
various higher-dimensional nonlinear phenomena in the ﬁeldof natural science, studies on synchronization phenomena areextensively carried out in various ﬁelds [5]-[13].
In our past studies [14][15], we investigated an interestingstate transition phenomenon observed in simple coupledchaotic circuits. In this study, we investigate the phasepatterns characterized by synchronization in a ring of crosscoupled chaotic circuits. Computer simulations show that thiscoupled system produces several phase patterns.

M

II. BASIC C IRCUIT [14][15]In this section, we review the phenomena observed from
simple two cross-coupled chaotic circuits. Figure 1 shows thebasic circuit model. In this model, two simple autonomouschaotic circuits [16][17] are cross-coupled via inductors L2 .
By using the following variables and the parameters,

Fig. 1.

Basic circuit model.

the normalized circuit equations are given as follows.
⎧ẋ = zk
⎪
⎪
⎪ k
⎪
⎪
⎪
⎨ ẏk = α{γyk − wk − βf (yk − zk )}żk = β f (yk − zk ) + wk+1 − xk⎪⎪
⎪ ẇk = δ(yk − zk+1 )⎪⎪⎪⎩(k = 1, 2),

(2)

(1)

where z3 indicates z1 and f are nonlinear functions corresponding to the v − i characteristics of the nonlinear resistorsof the diodes and are described as follows.
⎧yk − zk − 1(yk − zk > 1)
⎪
⎪
⎪
⎨ 0
(|yk − zk | ≤ 1)
(3)
f (yk − zk ) =⎪yk − zk + 1(yk − zk < −1)⎪⎪⎩(k = 1, 2).

Yumiko Uchitani and Yoshifumi Nishio are with the Department
of Electrical and Electronic Engineering, Tokushima University, 21 Minami-Josanjima, Tokushima 770-8506, JAPAN (email: {uchitani,
nishio}@ee.tokushima-u.ac.jp).

A typical example of the observed phenomena is shownin Fig. 2. Figure 2(a) is computer simulated results obtainedby integrating Eq. (2) with the Runge-Kutta method andFig. 2(b) is the corresponding circuit experimental results.
In this state, the two circuits exhibited chaos but almost
synchronized in in-phase in the sense that the attractor was
almost in the quadrant I or III on the y1 − y2 (or v11 − v21 )plane. The behaviors of the circuits are very interesting⎧⎪
L1 ik1L1 ik2⎪⎪xk =, wk =
,
⎪
⎪
C
V
C
⎪
2
2 V
⎪
⎪

v
v
⎪
⎪
⎪ yk = k1 , zk = k2 , t = L1 C2 τ,
⎨
V
V


⎪
C
L
L1
1
⎪
⎪ α = 2, β =
G,
γ
=
g,
⎪
⎪
C1
C2
C2
⎪
⎪
⎪
⎪
L
d
⎪
⎪
⎩ δ = 1 , “·”=(k = 1, 2)
L2
dτ

3854
c
978-1-4244-1821-3/08/$25.002008IEEEbecause the solutions on the yi − zi planes seem to be
attracted to the ﬁxed points located at around (yi , zi )=(±1.2,0). However, after converging to the ﬁxed points, the solution
abruptly moves toward the other ﬁxed point. When one circuit switches to/from the positive region from/to the negative
region in this way, the other follows the transition after a few
instants.

2.5

0

-2.5

0

2.5 -2.5

0

2.5

2.0

y1 0
-2.0
2.0

y2 0
2.5

-2.0
0

τ

1000

(a)
0

-2.5

0

2.5 -2.5

(a1)

0

2.5

(a2)

3.0

y1 0
-3.0
3.0

y2 0

(b)

-3.0
0

τ

(a3)

300

Fig. 3. State transition phenomenon around anti-phase synchronization.
(a) Computer calculated results. α = 2.0, β = 4.0, γ = 0.1,andδ = 0.0008. (b) Circuit experimental results. L1 = 9.93mH, L2 = 1.2H,
C1 =32.8nF, C2 =49.5nF, and g=495mS.

