BREAKDOWNOF IN-PHASESYNCHRONIZATION
OF TWOCIRCUITS COUPLED BY AN INDUCTOR
Masahiro

WADA,

Yoshifumi

NISHIOCHAOTICand Alcio USHIDA

Department of Electrical and Electronic Engineering, Faculty ofEngineering, Tokushima University, Tokushima, 770, JAPANTEL: +81-886-56-7470,
FAX: +81-886-54-9632
Email: wada@ee.tokushima-u. ac.jp

ABSTRACTIn this study, in-phasesynchronization
phenomena
observed from chaotic
circuits
coupled by an inductor
areinvestigatedespeciallypaying attention
to breakdownofchaos synchronization.
It is shown that breakdown of chaos
synchronizationoccurs when chaotic attractor
contains anunstable orbit splitting
the attractor
to two bands.
Further, it is also confirmed by calculated
results of Lyapunov
exponents that hyperchaos
appears at the same time.

1.

V2-rFig.

1 Circuit)model.

INTRODUCTION

Many nonlinear dynamicalsystems in various fields havebeen clarified to exhibit chaotic oscillations
and recently
applicationsof chaos to engineeringsystems are expected
such as chaos noise generators,
cent rol of chaos, synchronization of chaos, and so on. In those applications,
we arespecially interested in synchronizationof chaos. As far as
we know, synchronizationof chaos in a simple circuit was
first reported by Saito et al. [1]. Since Pecora et al. [2] studied theory of such phenomena,a large number of studies on
svnchronizat
ion of chaos have been reDorted.
However. the.
.
study on bifurcation
phenomenaof chaos synchronization
has been start ed recently [3] [4]. There remain a large number of problems related with bifurcationof synchronizationof chaos.
In this study, we investigate in-phase synchronization
observed from two autonomous
three-dimensional
chaotic circuits coupled by an inductor.
This circuit model is shownin Fig. 1. Each chaotic sub circuit is the circuit proposed
by Inaba et al. [5]. This system is a two sub circuits caseof a coupled system in Ref [4]. However, in Ref [4], we did
not analyze this circuit with exact solutions in detail.
In
this study, we analyze it in detail especially paying attention to breakdownof in-phase chaos synchronization.
For
the purpose of the detailed analysis, we consider the casethat diodes in the circuit are assumed to operate as idealswitches. In this case the circuit equation in a piecewise linear region is degenerated
to a four-dimensional
one. Therefore, Poincar6 map is derived as three-dimensional
map using exact solutions.
As a result, Lyapunov exponents can bealso calculated by exact solutions.
This simplified technique
was first proposed by Inaba et al. [6] and was confirmed to
be extremely
effective in analyzing
chaotic phenomena
indissipativeelectricalcircuits.
Further, this technique wasextended to higher-dimensional
circuits [7] or circuits including two diodes by Nishio [8]. However, it has not been

(a)

(b)
Fig.

2 Diodetains an unstable orbit splitting the attractor to two bands.
Further, it is also confirmed by calculated
results of Lyapunov exponents that hyperchaos appears at the same time,
We have to notice that two types of synchronizationmodes
coexist in this circuit model; in-phase and anti-phase.
We
can pro duce one of the two modes by inputting a certain initial conditions.
However, we investigate only the in-phasesynchronization
mode in this study.
2.

CIRCUIT

MODEL

Fig. 1 shows the circuit model used in this article. In ourmodel, two chaotic circuits which are simple chaotic sub circuits proposed by Inaba et al. [5] with the same parameters
are coupled by an inductor
Lo.
This sub circuit consistsof only three memory elements, one linear negative resistor and one diode, and is one of the simplest autonomouschaotic circuits.
At first we approximate
the characteristics
of two diodes
by the following two-segment
piecewise linear functions asFig. 2(a).

applied for the analysis of chaos synchronizationsin coupledchaotic circuits.
It is shown that breakdownof in-phasechaos synchronization
occurs when chaotic attractor
co,n-Copyright (C) 1996.

model.

