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IEICE TRANS. FUNDAMENTALS, VOL.E82–A, NO.9 SEPTEMBER 1999
PAPER Special Sec ion on Nonlinea Theo y and I s Applica ions
On he Takens-Bogdano Bi u ca ion in he Chua’s
Equa ion
An onio ALGABA†, Emilio FREIRE††, Es anislao GAMERO††,
and Alejand o J. RODR´
IGUEZ-LUIS††,Nonmembe s
SUMMARY The analysis o he Takens-Bogdano bi u ca-
ion o he equilib ium a he o igin in he Chua’s equa ion wi h
a cubic nonlinea i y is ca ied ou . The local analysis p o ides, in
fi s app oxima ion, diffe en bi u ca ion se s, whe e he p esence
o se e al dynamical beha iou s (including pe iodic, homoclinic
and he e oclinic o bi s) is p edic ed. The local esul s a e used
as a guide o apply he adequa e nume ical me hods o ob ain
a global unde s anding o he bi u ca ion se s. The s udy o
he no mal o m o he Takens-Bogdano bi u ca ion shows he
p esence o a degene a e (codimension- h ee) si ua ion, which is
analyzed in bo h homoclinic and he e oclinic cases.
key wo ds: bi u ca ions, oscilla ions, Chua ci cui
1. In oduc ion
The main objec i e o his wo k is o p o ide a deep un-
de s anding o some non i ial dynamical beha iou e-
la ed o he Takens-Bogdano bi u ca ion (double-ze o
eigen alue o he linea iza ion ma ix) in he Chua’s
equa ion wi h a cubic nonlinea i y. This equa ion mod-
els an elec onic ci cui , whose mos impo an ea u es
a e i s simplici y (only one nonlinea i y, which we ha e
aken as an odd cubic polynomial), and he complex
beha iou s ha can exhibi . Some o hese beha iou s
a e analy ically explained in he s udy we will pe o m.
The analysis o his ci cui has been sou ce o a
la ge bibliog aphy (see Ma sumo o e al. [9]). The
mos widely conside ed case co esponds o a piecewise
linea implemen a ion o he nonlinea de ice (see, e.g.,
Madan [8] and e e ences he ein). Unde his hypo h-
esis, he heo e ical analysis o he s a e equa ions can-
no p ofi om many esul s o diffe en iable dynamics
and, in pa icula , o bi u ca ion heo y (see, o in-
s ance, [5]).
We conside he e he Chua’s equa ion wi h a cubic
nonlinea i y:
˙x=α(y−ax3−cx),
˙y=x−y+z, (1)
˙z=−βy.
We a e in e es ed in hose bi u ca ion aspec s ela ed
Manusc ip ecei ed Decembe 1, 1998.
Manusc ip e ised Ap il 13, 1999.
†The au ho is wi h he Depa men o Ma hema ics,
Escuela Poli ´ecnica Supe io , Uni e si y o Huel a, Spain.
††The au ho s a e wi h he Depa men o Applied Ma h-
ema ics II, Escuela Supe io de Ingenie os, Uni e si y o
Se illa, Spain.
o he Takens-Bogdano bi u ca ion ha he equilib-
ium a he o igin in he abo e equa ion exhibi s. The
impo ance o his bi u ca ion lies in he possibili y o
finding global effec s (homoclinic and he e oclinic mo-
ions) om a local bi u ca ion s udy (see Ma sumo o
e al. [9]).
Chua’s equa ion (1) has been analyzed, e.g., by
Khibnik e al. [7], Huang e al. [6], Pi ka e al. [10].
In he wo las pape s, a new linea e m is included in
he las equa ion o (1), in o de o ake in o accoun
small esis i e effec s in he induc o . In Khibnik e al.
[7], he analysis is done by keeping fixed a,c, so ha
he e is no possibili y o a Takens-Bogdano bi u ca-
ion. The au ho s ca y ou he analysis o he Hop
bi u ca ion, and es ablish i s connec ion wi h a homo-
clinic bi u ca ion. This connec ion can be explained
wi h he analysis we will pe o m he e (compa e Fig. 10
o [7] wi h Fig. 1 he e).
In ou analysis o he Chua’s equa ion (1), we will
conside he non i ial cases a=0,α=0.
No e he symme y (x, y, z)→(−x, −y, −z) i ex-
hibi s. The o igin is always an equilib ium poin , and
he linea iza ion ma ix a his poin is:


