2013 International Workshop on Nonlinear Circuits,Communications and Signal Processing
NCSP'13, Isrand of Hawaii, Hawaii, USA. March 4-7, 2013

Detailed Investigation of Synchronizations in Three Cross-Coupled Chaotic Circuits
Naoto Kageyama, Yoko Uwate, Yoshifumi Nishio
Tokushima University
2-1 Minami-Josanjima, Tokushima, JAPAN
Phone/FAX:+81-88-656-7470
E-mail: {n-kage, uwate, nishio}@ee.tokushima-u.ac.jp

Abstract
Studies on chaos synchronization in coupled chaotic circuits
are extensively carried out in various fields. In this study,
we investigate synchronization phenomena observed in three
simple chaotic circuits cross-coupled by three inductors. Interesting synchronization phenomena can be confirmed bycomputer simulations and circuit experiments.

1. Introduction
Synchronization phenomena in complex systems are very
good models to describe various higher-dimensional nonlinFigure 1: Circuit model.
ear phenomena in the field of natural science. Studies on synchronization phenomena of coupled chaotic circuits are extensively carried out in various fields [1]- [5]. We consider
that it is very important to investigate the phenomena related segment piecewise-linear functions.
with chaos synchronization to realize future engineering ap
plication utilizing chaos. In our previous study, we have inG(v1n − v2n − V ) (v1n − v2n > V )
vestigated the cross-coupled chaotic circuits and reported var0(|v1n − v2n | ≤ V )idn =ious interesting synchronization phenomena [6]- [9].
In this study, we consider three Shinriki-Mori chaotic cirG(v1n − v2n + V ) (v1n − v2n < −V )cuits [9] cross-coupled via three inductors. The circuit is aspecial version of the ring considered in [7], but we have not(n = 1, 2, 3)
investigated the phenomena in detail because we did not noThe circuit equations are described as follows.
tice the phase differences of small oscillations until we noticed it in 2009 [8]. By computer simulations and circuit ex
di1nperiments, we investigate the synchronization phenomena in= v2nL1dt
detail, in particular, three-phase synchronizations of small os
cillations are confirmed to be generated.
dv1n
 C1 dt = gv1n − i2n − idndv2nC2= idn + i2,n+1 − i1n

dt



 L di2n = v − v
21n2,n−1 ,dt

2. Circuit Model

(1)

(2)

Figure 1 shows the circuit model. In the circuit, three
Shinriki-Mori chaotic circuits are cross-coupled via inductors
L2 .
(n = 1, 2, 3)First, we approximate the v − i characteristics of the nonlinear resistors consisting of the diodes by the following 3- where v20 = v23 and v24 = v21 . By using the following- 628 -parameters√√ α = C2 , β = L1 G, γ = L1 g, δ = L1 ,C1C2C2L2√
t = L1 C2 τ,(3)and variables
√√√C2C2C2Vx,i=Vx,i=V x3 ,i=
11221311


LL
L1

11


 v11 = V y1 , v12 = V y2 , v13 = V y3 ,v21 = V z1 , v22 = V z2 , v23 = V z3 ,


√√√


C2C2C2
 i21 =V ω1 , i22 =V ω2 , i23 =V ω3 ,L1L1L1(4)the normalized circuit equations are given as follows.
x˙n = zn y˙n = α(γyn − ωn − βf (yn − zn )(5)z˙n = βf (yn − zn ) + ωn+1 − xnω˙n = δ(yn − zn+1 )(a)(n = 1, 2, 3)(b)where z20 = z23 and z24 = z21 . The nonlinear function f (·)
corresponds to the v − i characteristics of the nonlinear resistors consisting of the diodes and are assumed to be describedby the following 3-segment piecewise-linear functions:(yn − zn > V ) yn − zn − 10(|yn − zn | ≤ V )f (yn − zn ) =(6)yn − zn + 1(yn − zn < −V )Figure 2: Typical examples of observed synchronization phenomena for α = 2.0, β = 4.0, γ = 0.15 and δ = 0.004. Upper figures
show the attractor on v11 −v12 phase plane, on v11 −v13 phase planeand on v12 − v13 phase plane from the left. Lower figure shows thetime waveforms of v11 . v12 and v13 .