2.5

0

(b1)

(b2)
-2.5

0

2.5 -2.5

0

2.5

2.0

y1 0
-2.0
2.0

y2 0
-2.0
0

τ

1000

(a)

(b3)
Fig. 2.
State transition phenomenon around in-phase synchronization.
(a) Computer calculated results. α = 1.5, β = 5.0, γ = 0.2, andδ = 0.005. (b) Circuit experimental results. L1 = 9.93mH, L2 = 800mH,
C1 =32.8nF, and C2 =49.5nF, and g=683mS. (a1) Attractor on y1 −z1 plane.
(a2) Attractor on y1 − y2 plane. (a3) Time waveform. (b1) Attractor onv11 −v12 plane. Horizontal and vertical: 1 V/div. (b2) Attractor on v11 −v21
plane. Horizontal and vertical: 1 V/div. (b3) Time waveform v11 and v21 .
Horizontal 1.0 ms/div and vertical: 2 V/div.

Similar transition phenomena can be observed around antiphase synchronization and quadrature-phase synchronization
as shown in Fig. 3 and Fig. 4, respectively.
III. R ING OF C ROSS -C OUPLED C HAOTIC C IRCUITS
Figure 5 shows the circuit model investigated in this study.
In the circuit, n chaotic circuits are cross-coupled with theirneighbors via inductors L2 . Note that the connections do notlook like crossed but are similar to the basic circuit in Fig. 1.

(b)Fig. 4. State transition phenomenon around quadrature-phase synchronization. (a) Computer calculated results. α = 2.0, β = 4.0, γ = 0.1,andδ = 0.0014. (b) Circuit experimental result. L1 = 9.93mH, L2 = 1.2H,
C1 =32.8nF, C2 =49.5nF, and g=495mS.

Namely a node of each circuit is connected to its neighbor’s
different node. Hence, we still call this connections as crosscouplings.
As well as the case of the two circuits, we approximate
the v − i characteristics of the nonlinear resistors consistingof the diodes by the following 3-segment piecewise-linear

2008 International Joint Conference on Neural Networks (IJCNN 2008)

3855

L1

C1

C2

i 1n

v n2

-g

i d n v n1
i 2n-1

L2-gi 42 L 1

L2

i 2n
v 11

C1

i 24 v 42i d4 C2C2 i d 1
v 12

L1

v 41

i 21
L2

i 11

i 22
v 21

-g

C1

v 22

idk =

id2
C2

L 1 -g⎪G(vk1 − vk2 + V ) (vk1 − vk2 < −V )
⎪
⎪
⎪
⎪
⎩
(k = 1, 2, 3, ..., n)(4)

The circuit equations are described as follows.

v 32

id3

i 13

C2

L1

are given as follows.
⎧ẋk = zk
⎪
⎪
⎪
⎪
⎪
⎪
⎨ ẏk = α{γyk − wk−1 − βf (yk − zk )}żk = β f (yk − zk ) + wk − xk
⎪
⎪
⎪
ẇ
⎪
k = δ(zk − yk+1 )⎪⎪⎩(k = 1, 2, 3, ..., n),(6)where yn+1 indicates y1 and f are the nonlinear functions
described by Eq. (3).
IV. S YNCHRONIZATION PATTERNS(5)

where i20 and v(n+1)1 indicate i2n and v11 due to the ringboundary condition, respectively. By using the same variables
and parameters as Eq. (1), the normalized circuit equations

3856

C1

-g

Ring of cross-coupled chaotic circuits.

⎧G(vk1 − vk2 − V ) (vk1 − vk2 > V )
⎪
⎪
⎪
⎪
⎪
⎨ 0
(|vk1 − vk2 | ≤ V )

⎧
di1k
⎪
L1
= vk2
⎪
⎪
⎪
dt
⎪
⎪
⎪
⎪
dvk1
⎪
⎪
= −idk + i2(k−1) + gvk1
C1
⎪
⎪
dt
⎨
dvk2
⎪
= idk − i2k − i1k
C2
⎪
⎪
dt
⎪
⎪
⎪
⎪
di2k
⎪
⎪
= vk2 − v(k+1)1
L2
⎪
⎪
dt
⎪
⎩
(k = 1, 2, 3, ..., n)

i 23

v 31

i 12

Fig. 5.

functions.

L2

L2

C1

The most common synchronization state observed from thering in Fig. 5 is the fully in-phase synchronization. Figure 6
shows an example of the fully in-phase synchronization statesobserved from 5 cross-coupled chaotic circuits. In this state,
the movement of all yi from/to the positive region to/from thenegative region almost synchronizes as shown in Fig. 6(g).
We consider that a movement of one circuit triggers theneighbors’ transition like as the two-circuit case. However,after the transient states we cannot distinguish the starting
circuit of the ring.

