Complex Behavior in a Ring of 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{uwate, nishio}@ee.tokushima-u.ac.jp
ABSTRACT: In this study, a complex behavior in a ring of chaotic circuits related with intermittency is investigated. When each chaotic circuit
generates three-periodic solution, various different types of synchronization states are observed. However, if a control parameter of each chaoticcircuit is varied to generate intermittency chaos near the three-periodic window, intermittency bursts interrupt the synchronizations and different synchronizations reappear after the bursts settle down.

1. Introduction
Synchronization and the related bifurcation of chaotic systems are good models to describe
various higher-dimensional nonlinear phenomena in the field of natural science. In particular,
the breakdown of chaos synchronization has attracted many researchers’ attentions and theirmechanisms have been gradually made clear [1]-[5]. However, a lot of phenomena aroundchaos synchronization are still veiled as well as other nonlinear problems. Hence, in order tounderstand and exploit such phenomena, it is important to discover them, to model them, andto investigate them.
On the other hand, intermittency chaos [6] is deeply related to the edge of chaos [7] andmany people suggest that such a behavior between order and chaos gains better performance forvarious kinds of information processing than fully developed chaos. Therefore, we consider thatunveiling various roles of the intermittency chaos is important to exploit it for future engineering
applications.
In this study, a complex behavior in a ring of chaotic circuits related with intermittency isinvestigated. At first, we analyze behavior in a basic system of two coupled chaotic circuits.
Next, we observe more complex behavior when the two coupled chaotic circuits are expanded
to a ring of chaotic circuits. In that case, we observe various different types of synchronization
states when each chaotic circuit generates three-periodic solution. And, we vary a control parameter of each chaotic circuit to generate intermittency chaos near the three-periodic window.
So, we can observe a complex behavior of the various synchronization states. Namely, intermittency bursts interrupt the synchronizations and different synchronizations reappear after thebursts settle down.

2. Basic Coupled Circuit
Figure 1 shows the basic coupled circuit. Each subcircuit is three-dimensional autonous one andconsists of three memory elements, one linear negative resistor and one diode. We can regard

the diodes as pure resistive elements, because operation frequency is not too high. Figure 2
shows three-periodic attractor observed from each subcurcuit.
LIILiCiL







LC



R-r-r

Figure 1: Basic coupled circuit.
vkZ k 1.5

0-1.5
-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(1) shows that three different types of synchronization states, when the two circuits
generating the three-periodic attractors are coupled. These three synchronization states canbe obtained by giving different initial conditions. As we can see from the figures, the twocircuits 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 threeperiodic solutions there exist three different peaks in the waveform. Hence, three different
synchronization states could coexist as shown in Fig. 3(1).
We also confirm the generation of the three different synchronization states in circuit experiments as shown in Fig. 3(2).
Next, we vary a control parameter of each subcurcuits to generate intermittency chaos near
the three-periodic window as shown in Fig. 4.
If we couple the two chaotic circuits when the intermittency chaos appear, we can observe acomplex behavior of the three synchronization states. Namely, intermittency bursts disturb thesynchronizations and different synchronizations appear and disappear in a chaotic way.
In order to investigate the complex phenomenon, we define the Poincaré section as z1 = 0and x1 < 0. The data of x1 on the Poincaré map is defined as x̂1 . Further we plot the discrete
data of x̂2 on the Poincaré map when x̂1 is smaller than −1.2. This threshold is introduced toextract only the data when x̂1 takes the largest peak (bullets in the waveform in Fig. 3). Figure 5
shows the discrete data of x̂2 obtained by the above-mentioned method. We can see that thesynchronization states are interrupted by the intermittent bursts and different synchronization
states reappear after the bursts settle down. Although the results can not be shown in the same

x1

x1

x1x2x2

(1)
x2
ttt

I1

I1

I2

I2

I1(2)
I2(2)

ttt

(a)

(b)

(c)

Figure 3: Time waveforms of three synchronization states. (1) shows Computer calculated
results. α = 7.0, β = 0.152, γ = 0.005 and δ = 100.0. (2) shows Circuit experimental results.
L1 = 300mH, L2 = 10mH, C = 33nF, r = 740Ω and R = 40.0Ω.
vk

Z k 1.5

0-1.5
0-1.5

1.5

Ikxk

(a)

(b)

Figure 4: 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. Ikvs. vk . L1 = 300mH, L2 = 10mH, C = 33nF, r = 735Ω and R = 0.0Ω.
manner, we also confirmed the same phenomenon in the circuit experiments. The changing ofthe synchronization states can be shown in a picture as Fig. 6.
1.2
1.1
1
0.9
0.8
x2 0.7
0.6
0.5
0.4
0.3
0.2

0200

400

600

800t

1000

1200

1400

Figure 5: Time series of synchronization states disturbed by intermittency chaos (computercalculated results). α = 7.0, β = 0.133682, γ = 0.005 and δ = 100.0.

