Publ. Ma . 45 (2001), 95–123
NONDEGENERATE LINEARIZABLE CENTRES OF
COMPLEX PLANAR QUADRATIC AND SYMMETRIC
CUBIC SYSTEMS IN C2
C. Ch is ophe and C. Rousseau
Abs ac
In his pape we conside complex diffe en ial sys ems in he plane,
which a e linea izable in he neighbo hood o a nondegene a e cen-
e. We find necessa y and sufficien condi ions o linea izabili y
o he class o complex quad a ic sys ems and o he class o
complex cubic sys ems symme ic wi h espec o a cen e.
The sufficiency o hese condi ions is shown by exhibi ing explic-
i ly a linea izing change o coo dina es, ei he o Da boux ype o
a gene aliza ion o i .
1. In oduc ion
This pape o igina ed om he in e es o he wo au ho s in isoch o-
nous cen es ([CD], [MRT] and [MMR]). Se e al au ho s ha e made
a sys ema ic s udy o he isoch onous cen es inside ce ain classes o
sys ems wi h cen e ([CJ], [L], [P], [RT1], [De], [Sa1] and [Sa2]). The
fi s ou pape s s udy isoch onous cen es inside quad a ic sys ems and
cubic sys ems symme ic wi h espec o i s cen e. The wo k o De-
lin [De] deals wi h hose cen es o which he e is an in eg a ing ac-
o (x2+y2)α. The pape s by Saba ini gi e condi ions o isoch onici y
in e ms o commu ing ec o fields. These s udies a e made possible by
he ac ha he cen e condi ions a e known in bo h hese cases.
Un o una ely, he e a e e y ew “na u al” amilies in which he cen-
e condi ions a e known, al hough a numbe o mechanisms p oducing
s a a o cen es a e well known.
In [CD] he Kukles amily o sys ems was s udied:
˙x=−y
˙y=x+a1x2+a2xy +a3y2+a4x3+a5x2y+a6xy2+a7y3.
(1.1)
2000 Ma hema ics Subjec Classifica ion. 34C, 58F.
This wo k was suppo ed by NSERC and FCAR.
96 C. Ch is ophe , C. Rousseau
An o iginal ea u e o his wo k is ha , al hough he cen e condi ions a e
no known, i is s ill possible o find necessa y and sufficien condi ions
o an isoch onous cen e in his amily.
The pape [CD] b ings o ligh o he in e es ing ques ions, one o
which is he s a ing poin o ou in es iga ions he e. In looking o he
necessa y condi ions o an isoch onous cen e in he Kukles sys em, a
new se o condi ions was ound which canno be sa isfied o eal sys ems
in he amily. Howe e i we allow he coefficien s ai o be complex, hen
he condi ions can be sa isfied, and he o igin is again isoch onous. The
no ion o isoch onici y s ill makes sense in his con ex as long as we
keep o eal ime.
In [CD] i was shown ha a singula poin is a complex isoch onous
cen e i and only i a whole punc u ed neighbo hood o he poin con-
sis s en i ely o closed ajec o ies. This in e es ing esul hus gi es
a geome ic cha ac e iza ion o isoch onici y. This is comple ely diffe -
en om he eal case, whe e any cen e can be made isoch onous by
mul iplying by a sui able posi i e unc ion. In he complex case, mul-
iplying by a non-cons an unc ion changes he geome y o he ( eal
ime) ajec o ies.
Simul aneous o hese in es iga ions, he e has been a g owing in e es
in he complex cen es o sys ems in C2, a cen e being a non-degene a e
singula poin wi h ze o ace and a local analy ic fi s in eg al. No e
ha in eg able saddles also come wi hin his ca ego y. Complex cen-
es in quad a ic sys ems ha e been s udied by se e al ma hema icians:
Dulac [Du], Liu and Li [LL] and mo e ecen ly Fa ell [F].