2008 International Joint Conference on Neural Networks (IJCNN 2008)

2.5

0

2.5
-2.5
-2.5

0

(a)

2.5 -2.50(b)

2.5 -2.50

2.5 -2.5

(c)

0

(d)

2.5 -2.50(e)

2.5 -2.5

2.5

0

(f)

0

2.0

y1 0

-2.5
-2.5

0

-2.0
2.0

2.5 -2.5(1a)0

2.5 -2.5(1b)0

2.5 -2.5(1c)0

2.5 -2.5

(1d)

0

2.5

(1e)

y2 0
2.0

-2.0
2.0

y1 0

y3 0

-2.0
2.0

-2.0
2.0

y2 0

y4 0

-2.0
2.0

-2.0
2.0

y3 0

y5 0
-2.0

τ

0

-2.0
2.0

1200

(g)
Fig. 6.
Fully in-phase synchronization of state transition (computer
calculated result). α = 1.5, β = 5.0, γ = 0.2, and δ = 0.001. (a) Attractor
on y1 − y2 plane. (b) Attractor on y2 − y3 plane. (c) Attractor on y3 − y4plane. (d) Attractor on y4 − y5 plane. (e) Attractor on y5 − y1 plane.
(f) Attractor on y1 − z1 plane. (g) Time waveform.

y4 0
-2.0
2.0

y5 0
-2.0

τ

0

1200

(1f)
2.5

0

-2.5
-2.5

By changing initial conditions, we can observe other typesof synchronization states. Figure 7 shows three different
kinds of synchronization states observe from 5 cross-coupledchaotic circuits. Because the adjacent two circuits can besynchronized in anti-phase as shown in Fig. 3, the phasestates of some circuits can be π with respect to the reference
circuit (circuit #1). Figure 7(a) shows the state that only oneof 5 circuits has π phase difference to the others. This state
can be written as [0, π, π, π, π] Figure 7(b) shows the statethat the adjacent two circuits has π phase difference to theother 3 circuits. This state can be written as [0, 0, π, π, π]Figure 7(c) shows the state that the second and the fourth
circuits has π phase difference to the other 3 circuits. This
state can be written as [0, π, 0, π, 0] We can expect muchlarger number of combinations of phase states for larger n.
Figure 8 shows how the characteristics of the synchronization state [0, π, π, π, π] change as the coupling parameter δ
increases. The horizontal axis is δ and the vertical axis is
the average length of the period of the transitions between
positive and negative. The curve of crosses shows the average
period of the state transitions of y1 . The curve of circles
shows the average delay time of the state transitions of y5
when the state transitions of y1 are considered to be thereference. We can see that the period decreases monotonically according to the increase of δ. Further, for δ larger thanabout 0.0021, the state [0, π, π, π, π] becomes unstable andonly the fully in-phase synchronization state can be observed.
The investigation of exact bifurcation scenario is our future
research subject.
Because the two cross-coupled circuits can generate thequadrature-phase synchronization as shown in Fig. 4, we
can expect some synchronization states including π/2 phasedifferences. Figure 9 shows two examples of such synchronization states. Figure 9(1) shows the state that 4 circuits
have π/2 phase difference to the reference circuit and hence0

2.5 -2.5(2a)0

2.5 -2.5(2b)0

2.5 -2.5(2c)0

2.5 -2.5

(2d)

0

2.5

(2e)

2.0

y1 0
-2.0
2.0

y2 0
-2.0
2.0

y3 0
-2.0
2.0

y4 0
-2.0
2.0

y5 0
-2.0

τ

0

1200

(2f)
2.5

0

-2.5
-2.5

0

(3a)

2.5 -2.50(3b)2.5 -2.50(3c)

2.5 -2.50(3d)

2.5 -2.5

0

2.5

(3e)

2.0

y1 0
-2.0
2.0

y2 0
-2.0
2.0

y3 0
-2.0
2.0

y4 0
-2.0
2.0

y5 0
-2.0

τ

0

1200

(3f)
Fig. 7. Several synchronization states including anti-phase synchronizations. (computer calculated result). α = 1.5, β = 5.0, γ = 0.2, andδ = 0.001. (1) [0,π,π,π,π] (2) [0,0,π,π,π] (3) [0,π,0,π,0] (a) Attractor on
y1 −y2 plane. (b) Attractor on y2 −y3 plane. (c) Attractor on y3 −y4 plane.
(d) Attractor on y4 − y5 plane. (e) Attractor on y5 − y1 plane. (f) Time
waveform.

