2013 International Symposium on Nonlinear Theory and its Applications
NOLTA2013, Santa Fe, USA, September 8-11, 2013

Investigation of Synchronizations in Five Cross-Coupled Chaotic Circuits
Naoto Kageyama, Yoko Uwate and Yoshifumi Nishio†Dept. of Electrical and Electronic Eng., Tokushima University
2-1 Minami-Josanjima, Tokushima, 770-8506 JAPAN
Email : {n-kage, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract—Studies on chaos synchronization in coupledchaotic circuits are extensively carried out in various fields.
In this study, we investigate synchronization phenomena
observed in five simple chaotic circuits cross-coupled by
five inductors. Interesting synchronization phenomena canbe confirmed by computer simulations.
1. Introduction
Synchronization phenomena in coupled oscillatory systems are very good models to describe various higherdimensional nonlinear phenomena in the field of naturalscience. Studies on synchronization phenomena of coupled chaotic circuits are extensively carried out in various fields [1]-[10]. We consider that it is very important
to investigate the phenomena related with chaos synchronization to realize future engineering application utilizing
chaos. In our previous study, we have investigated thecross-coupled chaotic circuits and reported various interesting synchronization phenomena [11]-[13]. Additionally,in Figs. 1(a) and (b), we have investigated the three crosscoupled chaotic circuits and considered the small oscillations of three voltages are synchronized clearly in threephase. We can expect a large number of steady states withlocal three-phase of small oscillations. As we expected, we
could find many diﬀerent steady states[14].
We extend the previous study, in this study we consider
five Shinriki-Mori chaotic circuits [15] cross-coupled via
five inductors. The circuit is a special version of the ringconsidered in [12], however we have not investigated thephenomena in detail because we did not notice the phase
diﬀerences of small oscillations until we noticed it in 2009[13]. By computer simulations, we investigate the synchronization phenomena in detail, in particular, five-phase synchronizations of small oscillations are confirmed to be generated.

(a)

(b)

2. Circuit Model
Figure 1:
Figure 2 shows the circuit model. In the circuit, five
Shinriki-Mori chaotic circuits are cross-coupled via inductors L2 .
First, we approximate the v − i characteristics of the nonlinear resistors consisting of the diodes by the following
3-segment piecewise-linear functions.

Three cross-coupled system.(a) Circuit model.
(b)Upper figures show the attractor on y1 − y2 phase plane, on
y1 −y3 phase plane and on y2 −y3 phase plane from the left. Lower
figure shows the time waveforms of y1 . y2 and y3 .

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(v1n − v2n > V)
-gi11

i15

L1

v11v21

v25
i25

id5

c2

L2

c1

i21

L2v 12(n = 1, 2, 3, 4, 5)

c2

c1i d2v15v 22i22-gi 12

L2i244i23v2i d3v 234idi 13
L1c24v1c1-g

where z20 = z25 and z26 = z21 . The nonlinear functionf (·) corresponds to the v − i characteristics of the nonlinear resistors consisting of the diodes and are assumed to
be described by the following 3-segment piecewise-linear
functions:yn − zn − 1 (yn − zn > V)0(|yn − zn | ≤ V)
(6)
f (yn − zn ) =  y − z + 1 (y − z < −V)
nnnnc21LL2c1i14(n = 1, 2, 3, 4, 5)v 13and variables
√
√
√


C2C2C2
i11 =V x1 , i12 =V x2 , i13 =V x3 ,


LL
L111






v = Vy1 , v12 = Vy2 , v13 = Vy3 ,


 11



v21 = Vz1 , v22 = Vz2 , v23 = Vz3 ,






√
√
√



C2C2C2Vω1 , i22 =Vω2 , i23 =Vω3 , i21 =L1L1L1(4)the normalized circuit equations are given as follows.
x˙n = zn
 y˙n = α(γyn − ωn − β f (yn − zn )
(5)



z˙
=βf(y−z)+ω−x
nnnn−1n ω̇ = δ(y − z )nnn+1L1L2(2)where v20 = v25 and v26 = v21 . By using the following
parameters√√
C2L1L1L1 α = C , β = C G, γ = C g, δ = L ,(3)
1222


 t = √L1C2 τ,(n = 1, 2, 3, 4, 5).