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where k = 1, 2. The circuit equations are explained
dimensional
piecewise
linear differentialequationslows.

by sixas fol(a)

(2)

(k=
By changing

the variables(c)and parameters,

rc

Fxk’

—

(3)

(2) is normalized

(b)

1,2.)

“o’’=:, iLlk = V

t=~T,

El+ii$d(d)as(e)(~k=~k(4)f(~k)=;{~yk+l-/&/k-11}(5)

where k = 1, 2. and ~(y~ ) is a piecewise linear function of~k corresponding
to the characteristicof the diode. When
we use this diode model, we can calculate
attractors
byusing exact solutionof Eq.
(4).
However, because it ishard to give analyticalrepresentationof the Poincar& map
and its Jacobian,
Lyapunov exponents cannot be calculated
analytically.
In order to calculate
Lyapunov
exponents
by using analytical form, we consider the case that the diodes in thecircuit are assumed to operate as ideal switches as shown in
Fig. 2(b). When this simplified technique is used, the circuit equation
is degenerated
by OFF state of the diode.
Namely, when a diode is in the OFF state, the current
iLzk is constrained
to zero. Hence, when both diodes arein the OFF state, the circuit equation is degenerated
toIt enables us to derive threefour-dimensional
equation.
dimensional
Poincar6 map analytically.
3.

SYNCHRONIZATION

(f)

MMM

(g)

(h)

PHENOMENA

Fig. 3 shows a typical example of chaotic attractors
obtained from computer
calculation.
The figures show, fromleft, attractors on the Z1 - zz plane, Z1 —z1 plane and Z2 —X2plane. For computer
calculations,the parameter values ofa and /? are fixed and ~ is varied as a control parameter.
Fig.
3(a)~(f ) shows that attractors
are synchronizedcompletely.
For these parameter values, the attract ors are

Copyright (C) 1996.

(i)

———

Fig. 3 Attractors
obtained from computer calculation.
a =0.52, /3 = 6.0. -y : (a) O.100, (b) O.125, (c) O.150, (d)O.175,(c)O.200,(f) O.205, (g) O.215, (h) O.235, (i) O.280. From left,attractors of z] vs. ZQ, Z1 vs. xl, Z2 vs. x2.

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split to two bands by an unstable orbit.
As a parametervalue increases, two bands of the chaotic attractor merges toone band and the attractor contains the unstable orbit. At
the same time, chaos synchronization
becomes incomplete
as shown in Fig. 3(g). The reason is explained as follows.
In the case that chaotic attractor contains the unstable orbit, synchronized
two chaotic solutions
can approachtheunstable orbit.
If two solutions go through different side
wit h respect to the unstable orbit, they go away to the opposite direction each other. This mechanism
makes stability of synchronization
weak. Though there exist infinitely
many unstable orbits in chaotic attractors,
another unstableorbits have little influence on the stability of synchronization. As a parameter~ increases, in-phase synchronization
becomes
unstableand only anti-phasesynchronization
isobserved as Fig. 3(h). Finally, both of synchronization
becomes unstable and self-switching
phenomenonof in-phaseand anti-phase is observed as shown in Fig. 3(i).
We can also observe similarments as shown in Fig. 4.

4.

POINCARE

results

fromcircuit

experi-MAP

We define the Poincar6 section in the four-dimensional
region where both diodes are in the OFF state. In that case
the Poincar6 map T can be derived as a three-dimensional
The projectionsof themap by using exact solutions.
Poincar6 map onto the u – X2 plane by numerical calculation are shown in Fig. 5. We can see that synchronizationof chaos breaks down in Fig. 5(c).
In order to investigate the stability of chaos synchronization in detail, we calculate the Lyapunov exponents of thethree-dimensional
Poincar6map.
Note that because thePoincar6 map can be represented
by analyticalform, wecan derive its Jacobian
DT analytically.
The first, second
and third Lyapunov exponents(Al > & > &j) can be calculat ed by using DT as follows.