Pu ing oge he he esul s o hese wo ks i becomes na u al o ask
wha a e he complex isoch onous quad a ic sys ems. This aises he
need o a good defini ion o a complex isoch onous cen e. Using a eal
ime o such a sys em is somewha a ificial, especially since i excludes
he na u al iden ifica ion o cen es and saddles when wo king o e he
complex numbe s. Ou s a ing poin he e, he e o e, is o in oduce he
equi alen no ion o linea izable cen e. Tha is, a sys em whose cen e
can be educed o i s linea pa by an analy ic change o coo dina es.
In his pape we gi e necessa y and sufficien condi ions o linea iz-
abili y o complex quad a ic and symme ic cubic cen es. The pa-
pe [MMR] in oduces a new me hod o p o e isoch onici y ia Da -
boux linea izing change o coo dina es. A efinemen o his me hod
allows us o gi e explici linea izing changes o coo dina es o each o
hese condi ions.
One impo an con ibu ion o hese in es iga ions o he s udy o
isoch onici y is ha he e seems o be a a wide ange o linea izable
Complex Linea izable Cen es 97
cen es i we allow he coefficien s o be complex. This seems o be in
con as o he case o in eg abili y, whe e he co esponding s a a a e
e y simila . Howe e , i we conside linea izable saddles ins ead, hen
he complex and eal cases a e compa able.
A simple eason o his si ua ion is ha , o eal cen es, he sepa-
a ices a e conjuga e and he e can be no independence in how a eal
linea izing ans o ma ion can ac on ei he one. In he complex case
and in he case o a eal saddle his independence is p ese ed.
I seems, he e o e, ha a be e unde s anding o isoch onici y as a
phenomena should be ob ained om conside ing linea izable saddles in
mo e de ail.
In all s udies o his na u e, he e is a need o ha e “ isible” phenom-
ena o spu on in es iga ion. In he case o a eal isoch onous cen e,
i is easy o see ha no fini e c i ical poin can lie on he bounda y o
he pe iod annulus a ached o he cen e; a esul which has been used
in se e al classifica ion p oblems. I we conside linea izable saddles,
hen his phenomena only emains alid wi h complex ime, and conse-
quen ly loses i s powe . Howe e , om ou in es iga ions we conjec u e
he ollowing obs uc ion o linea izabili y, which we hope will p o ide an
impe us o u he in es iga ion along hese lines: No linea izable sad-
dle can lie on a homoclinic loop o , mo e gene ally, on a monod omic
g aphic.
2. Gene ali ies
2.1. Complex cen es and linea izable cen es.
Defini ion 2.1. (1) A singula poin o a complex analy ic sys em
˙z=P(z,w)
˙w=Q(z,w)
(2.1)
in C2is a cen e i he sys em has an analy ic fi s in eg al in a
neighbo hood o he singula poin .
(2) A cen e is nondegene a e i he sys em has a non- anishing 1-je
a he singula poin which is a Mo se singula poin o an analy ic
fi s in eg al.
(3) A nondegene a e cen e is linea izable i he e exis s an analy ic
change o coo dina es in he neighbo hood o he singula poin ,
b inging he sys em o a linea sys em.
Rema k 2.2.A nondegene a e cen e necessa ily has wo opposi e eigen-
alues.
98 C. Ch is ophe , C. Rousseau
F om now on, we will always es ic ou sel es o nondegene a e cen-
es.
P oposi ion 2.3. Le us suppose ha he sys em (2.1) has a nondegen-
e a e cen e a he o igin and is o he o m
˙z=iz +p(z,w)=iz +o(|(z,w)|2)
˙w=−iw +q(z,w)=−iw +o(|(z,w)|2).
(2.2)
The ollowing a e equi alen :
(1) The o igin is linea izable;
(2) The e exis s a neighbo hood o he o igin such ha e e y ajec o y
inside ha neighbo hood (wi h ime as a eal pa ame e ) is pe iodic
(i his is so, hen he pe iod is a cons an ).
(3) The o igin is s able, in he sense s able in he u u e and s able in
he pas , o eal ime .
