2004 International Symposium on Nonlinear
Theory and its Applications (NOLTA2004)
Fukuoka, Japan, Nov. 29 - Dec. 3, 2004

Complex Behavior in Coupled Chaotic Circuits
Related with Intermittency
Yoko Uwate and Yoshifumi Nishio
Department of Electrical and Electronic Engineering
Tokushima University
2-1 Minami-Josanjima, Tokushima 770-8506, Japan
Email: {uwate, nishio}@ee.tokushima-u.ac.jp
Abstract—In this study, a complex behavior intwo coupled chaotic circuits related with intermittency
is investigated. When each chaotic circuit generates
intermittency chaos near the three-periodic window,three different synchronization states appear and disappear in a chaotic manner. This complex behavior ismodeled by a first-order Markov chain with four states.
Further, the phenomenon is confirmed by circuit experiments.
1. Introduction
Synchronization and the related bifurcation of
chaotic systems are good models to describe various
higher-dimensional nonlinear phenomena in the field ofnatural science. In particular, the breakdown of chaos
synchronization has attracted many researchers’ attentions and their mechanisms have been gradually madeclear [1]-[5]. However, a lot of phenomena aroundchaos synchronization are still veiled as well as othernonlinear problems. Hence, in order to understandand exploit such phenomena, it is important to discover them, to model them, and to investigate them.
On the other hand, intermittency chaos [6] is deeply
related to the edge of chaos [7] and many people suggest that such a behavior between order and chaos
gains better performance for various kinds of information processing than fully developed chaos. Therefore,
we consider that unveiling various roles of the intermittency chaos is important to exploit it for futureengineering applications.
In this study, a complex behavior in two coupledchaotic circuits related with intermittency is investigated. At first, we observe three different types ofsynchronization states when each chaotic circuit generates three-periodic solution. Next, we vary a control
parameter of each chaotic circuit to generate intermittency chaos near the three-periodic window. In thatcase, we can observe a complex behavior of the threesynchronization states. Namely, intermittency bursts
interrupt the synchronizations and different synchronizations reappear after the bursts settle down. We

model this interesting complex behavior by a firstorder Markov chain with four states. Transition probabilities between the states are obtained by counting
all of the transitions in plenty of computer simulation. The stationary probability distribution and theexpected sojourn time in each state are calculated fromthe transition probabilities. These statistical quantities are compared with those obtained from computer
simulations. We emphasize that the synchronizationsof the three-periodic solutions and those complex behavior caused by the intermittency are observed fromboth computer calculations and circuit experiments.
2. Circuit Model
Figure 1 shows the circuit model, which is the asymmetric version of the circuit investigated in [8]. In thecircuit, two identical chaotic circuits are coupled by aresistor R.
LIIL


iCiL





C



L


R-r-r

Figure 1: Circuit model.
At first, the i−v characteristics of the diodes are approximated by two-segment piecewise-linear functions
as1(1)
vd (ik ) = (rd ik + E − |rd ik − E|).
2
By changing the variables and parameters,rrCC
Eyk , vk = Ezk ,Exk , ik =Ik =L1
L1rpC
L1,(2), β=rt = L1 Cτ , α =L1L2589

r
γ=RC,L1r
δ = rdC,L1

x1

the normalized circuit equations are given as
2X
dxk


=β(x+y)−z−γxjkkk

dτ

j=1

dyk
= αβ(xk + yk ) − zk − f (yk )
dτ



 dzk = x + ykkdτ

(a)x2t(k = 1, 2)

x1(b)

(3)

x2twheref (yk ) = 0.5 (δyk + 1 − |δyk − 1|).

(4)

x1(c)

3. Synchronization Phenomena
Figure 2 shows three-periodic attractor observed
from the isolated subcircuit. For the computer calculations, the fourth-order Runge-Kutta method is usedwith step size h = 0.001.
vkZ k 1.5x2tFigure 3: Time waveforms of three synchronization
states (computer calculated results). α = 7.0, β =0.152, γ = 0.005 and δ = 100.0. (a) State T1 , (b)State T2 , and (c) State T3 .

0Next, we vary a control parameter of each chaotic
circuit to generate intermittency chaos near the threeperiodic window as shown in Fig. 5.