−αc α 0
1−11
0−β0

.(2)
I is a s aigh o wa d compu a ion o show ha , aking
c=cc=0,β=βc=α,
he linea iza ion ma ix a he o igin has a double ze o
eigen alue and a hi d eigen alue −1. Then, we ha e
a bidimensional cen e mani old and a one-dimensional
s able mani old. To analyze his linea codimension-
wo bi u ca ion, we ake cand βas bi u ca ion pa am-
e e s, and look o he bi u ca ion beha iou s co e-
sponding o pa ame e alues close o he c i ical ones:
cc,βc.
2. No mal Fo m
Fi s ly, we conside he Chua’s equa ion wi h he pa-
ame e e alua ed a hei c i ical alues. In o de o
pu he equa ion in an app op ia e o m, we make he
linea ans o ma ion:
ALGABA e al: BIFURCATIONS IN CHUA’S EQUATION
1723
(a) (c)
(b) (d)
Fig. 1 Bi u ca ion se o he Chua’s equa ion, o a=1,α=1.3. (a) Nume ical bi u -
ca ion se . (b) Quali a i e bi u ca ion se . The configu a ion o equilib ia and pe iodic
o bi s in each zone appea s in Fig. 4. (c) Zoom o he neighbou hood o Takens-Bogdano
poin . (d) Zoom o he neighbou hood o he degene a e homoclinic connec ion poin .


x
y
z

=

α0−α
011
−α1α



X
Y
Z

,
b inging he linea iza ion ma ix (2) o Jo dan o m:


01
00
−1

.
In he new a iables, a hi d-o de cen e mani old
(compu ed by using a ecu si e p ocedu e de eloped
in F ei e e al. [2]) is gi en by
Z=α4a(X3−3X2Y+6XY2−6Y3).
This app oxima ion enables us o ob ain he fi h-o de
educed sys em on he cen e mani old.
Nex , we will pu he educed sys em in an app o-
p ia e o m. Fo ha , we use nea -iden i y ans o -
ma ions leading o no mal o m. Using he algo i hm
de eloped in Game o e al. [3], we ob ain he ollowing
fi h-o de no mal o m o he educed sys em on he
cen e mani old:
˙
X=Y,
˙
Y=a3X3+b3X2Y+a5X5+b5X4Y, (3)
whe e
a3=−aα4,b
3=3aα3(α−1),
a5=3a2α8,b
5=−3a2α7(8α−5).
We obse e ha he coefficien a3is always nonze o
(al hough i may be posi i e o nega i e). Howe e ,
he coefficien b3 anishes o he alue α= 1, whe e a
degene a e Takens-Bogdano bi u ca ion akes place.
To know how he pa ame e s c,βaffec o he
no mal o m (3), we suspend he sys em (adding he
i ial equa ions ˙c=0, ˙
β= 0) and compu e he cen e
mani old o he suspended sys em. A e some com-
pu a ions, and neglec ing he highe -o de e ms in he
pa ame e s, we pu in co espondence he Chua’s equa-
ion (1) – aking c≈ccand β≈βc– wi h:
˙
X=Y,
˙
Y=1X+2Y+a3X3
+b3X2Y+a5X5+b5X4Y,
˙
Z=−Z,
whe e
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IEICE TRANS. FUNDAMENTALS, VOL.E82–A, NO.9 SEPTEMBER 1999
1=−α2c,
2=−(β−α)+α(α−1)c.
I is a s aigh o wa d compu a ion o show ha hese
un olding pa ame e s sa is y he ans e sali y condi-
ion:
∂(1,
2)
∂(c, β)