P oo : As poin ed by he e e ee he equi alence o (1) and (3) is a
consequence o a heo em o Ca an-Ca a hThe aodo y (1932) (see o
ins ance [M, Theo em 2.1]), s a ing ha a sys em o he o m (2.2) is
linea izable i and only i he solu ion (z,w)=(0,0) is s able. As any
ajec o y o he linea sys em is pe iodic, which implies he same esul
o he linea izable sys em, we ha e ha (1) implies (2). The las pa
ollows om (2) implies (3).
No e, ha he linea izabili y condi ion he e is s onge han in eg a-
bili y and is no equi alen o he linea izabili y o olia ions defined by
1- o ms. Mo e de ails on his can be ound in [CMR].
2.2. Da boux linea iza ion and i s gene aliza ion.
We s a wi h some defini ions.
Defini ions 2.4. (1) Fo equa ion (2.1) we define he diffe en ial op-
e a o Dac ing on analy ic unc ions F(z,w) defined in a neigh-
bo hood o (0,0) by
D(F)=∂F
∂z P(z,w)+∂F
∂wQ(z,w).(2.3)
(2) An in a ian algeb aic cu e o he sys em (2.1) is a cu e in C2
gi en by an equa ion F(z,w) = 0, wi h F(z,w)∈C[z,w] such ha
he e exis s K(z,w)∈Cn−1[z,w] sa is ying
DF(z,w)=F(z,w)K(z,w).(2.4)
Complex Linea izable Cen es 99
He e Cn−1[z,w] deno es he space o polynomials in zand wo
deg ee ≤n−1 and complex coefficien s. F(x, y) is also called a
Da boux ac o .
(3) Any analy ic unc ion F(z,w) sa is ying (2.4), o some K(z,w)∈
Cn−1[z,w], is a gene alized Da boux ac o . The polynomial K(F)=
K(z,w) is called he co ac o o he Da boux ac o .
(4) A noncons an unc ion F(z,w) sa is ying DF(z,w)≡0isafi s
in eg al.
(5) A Da boux unc ion is a unc ion Z(z,w) o he o m
Z=
k

j=0
Fαj
j,α
j∈C,(2.5)
wi h Fj∈C[z,w], j=0,... ,k.
(6) Gi en a sys em (2.1) and he diffe en ial ope a o Ddefined by
(2.3), a Da boux unc ion ( esp. gene alized Da boux unc ion) as-
socia ed wi h he sys em (2.1) is a unc ion Zo he o m (2.5),
wi h Fj= 0 in a ian algeb aic cu es ( esp. FjDa boux ac o s),
j=0,... ,k.
(7) A sys em is (gene alized) Da boux in eg able i i has a fi s in eg al
which is a (gene alized) Da boux unc ion associa ed o i .
Many o he s a a o polynomial sys ems wi h a cen e ha e a fi s
in eg al which is a Da boux unc ion, o a gene alized Da boux unc ion
(c . [C], [S1], [S2]). In p ac ice, he Da boux ac o s ha we a e mos
in e es ed in a ise as limi ing cases o Da boux unc ions, and can be
exp essed in he o m eD/E, whe e Dand Ea e polynomials [C]. Many
examples o isoch onous cen es ha ing Da boux fi s in eg als a e also
gi en in [MRT].
The concep o Da boux linea izabili y is in oduced in [MMR],
which howe e is only conce ned wi h eal sys ems. In he eal con-
ex a linea izing change o coo dina es is hence gi en by a unique unc-
ion Z=F(z,z). He e we mus adap he defini ions o he ac ha
we a e dealing wi h sys ems in C2.
Defini ion 2.5. The sys em (2.2) is (gene alized) Da boux linea izable
i he e exis s a (gene alized) Da boux change o coo dina es
(Z, W)=

k

j=0
Fαj
j,


j=0
Gβj
j
,(2.6)

100 C. Ch is ophe , C. Rousseau
egula a he o igin, i.e. o he o m (Z, W)=(z+o(|(z,w)|),w +
o(|(z,w)|)) linea izing (2.2). Such a unc ion Zis called a (gene alized)
Da boux linea izing change o coo dina es.