-1.5-1.50(a)1.5xkIk(b)

Figure 2: Three-periodic attractor observed form each
subcircuit. (a) Computer calculated result. xk vs. zk .
α = 7.0, β = 0.152, γ = 0.0 and δ = 100.0. (b) Circuit
experimental result. Ik vs. vk . L1 = 300mH, L2 =10mH, C = 33nF, r = 740Ω and R = 0.0Ω.
Figure 3 shows that three different types of synchronization states, when the two circuits generating thethree-periodic attractors are coupled. These three synchronization states can be obtained by giving different initial conditions. As we can see from the figures,the two circuits tend to be synchronized in anti-phase.
This is because the states minimizing the energy consumed by the coupling resistor R correspond to stable synchronization states. For three-periodic solutions there exist three different peaks in the waveform.
Hence, three different synchronization states could coexist as shown in Fig. 3. We name the three synchronization states as the states T1 , T2 , and T3 .
We also confirm the generation of the three different
synchronization states in circuit experiments as shownin Fig. 4.

If we couple the two chaotic circuits when the intermittency chaos appear, we can observe a complex
behavior of the three synchronization states. Namely,intermittency bursts disturb the synchronizations anddifferent synchronizations appear and disappear in achaotic way.
In order to investigate the complex phenomenon, we
define the Poincaré section as z1 = 0 and x1 < 0. Further we plot the discrete data of x2 on the Poincaré
map when x1 is smaller than −1.2. This threshold isintroduced to extract only the data when x1 takes thelargest peak (dots in the waveform in Fig. 3). Figure 6
shows the discrete data of x2 obtained by the abovementioned method. We can see that the synchronization states are interrupted by the intermittent bursts
and different synchronization states reappear after thebursts settle down. Although the results can not beshown in the same manner, we also confirmed the same
phenomenon in the circuit experiments. The changingof the synchronization states can be shown in a pictureas Fig. 7.

590vkZ k 1.5

I1(a)
0I2

t-1.50-1.5

I1Ikxk(b)Figure 5: Intermittency chaos near the three-periodic
window. (a) Computer calculated result. xk vs. zk .
α = 7.0, β = 0.133682, γ = 0.0 and δ = 100.0. (b) Circuit experimental result. Ik vs. vk . L1 = 300mH,
L2 = 10mH, C = 33nF, r = 735Ω and R = 0.0Ω.

(b)I2t  I1(c)    I2t
Figure 4: Time waveforms of three synchronization
states (circuit experimental results). L1 = 300mH,
L2 = 10mH, C = 33nF, r = 740Ω and R = 40.0Ω. (a)State T1 , (b) State T2 , and (c) State T3 .
4. Markov Chain ModelingIn this section we model the interesting complex behavior in the last section by a first-order Markov chain
with four states. The proposed Markov chain is shownin Fig. 8 where T1 . T2 , and T3 are the three synchronization states introduced in the last section, B is theburst state appearing in the transition between thethree. Further,P (B|B) = 1 −3X

P (B|Ti ).

0P (T2 |T2 )0
1 − P (T2 |T2 )00P (T3 |T3 )1 − P (T3 |T3 )(7)

can be calculated from the following equations
Q = PQ and3X
i=1Q(Ti ) + Q(B) = 1.

=∞
X

(8)



(9)

{n × PST (Q(T1 ), n)}

n=1∞
X

=

(1 − P (T1 |T1 ))=n=1
1.
1 − P (T1 |T1 ).

(6)
The stationary probability distribution describing
probability of the solution being in each phase state;

From (9) the expected sojourn time in Q(T1 ) is calculated asEST (Q(T1 ))Q = [Q(T1 ), Q(T2 ), Q(T3 ), Q(B)]TPST (Q(T1 )), n) = P (T1 |T1 )n−1 (1 − P (T1 |T1 )).