c=0,β=α
=α2=0,
and hen, he change o pa ame e s c, β by 1,
2is a
local diffeomo phism.
In nex sec ions we will add ess he s udy o bo h,
nondegene a e and degene a e cases.
3. Nondegene a e Cases
When bo h a3and b3a e nonze o, we a e dealing wi h
a nondegene a e Takens-Bogdano bi u ca ion [5]. I s
classifica ion depends on he sign o a3, and he coe -
ficien s a5,b5do no play any ole in he subsequen
analysis. So, in he s udy o he nondegene a e cases i
is enough o compu e he hi d-o de no mal o m, i.e.,
he no mal o m (3) unca ed up o hi d o de (any-
way, he whole fi h-o de no mal o m will be equi ed
la e in he analysis o he degene a e case).
Depending on he sign o a3(which is de e mined
by he sign o he pa ame e a), wo diffe en si ua ions
a ise:
3.1 Homoclinic Case
This case applies when he coefficien a3is nega i e,
ha is, o a>0. We can conside b3<0, whose
analysis can be ound in Guckenheime and Holmes [5]
( he s udy o b3>0 can be educed o he abo e one
by changing he sign o , X, 2).
In he bi u ca ion se a ound he o igin, he ol-
lowing codimension-one bi u ca ions a e p esen :
•A pi ch o k bi u ca ion o he o igin PI.
•A Hop bi u ca ion o he o igin H.
•A Hop bi u ca ion o he non i ial equilib ia h.
•A homoclinic connec ion o he o igin Hm (in ac ,
a pai o homoclinic connec ions, due o he sym-
me y).
•A saddle-node bi u ca ion o pe iodic o bi s SN1.
The local app oxima ions o hese bi u ca ions appea
in he quo ed book, and hey a e no included he e o
he sake o b e i y. I is impo an o no ice ha , as
we ha e compu ed 1,2 o fi s o de as a unc ion o
pa ame e s c,β, we ha e a use ul s a ing poin o he
nume ical analysis we will p esen la e .
3.2 He e oclinic Case
This si ua ion co esponds o a3>0. I occu s in
Chua’s equa ion (1) when a<0. We can es ic o
he case b3<0 ( he case b3>0 can be handled by
changing he sign o , X, 2). The ollowing bi u ca-
ions a e p esen (see Guckenheime and Holmes [5]):
•A pi ch o k bi u ca ion o he o igin PI.
•A Hop bi u ca ion o he o igin H.
•A he e oclinic connec ion be ween non i ial equi-
lib ia H .
As abo e, we ha e no included he local exp essions
o hese bi u ca ions.
4. Degene a e Cases
The complexi y o bi u ca ion beha iou s g ows in he
degene a e case co esponding o he anishing o b3,
which occu s a he c i ical alue αc= 1. In his case,
he fi h-o de e ms in (3) a e necessa y o de e mine
he local bi u ca ion beha iou .
Fo he c i ical alue o α, we find b5=−9a2α7=
0. This degene a e case is a codimension- h ee si ua-
ion, and a hi d un olding pa ame e is equi ed. We
will ake 3=α−αc=α−1, oge he wi h 1and 2,
o desc ibe his bi u ca ion.
In he degene a e cases, he knowledge o he whole
fi h-o de no mal o m (3) is equi ed. One o he fi h-
o de e ms in he no mal o m (3) — he coefficien
a5— can be elimina ed by escaling he ime in e ms