Rema k 2.6.A (gene alized) Da boux linea izable sys em is (gene al-
ized) Da boux in eg able wi h fi s in eg al F(Z, W)=ZW.
The ollowing heo em cha ac e izing Da boux linea izabili y is ob-
ained exac ly as he co esponding heo em in [MMR]:
Theo em 2.7. (i) The sys em (2.2) is Da boux linea izable i and
only i he e exis in a ian algeb aic cu es F0=0and G0=0
o he o m F0(z,w)=z+o(|(z,w)|),G0(z,w)=w+o(|(z,w)|)
and he e exis in a ian algeb aic cu es Fj=0,j∈J1,Gj=0,
j∈J2whe e J1and J2a e fini e subse s o N(possibly oid) such
ha Fj(0,0) =0,Gj(0,0) =0and
K0+
j∈J1
αjKj=i
L0+
j∈J2
βjLj=−i
(2.7)
whe e Kjis he co ac o o Fj,Ljis he co ac o Gjand αj,β
j∈
C. The Da boux linea izing change o coo dina es is hen gi en by
(Z, W)=
F0
j∈J1
Fαj
j,G
0
j∈J2
Gβj
j
.(2.8)
(ii) The sys em (2.2) is gene alized Da boux linea izable i and only i
(2.7) is sa isfied wi h he Kjand Ljco ac o s o Da boux ac o s Fj
and Gj. The linea izing change o coo dina es is again gi en by
(2.8).
P oo : The p oo goes exac ly as in [MMR]. I ollows om he ac
ha (Z, W)=(F(z,w),G(z,w)) is a linea izing change o coo dina es i
and only i he unc ions Fand Ga e Da boux ac o s sa is ying DF =i
and DG =−i.
I u ned ou , in se e al examples o Da boux in eg able sys ems, ha
we could only find algeb aic in a ian cu es o Da boux ac o s so ha
one equa ion o (2.7) is sa isfied. In o de o cons uc he linea izing
change o coo dina es in ha case we p o e he ollowing lemma.
Complex Linea izable Cen es 101
Lemma 2.8. Suppose ha he sys em (2.2) has a fi s in eg al H(z,w)=
zw +o(|(z,w)|2), and ha he e exis (gene alized) Da boux ac o s Fj
such ha he fi s equa ion o (2.7) is sa isfied. Then a linea izing change
o coo dina e is gi en by
(Z, W)=
F0
j∈J1
Fαj
j,H(z,w)
F0j∈J1Fαj
j
.(2.9)
P oo : Le us call F=F0j∈J1Fαj
jand G=H(z,w)
F0j∈J1
Fαj
j
. Then
H=FG. Since His a fi s in eg al we ha e DH = 0. F om DF =iwe
hen deduce ha DG =−i.
Rema k 2.9.A common me hod o exhibi explici ly a fi s in eg al is
o use he Da boux me hod (wi h in a ian algeb aic cu es o Da boux
ac o s). I may happen ha all Da boux ac o s occu ing in he exp es-
sion o he fi s in eg al do no anish a he o igin. In ha case he
in eg al ob ained H(z,w) does no anish a he o igin. The heo em
has o be applied o he fi s in eg al H(z,w)=H(z,w)−H(0,0). E en
i His a Da boux fi s in eg al i may occu ha H, and hence Gis no
a Da boux unc ion.
2.3. A class o Da boux linea izable cen es.
Theo em 2.10. The sys em
˙z=iz +zn+awn
˙w=−iw +1
nzn−1w
(2.10)
has a linea izable cen e a he o igin.