3. Ring of Chaotic Circuits
In this study, we consider a ring of the circuits as shown in Fig. 7. In the circuit adjacent two
subcircuits are coupled by one resistor R. Because such coupling systems tend to minimize theenergy consumed by the coupling resistors, every two adjacent subcircuits tend to synchronize
with anti-phase.

I1

I2

tFigure 6: 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.
2L I  I  2L

2L I I 2Li i CCL

RR-r

L

-r

Figure 7: Ring of Chaotic Circuits.
At first, thei − v characteristics of the diodes are approximated by two-segment piecewiselinear functions as
1(1)
vd (ik ) = (rd ik + E − |rd id − E|).
2
By changing the variables and parameters,
rrr
CCC
Eyk , vk = Ezk ,ExLk , ik =ExRk , ILk =IRk =L1L1L1(2)rrrpCCCL1,, δ = rd, γ=R, β=rt = L1 Cτ , α =L1L1L1L2

the normalized circuit equations are given as
1dxRk


= {β(xRk + xLk + yk ) − zk − γ(xRk + xL(k+1) )}


2dτ



1dxLk


 dτ = 2 {β(xRk + xLk + yk ) − zk − γ(xLk + xR(k+1) )}(k = 1, 2, 3, ..., N ))dyk
= α{β(xRk + xLk + yk ) − zk − f (yk )}dτ dzk = xRk + xLk + ykdτ(3)

wheref (yk ) =1(δyk + 1 − |δyk − 1|).
2(4)andxLN = xL1 , xR0 = xRN .

(5)

Note that when the coupling parameter γ, which is in proportion to R, is equal to zero, thecoupling term in (3) vanishes. For all of computer calculations, the fourth-order Runge-kutta
method is used with step size h = 0.005.
We carried out computer simulations for the case of N = 4 in a ring of chaotic circuits.
At first, we observed various different types of synchronization state by changing initial value
when each chaotic circuit generates three-periodic solution. Figure 8 shows data of x̂R2 + x̂L2 ,e map when x̂R1 + x̂L1 is smaller than −1.2.
x̂R3 + x̂L3 and x̂R4 + x̂L4 on the Poincar´
2
1.5
10.5
0-0.5
-1
-1.5
-2

2
1.5
10.5
0-0.5
-1-1.5
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2
1.5
10.5
0-0.5
-1-1.5-2000200

200

200

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400
time8002
1.5
10.5
0-0.5
-1-1.51000 -2

800

2
1.5
10.5
0-0.5
-1-1.51000 -2

800

2
1.5
10.5
0-0.5
-1-1.51000 -2(a)

0

0

0

200

200

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600

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time

(b)

800

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1000

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(c)

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10.50-0.5-1-1.5-221.510.50-0.5-1-1.5-20200

400

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0200400600

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time(d)Figure 8: Various different types of synchronization state disturbed by changing initialvalue (computer calculated results). α = 7.0,β = 0.16,γ = 0.005 and δ = 50.0. Upper
figures: x̂R2 + x̂L2 . Middle figures: x̂R3 + x̂L3 . Lower figures: x̂R4 + x̂L4 .
Next, we vary a control parameter of each chaotic circuit to generate intermittency chaos
near the three-periodic window. We can observe a complex behavior of the various synchronization state in a ring of chaotic circuits when the intemittency chaos appear. Figure 9 showse map when x̂R1 + x̂L1 is smaller
data of x̂R2 + x̂L2 , x̂R3 + x̂L3 and x̂R4 + x̂L4 on the Poincar´than −1.2. We can see that the synchronization states are interrupted by the intermittent bursts
and different synchronization sates reappear after the bursts settle down.

4. Conclusions
In this study, we investigated a complex behavior in two coupled chaotic circuits related withintermittency chaos near the three periodic window. At first, we observed three different typesof synchronization states when each chaotic circuit generates three-periodic solution. Next, wevaried a control parameter of each chaotic circuit to generate intermittency chaos near the threeperiodic window. In that case, we observed a complex behavior of the three synchronizationstates. Namely, intermittency bursts interrupt the synchronizations and different synchronizations reappear after the bursts settle down. Furthermore, we observed these complex behavior
when two coupled chaotic circuits is expand to a ring of chaotic circuits.

21.5
10.5
0-0.5
-1-1.5-205000

10000

15000

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05000

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20000

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2
1.5
10.5
0
-0.5
-1
-1.5
-2

2
1.5
10.5
0
-0.5
-1
-1.5
-2timeFigure 9: Time series of synchronization states disturbed by intermittency chaos. α = 7.0,β =0.152,γ = 0.005 and δ = 50.0. Upper figures: x̂R2 + x̂L2 . Middle figures: x̂R3 + x̂L3 . Lowerfigures: x̂R4 + x̂L4 .
Acknowledgement
This works was partly supported by Kayamori Foundation of Informational Science Advancement’s Research Grant.

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