o he s a e a iables. This ope a ion do no al e he
alues o he emaining coefficien s in he fi h-o de
no mal o m.
As in he nondegene a e cases, wo diffe en si ua-
ions a e possible. Each one will be conside ed in nex
subsec ions.
4.1 Homoclinic Case
This is he case when a3<0. We can assume b5>
0 ( he case b5<0 can be educed o he abo e by
changing he sign o , Y, 2,
3). In Rousseau and Li [14]
and Rod ´ıguez-Luis e al. [12], i is shown ha , besides
he bi u ca ions p esen in he nondegene a e case (see
Sec . 3.1), he ollowing ones appea :
•A degene a e Hop bi u ca ion o he o igin Hd.
F om he e, a saddle-node bi u ca ion o pe iodic
o bi s SN2 eme ges.
•A degene a e Hop bi u ca ion o he non i ial
equilib ia hd. F om he e, a saddle-node bi u ca-
ion o pe iodic o bi s sn eme ges.
•A degene a e homoclinic connec ion Hmd. F om
he e, wo saddle-node bi u ca ions o pe iodic o -
bi s sn and SN3 eme ge.
•A cusp o saddle-node bi u ca ions o pe iodic o -
bi s C1, whe e SN1 and SN2 collapse.
ALGABA e al: BIFURCATIONS IN CHUA’S EQUATION
1725
4.2 He e oclinic Case
This co esponds o he case a3>0. We can educe
ou s udy o he case b5>0, by changing he sign o
, Y, 2,
3i necessa y. Beyond he bi u ca ions p esen
in he nondegene a e case (see Sec . 3.2), he ollowing
bi u ca ions appea (see Rousseau [13]):
•A degene a e Hop bi u ca ion Hd. F om he e,
a saddle-node bi u ca ion o pe iodic o bi s SN
eme ges.
•A degene a e he e oclinic connec ion H d, whe e
he he e oclinic connec ion changes i s s abili y.
He e, he abo e saddle-node bi u ca ion o pe iodic
o bi s SN ends.
5. Nume ical S udy
Now, we will look o his ich bi u ca ion beha iou
p edic ed by he heo y, a ound his degene a e Takens-
Bogdano bi u ca ion, in he Chua’s equa ion. Fi s ly,
we ha e selec ed a= 1 in o de o p esen bi u ca ion
se s co esponding o he homoclinic case. We will o-
cus on he degene a e case ha occu s a cc=0,βc=1,
αc=1.
The nume ical esul s a e p esen ed in Figs. 1 and
3, co esponding o fixed alues o α=1.3 and 0.8, e-
spec i ely. They a e loca ed on bo h sides o he c i ical
alue αc= 1 whe e he degene acy akes place.
The local esul s achie ed in p e ious sec ions ha e
been essen ials as a guide in he use o he adequa e
nume ical con inua ion me hods (see Doedel e al. [1],
Rod ´ıguez-Luis e al. [11]), in o de o ex end globally
he local in o ma ion.
In Fig. 1 we ha e aken α=1.3. This is he iches
si ua ion om he poin o iew o diffe en bi u ca ion
beha iou s. In he bi u ca ion se d awn in (a) we ha e
conside ed he ange β∈(0,1.4), c∈(−0.8,1). The
nume ical esul s o he quo ed pic u e ha e been ob-
ained wi h AUTO [1]. Se e al cu es a e so close ha
a e almos indis inguishable. We e e o he quali-
a i e cu es (Fig. 1(b)) and he wo zooms ((c)–(d))
o cla i y he bi u ca ion se . All he bi u ca ions p e-
dic ed by he analysis o Sec . 3.1 a e p esen . Mo e-