P oo : Fo each sys em, we ha e he in a ian algeb aic cu es F1(z,w)=
w= 0 and F2(z,w)=z−iawn/(n+ 1) = 0, wi h co ac o s −i+
zn−1/n and i+zn−1 espec i ely. A hi d in a ian cu e wi h co ac-
o K3(z,w)=(n−1)zn−1is gi en by F3(z,w)=1+h(z,wn), whe e
h(z,W)=n−1
j=0 ajzjWn−1−j, wi h
an−1=−i, aj=−i(j+1)a
n(n−1−j)−jaj+1 j=0,...,n−2.(2.11)
The Da boux linea izing change o coo dina es is hen gi en by:
(Z, W)=F2(z,w)
(F3(z,w))1/(n−1) ,w
(F3(z,w))1/(n(n−1)) .
(2.12)
102 C. Ch is ophe , C. Rousseau
3. Linea izable complex quad a ic sys ems
Theo em 3.1. We conside a quad a ic sys em in C2:
˙z=iz +c20z2+c11zw +c02w2
˙w=−iw +d20z2+d11zw +d02w2.
(3.1)
The sys em has a linea izable cen e i and only i one o he ollowing
condi ions is sa isfied
Ic11 =d20 =d11 =0(3.2)
II c11 =c02 =d11 =0(3.3)
III c02 =d20 =0,c
20 −d11 =d02 −c11 =0(3.4)
IV 7c11 −6d02 =7d11 −6c20 =7d2
11 −12d20d02
(3.5)
=14d20c02 −3d11d02 =49d11c02 −18d2
02 =0
V2c20 −5d11 =2d02 −5c11 =15d2
11 +4d20d02
(3.6)
=6d2
02 +25c02d11 =10d20c02 −9d11d02 =0
VI c11 =c02 =d02 =0(3.7)
VII c20 =d20 =d11 =0(3.8)
VIII c11 =d20 =d02 =c20 −2d11 =0(3.9)
IX c20 =c02 =d11 =d02 −2c11 =0.(3.10)
In all cases one sepa a ix o he o igin is an algeb aic cu e.
P oo : To p o e he necessi y o he condi ions, we b ing he sys em (3.1)
o no mal o m
˙
Z=iZ +
j≥1
cjZj+1Wj
˙
W=−iW +
j≥1
djZjWj+1
(3.11)
up o e ms o o de 7 unde a change o coo dina es (z,w)=(Z+
o(|Z, W|),W +o(|Z, W|)). I he sys em is linea izable hen we mus
ha e a1=a2=a3=b1=b2=b3= 0. The compu a ions o ajand
bj, o j=1,2,3 we e pe o med in Maple and Reduce and a ac o ised
G ¨obne basis p oduced. The esul s we e checked ca e ully be ween
bo h packages, and yield he condi ions I–IX abo e.
The sufficiency o he condi ions is gi en below: o each case (3.2)–
(3.10) we gi e a linea izing change o coo dina es. All Da boux ac o s
a e no ed Fiand hei espec i e co ac o s Ki.
Complex Linea izable Cen es 103
(1) c11 =d20 =d11 = 0. Le us fi s suppose ha c20,d
02 =0. We
hen scale c20 =d02 = 1. (The case c02 = 0 co esponds o he Loud
sys em (S1) in he no a ion o [MRT].) The sys em has ou in a ian
lines:
F1(z,w)=wK
1(z,w)=−i+w
F2(z,w)=1+iw K2(z,w)=w
F3,4=1−iz +B3,4wK
3,4(z,w)=z−iB3,4w,
(3.12)
whe e B3,4a e he oo s o B2−iB −c02 = 0. The fi s in eg al is gi en
by
H(z,w)=1−iz +B3w
1−iz +B4w(1 + iw)i(B3−B4).(3.13)
Choosing H(z,w)=i1−H(z,w)
B3−B4=zw +o(|z,w|2), he linea izing change
o coo dina es is gi en by
(Z, W)=H(z,w)(1 + iw)
w,w
1+iw.(3.14)
We now conside he case d02 = 0 and c20 = 1 (a e scaling). Da boux
ac o s and co ac o s a e gi en by:
F1(z,w)=wK
1(z,w)=−i
F2,3(z,w)=1−iz ±wK
2,3(z,w)=z∓iw
F4(z,w)=ewK4(z,w)=−iw.