5. CONCLUSIONSIn this study, we investigatedbreakdown of in-phase synchronizationof chaotic circuits coupled by an inductor.
Itwas shown that breakdown of chaos synchronization
occurswhen chaotic attractor contains an unstable orbit splitting
the attractor to two bands. Further it was confirmed that
hyperchaos
appeared at the same time.

REFERENCES[1] Y. Ohmori, M. Nakagawa and T. Saito : “Mutual Coupling of Oscillatorswith Chaos and Period Doubling
Bifurcation,”
Proc.
of ISCAS’86,
pp. 61–64, 1986.
[2] L. M. Pecora and T. L. Carroll : “Synchronizationin Chaotic Systems,”
Phys.Rev.Lett.,
VO1.64, no.4,
pp. 821-824, 1990.
[3] P. Ashwin, J. Buescu and I. Stewart : “Bubblingof attractors and synchronizationof chaotic oscillators,” Phys. Lett., A 193 pp. 126-139, 1994.
[4] Y. Nishio and A. Ushida : “On a Ring of Chaotic
IEICE Trans. on FundaCoupled by Inductors,”
mentals, vol. E78-A, no. 5, May. 1995
in a[5] N. Inaba and S. Mori : “Chaotic phenomena
Circuit with a Diode due to the Change of the Oscillation Frequency,”
Trans. oj IEICE, vol. E71,pp. 842-849, Sep. 1988.
Phenomenain[6] N. Inaba and T. Saito : “Chaotica Circuit with a Negative Resistance and an Ideal
Switch of Diodes,”Trans.
of IEICE, vol.
E70,pp. 744–754, Aug. 1987.
Conductor
Hyper[7] T. Saito : “The Dead-Zone
chaos Generator,” (in Japanese)
Trans. oj IEICE,
A vol. J72-A, no. 7, pp. 1084-1092,Jul. 1989.
N. Inaba and S. Mori :“Chaos[8] Y. Nishio,
in a Four-Dimensional
Autonomous
Circuit with
Two Diodes,”Electronicsand Communication
inJapan, part 3, vol. 74, no. 11, pp. 77-89, 1991.

(a)where A is an exterior✌✎✌❑,$..””,.*.

..!<product.

Calculated
results are shown in Fig. 6. We can classify
characteristicof chaos as follows.
The attractors
in Fig.
3(b)~(i)are classified to chaos, because Lyapunov
exponents are as Al > 0. While the attractor
in Fig. 3(g) ishyperchaoswith 2 positive
Lyapunov
exponents,
becauseLyapunov exponents
are as Al > J2 >0>
A3. We can seethat the hyperchaos
appears for the value of ~ ~ 0.210. It
just corresponds
to breakdownof in-phase synchronizationof chaotic attractors.
Namely, we can confirm that breakdown of chaos synchronizationmeans the generation of hyperchaos.
In this circuit, we can make sure that hyperchaoswith three positive Lyapunov
exponents
exists 7 ~ 0.230.
However, such attractors
cannot be observed
at m-phasemode.

Copyright (C) 1996.

(b)

(c)

4 Experimental
Fig.
.56~8mH, L2 = 10.OmH,1.89M2, (c)r = 1.61KL

results.
LO =114.8mH, L1
C = 68.2nF,(a)r= 2.1UK2, (b)r

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==

/: hyperchaos

aO.L
..#’-

“

...
../“.+,,...................’...
.............. ....

0.00

:I

(c)
Fig. 5 Poinca&map.
(c) O.215, (d) O.235.

xl

vs. X2. y : (a) O175,

Fig. 7
nents.

,(b) O.20b,0.190

Expanded

OJOO

boxed

:1
0.210

portionLyapunov
alIA‘?1+’?2...............

0.00 —

al+a~+a3

-1.00

-2.00

-3.00

-4.00

-5.00

0.150
Fig.

6 Lyapunov

exponents

obtained

0.250

0.200

from computer

0,Y

calculation.

Copyright (C) 1996.

All Rights Reserved.

21.—..
—.......’-”-”””””............. ...

:}::jij;: ....
:..”:-0.50
0.180

h‘

Fig.

0.220

y6 Lyapunov

expo-