o e , a cusp o saddle-node bi u ca ions o pe iodic o -
bi s C2, whe e SN3 and SN4 collapse, also appea s.
The p esence o his cusp may be ela ed o a highe
degene acy in he homoclinic Hm. Fu he s udy will
be needed o unde s and i .
Fo he momen , we show nume ically ha , in-
c easing α, he wo cusps C1 and C2 mee s in a beak-
o-beak singula i y. This ac is p esen ed in he bi-
u ca ion se s o Fig. 2, whe e we only d aw he ou
saddle-node bi u ca ions o pe iodic o bi s. No ice ha ,
o α=1.45, he saddle-node bi u ca ions SN1and
SN2collapse a C1, and SN3and SN4a C2. La e ,
(a)
(b)
Fig. 2 Beak- o-beak singula i y o a= 1. Pa ial bi u ca ion
se including only he saddle-node bi u ca ion o pe iodic o bi s
cu es. (a) Si ua ion a α=1.45, be o e he collision o he
cusps. (b) Si ua ion a α=1.5, a e he collision.
Fig. 3 Nume ical bi u ca ion se o he Chua’s equa ion o
a=1,α=0.8. The configu a ion o equilib ia and pe iodic
o bi s in each zone appea s in Fig. 4.
o α=1.5, a saddle-node cu e (SN1–SN3) joins TB
and Hmd, whe eas SN2–SN4a e no he same cu e.
In Fig. 3, he si ua ion is simple , and only he bi-
u ca ions p edic ed by he local analysis appea . This
bi u ca ion se also illus a es he homoclinic nonde-
gene a e case, conside ed in Sec . 3.1.
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IEICE TRANS. FUNDAMENTALS, VOL.E82–A, NO.9 SEPTEMBER 1999
Fig. 4 Configu a ion o equilib ia and pe iodic o bi s in he homoclinic case.
Fig. 5 Nume ical bi u ca ion se o he Chua’s equa ion o
a=−1, α=1.3. The configu a ion o equilib ia and pe iodic
o bi s in each zone appea s in Fig. 7.
The configu a ions o equilib ia and pe iodic o bi s
p esen in each zone o he bi u ca ion se s, o his
homoclinic case, a e depic ed schema ically in Fig. 4.
Now, we selec a=−1, o deal wi h he he e-
oclinic case. As abo e, we ha e aken α=1.3 and
α=0.8. Each alue co esponds o a side o he c i -
ical alue αc= 1 whe e he degene acy akes place.
The nume ical esul s a e p esen ed in Figs. 5 and 6.
The configu a ions o equilib ia and pe iodic o bi s a e
ske ched in Fig. 7. All he cu es and configu a ions a e
explained by he p e ious heo e ical analysis. In pa -
Fig. 6 Nume ical bi u ca ion se o he Chua’s equa ion o
a=−1, α=0.8. The configu a ion o equilib ia and pe iodic
o bi s in each zone appea s in Fig. 7.
icula , he configu a ion in he icini y o a he e oclinic
nondegene a e Takens-Bogdano bi u ca ion coincides
wi h he si ua ion o Fig. 6.
The analysis ca ied ou is a s aigh way o de ec
global dynamics om a local analysis. Fo ins ance,
in Fig. 8 we show wo global connec ions. Namely, a
homoclinic o bi loca ed on he cu e Hm o Fig. 1. I
co esponds o he pa ame e alues a=1,α=1.3,
β=0.892 and c≈−0.435. We ha e ep esen ed i s
p ojec ion on o he xy plane. Also, a he e oclinic o bi
loca ed on he cu e H o Fig. 5, co esponding o he