(3.15)
This yields a fi s in eg al
H(z,w)=e−2w1−iz +w
1−iz −w.(3.16)
We le H(z,w)= 1
2i(H(z,w)−1) = zw+o(|z,w|2) yielding he linea iz-
ing change o coo dina es
(Z, W)=H(z,w)
w,w.(3.17)
The case c20 =d02 = 0 is con ained in (7) below.
(2) c11 =c02 =d11 = 0 is dual o (1) unde (z,w, )→ (w,z,− ).
(3) c02 =d20 =0,c
20 −d11 =d02 −c11 = 0. In he case c20d02 =0
we can scale c20 =d02 = 1, yielding he Loud sys em (S2) in he no a ion
110 C. Ch is ophe , C. Rousseau
Con e sely suppose ha he saddle poin o he sys em is no in-
eg able. Modulo an analy ic change o coo dina es and di ision by a
locally non- anishing unc ion we can b ing he sys em o a no mal o m
˙x=x, ˙y=−y+Axkyk+1 +o(|(x, y)|2k+1)(5.2)
wi h A= 0. We will see ha his is an obs uc ion o find an in olu-
ion. Indeed i can be a gued ha such an in olu ion Thas a linea
pa o he o m T1(x, y)=(by, x/b), wi h b= 0. We look o he
in olu ion as a powe se ies T(x, y)=(X, Y )=(by +∞
=2 h (x, y),
x/b+∞
=2 k (x, y)), whe e h (x, y) and k (x, y) a e homogeneous poly-
nomials in xand yo deg ee . F om he hypo hesis we mus ha e he
ela ion ˙
X=−X. Howe e he e m in xkyk+1 o deg e 2k+ 1 in he
exp ession ˙
X+Xis always o he o m bA = 0, which con adic s he
exis ence o such a T.
Rema ks 5.2.(1) In ac he hypo hesis o he heo em can be elaxed.
Fo example, by conside ing he o de (2k+1)- e ms o equa ion ˙
Y
as well as ˙
Xwe can assume only ha he linea pa s o T(in
he expansion abou he c i ical poin ) a e in olu i e and ha he
ans o med sys em is equal o he o iginal sys em mul iplied by
some nega i e unc ion. De ails a e le o he eade .
(2) I is also possible o p o e he con e se pa o he heo em di ec ly,
wi hou using no mal o ms. We ske ch he idea below.
I T:(x, y)→ (T1(x, y),T
2(x, y)), hen we ake new a iables
φ(x, y)=(X, Y )=(T1(x, y)−x, T1(x, y)+x). We wan o show
ha in he coo dina es (X,Y ) he in olu ion becomes S(X,Y )=
(−X,Y ), wi h S=φ◦T◦φ−1. I we call R(X,Y )=(−X,Y ) he
symme y wi h espec o he Y-axis, his amoun s o showing ha
R◦φ◦T(x, y)=φ(x, y), which is a s aigh o wo d consequence
o he ac ha Tis an in olu ion.
We hus ob ain a new sys em wi h a co esponding e e sing
ans o ma ion S, and so he sys em mus be o he o m
˙
X=−Y−P(X2,Y),˙
Y=−X+XQ(X2,Y).(5.3)
Now such a sys em a ises om he sys em
˙
W=−2Y−2P(W, Y ),˙
Y=−1+Q(W, Y ),(5.4)
ia he ans o ma ion W=X2. Since his la e sys em is non-
singula a he o igin, i has a fi s in eg al H(W, Y )=W−
Y2+o(W)+o(Y2), which can be pulled back o a fi s in e-
g al K(X,Y )=X2−Y2+o(|(X,Y )|2) o he o iginal sys em.

Complex Linea izable Cen es 111
Theo em 5.3. We conside a quad a ic sys em in R2wi h a saddle
poin a he o igin wi h opposi e eigen alues
˙x=x+c20x2+c11xy +c02y2
˙y=−y+d20x2+d11xy +d02y2.