2008 International Joint Conference on Neural Networks (IJCNN 2008)3857

τ 1200

2.5

0

1000

-2.5
-2.5

0

800

2.5 -2.5(1a)0

2.5 -2.5(1b)0

2.5 -2.5(1c)0

2.5 -2.5

(1d)

0

2.5

(1e)

2.0

600

y1 0
-2.0
2.0

y2 0

400

-2.0
2.0

y3 0

200

-2.0
2.0

y4 0
0
0.00040.00060.00080.001

0.00120.00140.00160.00180.002

0.00220.0024

δ

-2.0
2.0

y5 0
-2.0

Fig. 8.
Average period of the transitions of synchronization states[0, π, π, π, π]. α = 2.0, β = 4.0, and γ = 0.1.

τ

0

1200

(1f)
2.5

this state can be written as [0, π/2, π/2, π/2, π/2]. In thisstate, the solutions on y1 − y2 (or y5 − y1 ) plane move tothe quadrant I, II, III, and IV (or IV, III, II, and I) in thisorder as shown in Fig. 9(1a) (or Fig. 9(1e)). While, Fig. 9(2)shows the state that only the ﬁrst and the ﬁfth circuits haveno phase difference and hence this state can be written as[0, π/2, 0, π/2, 0]. In this state, the solutions on all yi − yi+1
planes except y5 − y1 plane move to the quadrant I, II, III,and IV (or IV, III, II, and I) as shown in Figs. 9(2a)-(2e).
sectionLarge Number of Coupled Circuits
Finally, in this section, some computer simulated results
for the case of 20 circuits are introduced. Figure 10 shows
two examples of the synchronization states obtained bygiving different initial conditions. As we can expect, a huge
number of combinations of phase states can be generated.
Classiﬁcations and stability analysis of these phase states are
our future research.
V. C ONCLUSIONSIn this study, we have investigate the phase patterns characterized by the synchronization states of the interesting statetransitions in a ring of cross-coupled chaotic circuits. We
conﬁrmed that several patterns could be observed by giving
different initial conditions to the circuits. Our future research
includes the investigation of the larger size circuits, theclariﬁcation of the generation mechanism of the transition,and the application to signal processing using the proposed
circuit.
ACKNOWLEDGMENTS
This work was partly supported by Yazaki Memorial
Foundation for Science and Technology.
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3858

0

-2.5
-2.5

0

(2a)

2.5 -2.50(2b)2.5 -2.50(2c)

2.5 -2.50(2d)

2.5 -2.5

0

2.5

(2e)

2.0

y1 0
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2.0

y3 0
-2.0
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y4 0
-2.0
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y5 0
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τ

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1200

(2f)
Fig. 9. Two examples of synchronization states including quadrature-phasesynchronizations (computer calculated result). α = 1.5, β = 5.0, γ = 0.2,
and δ = 0.001. (1) [0,π/2,π/2,π/2,π/2]. (2) [0,π/2,0,π/2,0]. (a) Attractor
on y1 − y2 plane. (b) Attractor on y2 − y3 plane. (c) Attractor on y3 − y4plane. (d) Attractor on y4 − y5 plane. (e) Attractor on y5 − y1 plane. (f)Time waveform.

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2008 International Joint Conference on Neural Networks (IJCNN 2008)

2.0

y1
y2
y3
y4
y5
y6
y7
y8
y9
y10
y11
y12
y13
y14
y15
y16
y17
y18
y19
y20
0

τ

1200

τ

1200

(a)
2.0

y1
y2
y3
y4
y5
y6
y7
y8
y9
y10
y11
y12
y13
y14
y15
y16
y17
y18
y19
y20
0

(b)
Fig. 10. Two examples of synchronization states observed from 20 cross-coupled circuits (computer calculated result). α = 1.5, β = 5.0, γ = 0.2, andδ = 0.001.

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