(5)

P (B|T1 )P (B|T2 )P (B|T3 )P (B|B)

It is also possible to estimate expected sojourn timein each phase state by using the transition probabilities. For example, probability density function of thesojourn time in Q(T1 ) is given bymust be satisfied.
From the state-transition diagram in Fig. 8, we canobtain the transition probability matrix P asP (T1 |T1 )00
1 − P (T1 |T1 )Figure 6: Time series of synchronization states disturbed by intermittency chaos (computer calculatedre sults). α = 7.0, β = 0.133682, γ = 0.005 andδ = 100.0.

i=1

1.5(a)nP (T1 |T1 )n−1(10)5. Results and Discussions
Because we cannot get the transition probabilities
theoretically, we obtain the data by counting all of
the transitions in plenty of computer simulations. For
the parameter values in Fig. 6, we obtained those as
P (T1 |T1 ) = 0.91641, P (T2 |T2 ) = 0.33261, P (T3 |T3 ) =0.79080, P (TB |TB ) = 0.84255, P (TB |T1 ) = 0.03922,P (TB |T2 ) = 0.05680, and P (TB |T3 ) = 0.06142. Byusing these probabilities, we calculated the stationary591

Table 2: Expected sojourn times EST .

I1

State
EST (Q(T1 ))EST (Q(T2 ))EST (Q(T3 ))EST (Q(B))

I2t
Figure 7: Time series of synchronization states disturbed by intermittency chaos (circuit experimental results). α = 7.0, β = 0.133682, γ = 0.0 and δ = 100.0.
1-P(T1 |T1 )1-P(T2 |T2 )Simulation
11.9632
1.49844.7801
6.3391

Markov
11.9198
1.49844.75006.3490mittency bursts interrupted the synchronizations anddifferent synchronizations reappeared after the bursts
settle down. This complex behavior was modeled by afirst-order Markov chain with four states. Further, thephenomenon was confirmed by circuit experiments.

1-P(T3 |T3 )P(T3 |T3 )P(T2 |T2 )P(T1 |T1 )T1

T2

Acknowledgments

P(B |B )B

T3

This work was partly supported by Kayamori Foundation of Informational Science Advancement’s Research Grant.

P(B |T3 )P(B |T2 )P(B |T1 )References[1] N. Platt, E.A. Spiegel and C. Tresser, ”On-Off Intermittency: A Mechanism for Bursting,” Phys.
Rev. Lett., vol. 70, no. 3, pp. 279-282, 1993

Figure 8: Markov chain model.
probability distribution and the expected sojourn timein each state.
The results are summarized in Tables 1 and 2. In thetables, “Simulation” is the data obtained by counting
the states in computer simulations and “Markov” isthe results calculated by using Eqs. (6)-(10).
We can see the results obtained from Markov chain
agree very well with computer simulated results.
6. Conclusions
In this study, we have investigated a complex behavior in two coupled chaotic circuits related with intermittency. When each chaotic circuit generated threeperiodic solution, three different types of synchronization states were observed. However, if a control parameter of each chaotic circuit was varied to generate intermittency chaos near the three-periodic window, inter-Table 1: Stationary Probability Q.
State
Q(T1 )
Q(T2 )
Q(T3 )
Q(B)

Simulation
0.25470.04600.15870.5406

Markov0.25397
0.04606
0.15888
0.54116[2] E. Ott and J.C. Sommerer, ”Blowout Bifurcations:
the Occurrence of Riddled Basins and On-Off Intermittency,” Phys. Lett., vol. A188, pp. 39-47,
1994.
[3] P. Ashwin, J. Buescu and I. Stewart, ”Bubbling ofAttractors and Synchronisation of Chaotic Oscillators,” Phys. Lett., vol. A193, pp. 126-139, 1994.
[4] T. Kapitaniak and L.O. Chua, ”LocallyIntermingled Basins of Attraction in Coupled
Chua’s Circuits,” Int. J. Bifurcation and Chaos,vol. 6, no. 2, pp. 357-366, 1996.
[5] M. Wada, Y. Nishio and A. Ushida, ”Analysis of
Bifurcation Phenomena on Two Chaotic Circuits
Coupled by an Inductor,” IEICE Trans. Fundamentals, vol. E80-A, no. 5, pp. 869-875, May 1997.
[6] Y. Pomeau and P. Manneville, “Intermittent Transition to Turbulence in Dissipative Dynamical Systems,” Comm. Math. Phys., vol. 74, pp. 189-197,
1980.
[7] C.G. Langton, “Computation at the Edge ofChaos: Phase Transitions and Emergent Computation,” Physica D, vol. 42, pp. 12-37, 1990.
[8] Y. Nishio, K. Suzuki, S. Mori and A. Ushida, ”Synchronization in Mutually Coupled Chaotic Circuits,” Proc. ECCTD’93, vol. 1, pp. 637-642, Aug.
1993.

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