ALGABA e al: BIFURCATIONS IN CHUA’S EQUATION
1727
Fig. 7 Configu a ion o equilib ia and pe iodic o bi s in he he e oclinic case.
(a)
(b)
Fig. 8 (a) P ojec ion on he xy plane o he homoclinic con-
nec ion ha appea s, o a=1,α=1.3, a β=0.892 and
c≈−0.435. (b) P ojec ion on he xy plane o a pai o he e o-
clinic connec ions ha appea s, o a=−1, α=1.3, a β=1.295
and c≈0.2052.
alues a=−1, α=1.3, β=1.295 and c≈0.2052.
These global connec ions, ha nea he Takens-
Bogdano bi u ca ion a e plana phenomena, de elop
a idimensional s uc u e by mo ing, o example, he
pa ame e α. In he plana si ua ion, he pe iodic o -
bi s ha e a mono onically inc easing pe iod when ap-
p oaching homoclinici y. On he o he hand, when he
homoclinic connec ion en e s in he Shil’niko egion, a
wiggling beha iou appea s, wi h a sequence o saddle-
(a)
(b)
Fig. 9 (a) Bi u ca ion diag am (pe iod e sus β) o a=1,α=
10, c=−0.5. (b) P ojec ion on he xy plane o he homoclinic
connec ion ha appea s, o a=1,α= 10, c=−0.5, β≈0.7705.
node and pe iod-doubling bi u ca ions o pe iodic o -
bi s (see Glendinning and Spa ow [4]). This beha iou
is p esen ed in Fig. 9, whe e we ha e included a bi u -
ca ion diag am, plo ing he pe iod agains β. Also,
a phase po ai o he Shil’niko homoclinic connec-
ion is d awn. No ice he saddle- ocus cha ac e o he
equilib ium a he o igin.
6. Conclusions
The s udy o he Takens-Bogdano bi u ca ion is a pow-
1728
IEICE TRANS. FUNDAMENTALS, VOL.E82–A, NO.9 SEPTEMBER 1999
e ul me hod ha p o ides aluable in o ma ion abou
pe iodic beha iou and global dynamics.
Nea he Takens-Bogdano bi u ca ion, he phe-
nomena a e plana , bu mo ing he pa ame e s a
away, we can expec ha hey de elop a idimensional
s uc u e, and hen hey can p o ide a ou e o chao ic
dynamics.
In his pape , we ha e ca y ou he analysis o he
Takens-Bogdano bi u ca ion o he equilib ium a he
o igin in he Chua’s equa ion wi h a cubic nonlinea -
i y. De i ing he co esponding no mal o m, we pu
in e idence he p esence o degene a e cases. Then, we
ob ain heo e ically local and global bi u ca ions, ha
p o ide in o ma ion abou pe iodic beha iou s and ho-
moclinic and he e oclinic mo ions. The comple ion o
he bi u ca ion se equi es nume ical me hods. These
allow us o de ec he p esence o a cusp o saddle-node
bi u ca ion o pe iodic o bi s, and also o a beak- o-
beak singula i y. Mo eo e , ou analysis explains he
p esence o se e al codimension- wo bi u ca ions de-
ec ed nume ically in Khibnik e al. [7]. Namely, a
degene a e Hop bi u ca ion o he o igin, a degene a e
homoclinic and a cusp o saddle-node o pe iodic o bi s
(see Fig. 10 o [7]).
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[4] P. Glendinning and C. Spa ow, “Local and global beha -
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and homoclinic bi u ca ions in Chua’s ci cui wi h a smoo h
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384, 1993.
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An onio Algaba was bo n in
Pe˜na oya-Pueblonue o (Spain). He
ecei ed he Licenciado en Ciencias
Ma em´a icas deg ee in 1979 and he Doc-
o en Ciencias Ma em´a icas deg ee in
1996, bo h om he Uni e si y o Se ille.
Since Oc obe 1989, he has been wi h he
Depa men o Applied Ma hema ics a
he Uni e si y o Huel a, whe e he s a ed
wi h a esea ch g an in 1989 and became
P o eso Ti ula in 1998. His esea ch in-
e es lies in he fields o plana ec o fields, bi u ca ions in
dynamical sys ems wi h applica ions o elec onic ci cui s.
Emilio F ei e was bo n in Reus
(Spain). He ecei ed he B.S. deg ee
in elec ical enginee ing in 1975 and he
Ph.D. deg ee in 1982, bo h om he Uni-
e si y o Se ille. Since Oc obe 1980, he
has been wi h he Depa men o Applied
Ma hema ics a he Uni e si y o Se ille,
whe e he became P o esso in 1990. His
esea ch in e es lies in he fields o bi-
u ca ions in dynamical sys ems (ODEs)
wi h applica ions o elec onic ci cui s.
Es anislao Game o was bo n in
Se ille (Spain). He ecei ed he Li-
cenciado en Ciencias Ma em´a icas de-
g ee in 1986 and he Doc o en Ciencias
Ma em´a icas deg ee in 1990, bo h om
he Uni e si y o Se ille. Since Oc obe
1986, he has been wi h he Depa men o
Applied Ma hema ics a he Uni e si y o
Se ille, whe e he became P o eso Ti ula
in 1993. His esea ch in e es lies in he
fields o bi u ca ions in low-dimensional
dynamical sys ems (ODEs) and hei applica ions (mainly o
elec onic sys ems).
Alejand o J. Rod ´ıguez-Luis was
bo n in Le´on (Spain). He ecei ed he
Licenciado en F´ısica deg ee in 1984 and
he Doc o en Ciencias F´ısicas deg ee in
1991, bo h om he Uni e si y o Se ille.
Since Oc obe 1986, he has been wi h he
Depa men o Applied Ma hema ics a
he Uni e si y o Se ille, whe e he be-
came P o eso Ti ula in 1993. His e-
sea ch in e es lies in he fields o bi u ca-
ions in low-dimensional dynamical sys-
ems (ODEs) and hei applica ions (mainly o elec onic sys-
ems).
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