(n = 1, 2, 3)3. Synchronization Phenomenon
The coupled circuits generate various synchronization
states. Two examples of steady states are shown in Fig. 2.
These two synchronization states are obtained for the same
parameter set of α = 2.0, β = 4.0, γ = 0.15 and δ = 0.004
however with different initial conditions. In Fig. 2(a), two circuits are completely synchronized in in-phase and the othercircuit exhibits some phase difference (time waveform showsthat the switching from plus to minus or from minus to plus
delays 90 degrees from the other circuits. On the other hand,in Fig. 2(b), all three circuits seem to be synchronized in inphase from the time waveforms, however, the attractors do
not show in-phase synchronizations.
In order to investigate how many steady states coexist andwhy the shape of attractors in Fig. 2(b) does not show inphase, we investigate the time waveforms of the voltages

more carefully, because we realized in 2009 that small oscillations could synchronize when two circuits are crosscoupled [13]. Figure 3 shows closer looks of the time waveforms of the three voltages. Figures 3(a) and 3(b) correspondto the steady states in Figs. 2(a) and 2(b), respectively. Because v11 and v12 are completely synchronized in in-phasein Fig. 2(a), we can see only two waveforms in Fig. 3(a).
From this figure we can notice that the small oscillation ofv13 is synchronized in anti-phase with the small oscillationsof v11 and v12 . Although all three waveforms looked like tobe synchronized in in-phase in Fig. 2(b), the closer look in
Fig. 3(b) revealed that the small oscillations of these threevoltages are synchronized clearly in 3-phase. This is why theattractors in Fig. 2(b) are not on the in-phase synchronization
space completely. Also we can understand why the shapesof the attractors show some phase differences in the first andthe third quadrants of the phase space in Fig. 2(b). This factencouraged us to search many more coexisting steady states,
because we can expect a large number of steady states withlocal 3-phase of small oscillations. As we expected, we could- 629 -

v13

v12
v11(a)
(a)

v13

v11

v12(b)
(b)

v13

v12
v11(c)(c)Figure 3: Magnified time waveforms of three voltages obtained
from three circuits. (a) and (b) correspond to Fig. 2(a) and (b), respectively.

Figure 4: Attractors in 3-dimensional phase space (v11 −v12 −v13 ).
(a), (b) and (c) correspond to Fig. 3(a), (b) and (c), respectively.

journ time in the plus or the minus regions becomes shorter.
find many different steady states and one example is shown in
Fig. 3(c). Combinations of the timing shift of the switchingfrom minus to plus or from plus to minus with local 3-phaseof small oscillations make it possible to generate a large number of steady states.
The amount of the timing shift of the switching influences
the shape of the attractors. Figure 4 shows the attractors in
3-dimensional phase space. Attractors in Figs. 4(a), (b) and(c) correspond to the time waveforms in Figs. 3(a), (b) and(c), respectively. We can see that the sojourn time of the solution in the plus or the minus region decides the shape of theattractors.

5. Circuit Experiments
Finally, circuit experimental results are shown in Fig. 6.
We can say that this interesting phenomenon can be observed
from both computer calculations and circuit experiments.
6. Conclusions

In this study, we have investigated the synchronization phenomena observed from three simple chaotic circuits crosscoupled by inductors. The detailed investigation of time
waveforms clarified several interesting phenomena.
Investigating the coexistence of the states and statistical
4. Parameter Dependency
analysis of the observed phenomena are our important future
Figure 5 shows the parameter dependency of the observed work as well as more detailed explanation of the mechanism
phenomena. As increasing the coupling parameter δ, the so- of the generations.

- 630 -(a)

(a)
(b)

(c)
Figure 6: Circuit experimental results. From the left: v11 −v12 , v11 − v13 and v12 − v13 . C1 =22nF, C2 =47nF, L2 =500mH,g=1.78mS. (a) L1 =0.9mH. (b) L1 =1.35mH. (c) L1 =1.8mH.

(b)

(c)
Figure 5: Parameter dependency of synchronization. (a) δ=0.002.
(b) δ=0.003. (c) δ=0.005.

Acknowledgment
This work was partly supported by JSPS Grant-in-Aid for
Scientific Research 22500203.
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