L1id1(v1n − v2n < −V)

The circuit equations are described as follows.
di1n

= v2nL1



dt





dv1n



C1= gv1n − i2n − idn

dt




dv2n



C2= idn + i2,n−1 − i1n


dt





di

 L2 2n = v1n − v2,n+1 ,dt(n = 1, 2, 3, 4, 5)

c2c1(1)-g(|v1n − v2n | ≤ V)-g


G(v1n − v2n − V)0idn = 
 G(v − v + V)
1n2nFigure 2: Circuit model.

3. Equal Interval Phase Diﬀerences
The proposed circuit model generates various synchronization states. Figure 3 shows some examples of attractors and magnify the time waveforms of the five voltages.
These synchronization states are obtained for the same parameter set of α = 2.0, β = 4.0, γ = 0.15 and δ = 0.004 butwith diﬀerent initial conditions.
In Fig. 3(a), all five circuits seem to be synchronized inin-phase from the time waveforms, however, the attractors
do not show in-phase synchronizations. The small oscillations of these five voltages are synchronized clearly in
five-phase. This local five-phase synchronization is thereason of that the attractors in Fig. 3(a) do not show thein-phase synchronization completely in the phase space.
Also we can understand why the shapes of the attractors
show some phase diﬀerences in the first and the third quadrants of the phase space in Fig. 3(a). This result encouraged us to search many more coexisting steady states. As
we expected, we could find many diﬀerent steady statesand some examples are shown in Figs. 3(b) and 3(c). Thesynchronization state depends on the initial conditions. InFigs. 3(b) and 3(c), the local five-phase state of the small
oscillations varies by setting the diﬀerence initial conditions and the phase diﬀerence of these local five-phase of
the small oscillations is shifted with the equal interval. Table 1 shows the phase diﬀerences of the voltage waveforms
in Figs. 3.
We confirm combinations of the timing shift of theswitching from minus to plus or from plus to minus withlocal five-phase of small oscillations make it possible to- 163 -generate a large number of steady states.
Table 1: Phase diﬀerence of the voltages in Fig. 3.
Phase diﬀerence

(a)
3.57◦

(b)
11.21◦(c)
27.57◦

4. Unequal Interval Phase Diﬀerences
Figures 4 shows some examples of attractors and magnify the time waveforms of the five voltages. These synchronization states are obtained for the same parameter setof α = 2.0, β = 4.0, γ = 0.15 and δ = 0.004 but with
diﬀerent initial conditions.
The amount of the timing shift of the switching fromminus to plus or from plus to minus with local five-phase ofsmall oscillations influences the shape of the attractors andsynchronization states. In Fig. 3(a), the small oscillationsof these five voltages are synchronized clearly in five-phaseand in Figs. 3(b), (c), local five-phase of small oscillations
rise up at equal interval. By contrast, in Fig. 4, each local
small oscillations rise up at unequal interval.
These result we can expect a large number of steadystates with local five-phase of small oscillations.

(a)5. Conclutions
In this study, we have investigated the synchronization phenomena observed from five simple chaotic circuits
cross-coupled by inductors. The detailed investigation oftime waveforms clarified several interesting phenomena.
Investigating the coexistence of the states and statisticalanalysis of the observed phenomena are our important future work as well as more detailed explanation of the mechanism of the generations.

(b)

Acknowledgment
This work was partly supported by JSPS Grant-in-Aidfor Young Scientists 23700269.
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(c)
Figure 3: State transition phenomena around for constant phase
diﬀerence. Upper figures show the attractor on y1 − y2 , y1 − y3 ,y1 −y4 , y1 −y5 and y2 −y3 phase plane from the left. Middle figures
show the attractor on y2 − y4 , y2 − y5 , y3 − y4 , y3 − y5 and y4 − y5phase plane from the left. Lower figure shows the magnified time
waveforms of y1 , y2 , y3 , y4 and y5 .

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(c)Figure 4: State transition phenomena around for indeterminate
phase diﬀerence. Upper figures show the attractor on y1 − y2 ,y1 − y3 , y1 − y4 , y1 − y5 and y2 − y3 phase plane from the left.
Middle figures show the attractor on y2 − y4 , y2 − y5 , y3 − y4 ,y3 − y5 and y4 − y5 phase plane from the left. Lower figure shows
the magnified time waveforms of y1 , y2 , y3 , y4 and y5 .

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