(5.5)
The sys em is linea izable a he o igin i and only i one o he con-
di ions (3.2)–(3.10) is sa isfied. The phase po ai s a e gi en in Fig-
u es 1–6.
P oo : The sys em (5.5) is ob ained om (3.1) by means o he ans-
o ma ion (x, y, T)=(−iz, −iw, i ).
We ha e a eal sys em whene e he cjk a e eal. As he condi ions
a e in a ian unde (x, y)→ (ax, by), he amilies I, II, VI–IX can be
educed o one-dimensional amilies, while he cases III–V can be educed
o 0-dimensional amilies. Howe e o p ac ical easons i is simple o
cases I, II, VI and VII o d aw he bi u ca ion diag am on one ou h o
a 2-sphe e.
Fo case I we suppose c2
20 +c2
02 +d2
02 = 1 and c20,d
02 ≥0. The bi u -
ca ion diag am appea s in Figu e 1. The case II can be deduced easily
om i .
d02 =0
c20
c02
Figu e 1
112 C. Ch is ophe , C. Rousseau
The phase po ai s o case III, IV, V appea in Figu es 2, 3, and 4
espec i ely.
Figu e 2
Figu e 3
Complex Linea izable Cen es 113
Figu e 4
Fo case VI we suppose c2
20+d2
20+d2
11 =1,c20,d
20 ≥0. The bi u ca ion
diag am appea s on Figu e 5. Case VII can be easily deduced om i .
c20 =0
d20
d11
d20 +d11 =0
d20 −d11 =0
Figu e 5
114 C. Ch is ophe , C. Rousseau
Fo case VIII we can scale d2
11 +c2
02 =1,d11 ≥0. The bi u ca ion
diag am appea s in Figu e 6. Case IX ollows om i .
c20 =0
c20 >0c20 <0
d11 =0d11 =0
Figu e 6
A na u al ques ion o us was o see whe e hese linea izable saddles
lie inside he s a a o in eg able saddles. The in eg abili y condi ions o
a quad a ic sys em wi h a saddle a he o igin fi s appea in he wo k
o Dulac [Du] in a case by case p ocedu e. They ha e been simplified
in [LL] and hen u he mo e in [Z]. The phase po ai s o in eg able
saddle poin s o quad a ic sys ems a e sys ema ically s udied in [DGS].
Theo em 5.4 ([LL] and [Z]).The s a a o in eg able saddles o (5.5)
a e gi en by
(A)c11 =d11 =0,
(B)d11 +2c20 =c11 +2d02 =0,
(C)c20c11 −d11d02 =c3
11d20 −d3
11c02 =0,
(D)2c11 −d02 =c20 −2d11 =c02d20 −d11c11 =0.
(5.6)
The s a um (A)consis s o sys ems gene ically ha ing h ee in a ian
lines, allowing a Da boux fi s in eg al (called Lo ka-Vol e a in [Z]and
ha monic in [LL]).
Complex Linea izable Cen es 115
The s a um (B)consis s o Hamil onian sys ems.
The s a um (C)consis s o sys ems gene ically ha ing an in a ian
line and an in a ian conic allowing a Da boux fi s in eg al. I is called
“symme ic” in [LL]( o easons o symme y in he calcula ions o
Lyapuno quan i ies) and e e sible in [Z].
The s a um (D)consis s o sys ems ha ing a Malkin fi s in eg al
gene ically cons uc ed om an in a ian conic and an in a ian cubic.
Rema k 5.5.When c11d11 = 0 i is no possible o find a symme y axis
in he usual sense and we mus in oduce a gene aliza ion o e e sibili y,
in he same way as exponen ial ac o s appea as he limi s o Da boux
in eg als.
The effec o app oaching hese limi ing cases wi hin he s a um (C)
is ha he eigen ec o s o he e e sing ans o ma ion coalesce and he
ans o ma ion becomes singula . Thus, in o de o esol e his difficul y,
we a e led o conside he sys em as a blow-up o a simple sys em.
