2012 International Symposium on Nonlinear Theory and its Applications
NOLTA2012, Palma, Majorca, Spain, October 22-26, 2012

Synchronization of Coupled Chaotic Circuits in Regular Tetrahedron Form
Takahiro Nagai, Yoko Uwate and Yoshifumi Nishio
Dept. of Electrical and Electronic Engineering, Tokushima University,
2-1 Minamijosanjima, Tokushima, 770-8506 Japan
Email: {nagataka, uwate, nishio}@ee.tokushima-u.ac.jp
Abstract—This article presents synchronization phenomena of coupled chaotic circuits by regular tetrahedral
form. We pay attentions to the behavior of each chaotic
circuit and observe the synchronization states between adjacent circuits. In this circuit model, we can observe several
patterns of synchronization phenomena.
1. Introduction
There are a lot of synchronization phenomena in naturalenvironment. It is one of the nonlinear phenomena that we
can actually observe. For example, swing of the pendulum,
firefly luminescence, cardiac heartbeat etc. are well known
as the synchronization phenomena. Synchronization phenomena have a feature that the set of small power can produce very big power by synchronizing at a time. Therefore,studies of synchronization phenomena have been widelyreported not only engineering but also the physical and biological fields. Coupled chaotic circuits systems are suitablemodels to express many kinds of high-dimensional nonlinear phenomena. Especially, many researchers are attracted
to chaotic mechanisms because chaotic synchronizations
include complex behaviors [1]-[4].
In our basic investigations, we use several van der Pol oscillators. Van der Pol oscillators have been coupled in various form and have been investigated about their synchro-Figure 2: Chaotic circuit.
nization phenomena. Especially, we can observe very interesting synchronization phenomena when we couple eachinductor and ground by coupling resistor R. For example,
the circuit model which coupled three van der Pol oscillators as a ring topology has been found out the three phasesynchronization (phase shift 120 degrees) by the interaction of each oscillator [5]. However, the three-phase synchronization was always observed stably in that system. In
our previous study, we have investigated synchronization
phenomena of coupled van der Pol oscillators in regulartetrahedral form (see Fig. 1) [6]. We could observe that thephase diﬀerence between adjacent oscillators changed andthat the synchronization was destroyed after the adjacentoscillators synchronize with anti-phase. In other words,
this circuit model has the feature of repeated anti-phasesynchronous and asynchronous states.
In this study, we investigate synchronization phenomena in coupled chaotic circuits with tetrahedron form. We
pay attentions to the behavior of each chaotic circuit andobserve the synchronization states between adjacent circuits. Also, we make comparison between coupled oscillators and coupled chaotic oscillators and confirm the rotation of phase diﬀerences between adjacent chaotic circuits.
Moreover, we research synchronization states by changing
chaotic parameters.
2. Circuit Model

Figure 1: Four coupled oscillators.

We show the chaotic circuits using in this investigationin Fig. 2. This circuit is composed of two types of inductors, a capacitor, a negative resistor and interactive diodes.
We couple each chaotic circuit via L1 and ground by coupling resistor R in this study. We describe this circuit diagram of coupled chaotic circuits in Fig. 3. In the computer- 844 -Figure 4: Attractor of each chaotic circuit (horizontalaxis:xk , vertical axis:yk (k = 1, 2, 3, 4).

Figure 5: Time waveform of the each circuit.
Figure 3: Circuit model for tetrahedron form.
simulations, we assume that the vd − ik characteristics ofthe nonlinear resistor consisting of diodes by the followingfunction.
√vd (ik ) = 9 rd ik(1)

The normalized circuit equations are expressed as: dx}
1{ak
= (xak + xbk + xck + yk ) − zk − ηxak − γ(xak + xn )
dτ3

}
dxbk1{= (xak + xbk + xck + yk ) − zk − ηxbk − γ(xbk + xn )
dτ3

}
 dxck 1 {= (xak + xbk + xck + yk ) − zk − ηxck − γ(xck + xn )
dτ3





dyk


= αβ(xak + xbk + xck + yk ) − zk − f (yk )
dτ




 dzk = (1 + ∆ωk )(xak + xbk + xck + yk ).

(2)dτwhere√√√CCCIk = axk , iak = ayak , ibk = aybk ,
L1
L1L1
√
√
Cyck , vk = azk , t = L1Cτ ,ick = aL1√√
L1CC, γ=R
α= , β=rL2L1L1√ √√8CC, a = rd
η = rmL
L1

elements, we suppose the frequency error ∆ωk and calculate the phase diﬀerences between adjacent chaotic circuits.
3. Synchronization Phenomena
We calculate Eq. (2) using the fourth-order Runge-Kutta
method with the step size h = 0.001. In this simulation, wefix the parameters α = 20.0, η = 0.00001, ∆ω1 =0.0, ∆ω2=0.001, ∆ω3 =0.002, ∆ω4 =0.003, and change the bifurcation parameter or the coupling strength. As a result, wecan find out very interesting phase synchronizations which
could not been observed in four coupled oscillators. For
example, we show the simulation result of the synchronization phenomena in Fig. 4 and Fig. 5. In Fig. 4, weshow the attractor of each chaotic circuit and the horizontal
axis is the electric current, and the vertical axis is the voltage. We set the parameters β = 0.330 and γ = 0.250. In
this case, we can observe complex behaviors of time waveform and phase diﬀerences between adjacent chaotic circuits are shown as Fig. 6. In this figure, first and fourth
circuit, second and third circuit synchronize in-phase andphase diﬀerences between adjacent other circuits are going to approach anti-phase. These synchronization states(k= 1, 2, 3, 4),and√f (yk ) = 9 yk .

(3)

β is bifurcation parameter, γ is coupling strength, η indicates the resistive component of the inductor and yn denotes the current of adjacent oscillator. For computer simulations, in order to consider the diﬀerence of real circuit- 845 -Figure 6: Lissajous figures.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 7: Phase shifts for β = 0.330 and γ = 0.250 (red isphase diﬀerence between 1st-2nd circuit, green is between
1st-3rd, blue is between 1st-4th).
keep changing in-phase and anti-phase by progress of time.
Figure 7 shows the simulation result of long span which
represented change of phase diﬀerences. Here we showthat red dot indicates phase diﬀerence between 1st and 2nd
circuit, green one is between 1st and 3rd, blue one is between 1st and 4th. Even if initial conditions are changed,
these synchronization phenomena do not have the qualitative change and synchronization states of beginning onlychange in-phase or anti-phase.
Next, we present the relationship of coupling strengthand synchronization phenomena. We can see chaotic attractors in the case of week coupling strength and thesechange from chaos to periodic solution by becoming strong
coupling strength. Change of attractors also aﬀects synchronization phenomena between adjacent chaotic circuits.
If the coupling strength become smaller than 0.200, we canobserve asynchronous between all of adjacent circuits.
Also, the four-phase synchronization appears when thecoupling strength γ = 0.280. At this time, phase diﬀerences synchronize in a particular area clearly as compared

Figure 8: Phase shifts for β = 0.330 and γ = 0.280 (red isphase diﬀerence between 1st-2nd circuit, green is between
1st-3rd, blue is between 1st-4th).

Figure 9: First attractor and phase shifts between 1st and2nd circuits. (a) τ p = 3400. (b) τ p = 14000. (c) τ p =120000. (d) τ p = 142000. (e) τ p = 182000. (f) τ p =200000.
to switching of in-phase and anti-phase. Figure 8 showsthat synchronization states between first and the other circuits change by time. Namely, the red point shows the synchronization phenomena between first and second circuits.
This means that two circuits synchronize 270 degrees first,
after that phase diﬀerence shifts anti-phase and 90 degrees.
4. Dependence on Bifurcation Parameter
We can observe the phase shifts which are diﬀerent fromsynchronization phenomena of former section by changing bifurcation parameter. In this section, we fix the coupling strength γ = 0.300 and research the behaviors of synchronization states for each circuit. When β is same value
and the coupling strength diﬀers, the chaotic attractors do
not occur but the periodic attractors occur. For example,
one-periodic attractor can be described when the parameter is β = 0.280. Synchronization phenomena between
adjacent circuits can be distinguished one anti-phase synchronization and two asynchronous. Also, we can observe
switching of three diﬀerent phase shift when β = 0.290(see Fig. 10(a)). This means that red point describes phaseshift from 180 to 330 degrees, green one is from 30 to
180 degrees and blue one is switching of anti-phase andasynchronous. The other case, we can find out the phaseshift which occurs alternately anti-phase synchronization
and asynchronous in β = 0.305 (see Fig. 10(b)). After that,four-phase synchronization occurs (see Fig. 10(c)).
Next, the chaotic parameter is over β = 0.335, the attractors shift from one-periodic solution to two-periodic solution. Therefore we can see two types of phase diﬀerences
over one synchronization phenomena between adjacent circuits. These phase diﬀerences appear alternately and keep- 846 -Table 1: Change of synchronization states by β.
βSynchronization States∼ 0.289
Anti-phase and asynchronous
0.289 ∼ 0.300
Switching of diﬀerent phase shift
0.300 ∼ 0.305
Anti-phase and asynchronous
0.305 ∼ 0.316 Switching of anti and asynchronous
0.316 ∼ 0.335
Four-phase synchronization0.335 ∼ 0.336
Phase synchronization
0.336 ∼ 0.345 Switching of in-phase and anti-phase
0.345 ∼Divergence(a)synchronizing while maintaining the states. Also, asynchronous states are observed in whole chaotic networks before two-periodic solution appear. The time which shifts
from asynchronous until periodic solution depends on theinitial conditions. Two-periodic solution can be seen until β= 0.336, the attractors change chaos (see Fig. 10(d)). Then,
we can see week switching of in-phase and anti-phase synchronization without phase synchronization.

(b)

5. Conclusion
This article presents synchronization phenomena which
coupled chaotic circuits by regular tetrahedral form. In thiscircuit model, we have been able to observe several patterns synchronization phenomena. The observed switchingin the former study about four coupled van der Pol oscillators sojourned only anti-phase area but we could confirm
sojourn of in-phase area in this coupled chaotic circuits.
Also, the four-phase synchronization are observed newly
and we have found the feather which four-phase synchronization keeps shifting by 90 degrees depending on the bifurcation parameter. In the whole of coupled circuits, ifbifurcation parameter is too strong or coupled strength is
too week, this system is always asynchronous.

(c)

Acknowledgments
This work was partly supported by JSPS Grant-in-Aidfor Young Scientists 23700269.
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(d)Figure 10: The phase shifts (red is phase diﬀerence between 1st-2nd circuit, green is between 1st-3rd, blue is between 1st-4th). (a) β = 0.295. (b) β = 0.305. (c) β =0.330. (d) β = 0.336.

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