Fo example he subs a um (C1)o (C):
˙x=x+c20x2
˙y=−y+d20x2+d11xy,
(5.7)
is a pa icula case o he ollowing heo em which has been ob ained by
an algeb aic me hod in [LL] (symme ies in he calcula ions o Lyapuno
cons an s).
Theo em 5.6. The ollowing class o in eg able saddles lies in he clo-
su e o he s a a o e e sible sys ems:
˙x=x+P(x, y)=x(1 + p(x, y))
˙y=−y+Q(x, y),
(5.8)
whe e we ha e ei he
P(x, y)= 
i>j
i+j≥2
cijxiyj,Q(x, y)= 
i>j−2
i+j≥2
dijxiyj,(5.9)
wi h c +1, +d , +1 =0, o he conjuga e sys em unde (x, y, )→
(y,x,− ).

116 C. Ch is ophe , C. Rousseau
P oo : Taking X=xand Y=xy, we ob ain he sys em
˙
X=X1+pX, Y
X
˙
Y=YpX, Y
X+XQX, Y
X.
(5.10)
The condi ions imply ha a common ac o o Xcan be emo ed o lea e
an analy ic sys em wi h a non c i ical poin a he o igin. I he e o e
has a local fi s in eg al Φ(X,Y )=Y+o(|(X,Y )|) which can be pulled
back o a local fi s in eg al o he o iginal sys em, Φ(x, xy)=xy +
o(|(x, y)|2).
Rema ks 5.7.(1) In he co esponding case o eal cen es, such sys-
ems do no a ise excep in he i ial case
˙x=−y1+αi(x2+y2)i
˙y=x1+αi(x2+y2)i.
(5.11)
(2) I would be in e es ing o see i he co esponding no ion o a limi
o a ional e e sible sys ems would also gi e some new in eg abil-
i y condi ions.
P oposi ion 5.8. The linea izable saddles desc ibed in Theo em 5.3
lie in he s a um (A) o he cases I–II, he s a um (C) o he ca-
ses III–VII, and he s a um (D) o he cases VIII–IX. In he la e
case he in a ian cubic is educible, yielding a line h ough he o igin
and a conic.
Theo em 5.9. We conside a cubic sys em in R2symme ic wi h e-
spec o a saddle poin a he o igin wi h opposi e eigen alues
˙x=x+c30x3+c21x2y+c12xy2+c03y3
˙y=−y+d30x3+d21x2y+d12xy2+d03y3.
(5.12)
The sys em is linea izable a he o igin i and only i (4.2) and one
o he condi ions (4.3)–(4.9) is sa isfied. The phase po ai s appea in
Figu es 7–12 (only he gene ic cases).
P oo : Fo case I we suppose c2
12 +c2
03 +d2
03 =1,c03 ≥0, which yields
a hal 2-sphe e. The bi u ca ion diag am appea s in Figu e 7. (Case II
is dual.)
Complex Linea izable Cen es 117
d03
c12 =d03
c12 =−d03
c12
c03 =0
Figu e 7
Fo case III we suppose c2
30+c2
12 = 1. The bi u ca ion diag am appea s
in Figu e 8.
c12
c30
Figu e 8
118 C. Ch is ophe , C. Rousseau
Fo case IV we suppose c2
30+d2
03 = 1. The bi u ca ion diag am appea s
in Figu e 9.
c30
d03
Figu e 9
Case V is ze o-dimensional. We ha e he condi ions c30c12 <0,
d30c03 <0, c30d21 >0, c12d03 >0 which yields, a e scaling, he wo
cases
˙x=x+7x3−3xy2−4y3
˙y=−y+4x3+3x2y−7y3
(5.13)
and
˙x=x−7x3+3xy2+4y3
˙y=−y−4x3−3x2y+7y3.
(5.14)
Thei espec i e phase po ai s appea in Figu es 10 and 11.
Figu e 10
Complex Linea izable Cen es 119
Figu e 11
Fo case VI we suppose c2
03 +d2
21 = 1 and c03 ≥0. The bi u ca ion
diag am appea s in Figu e 12. (Case VII is dual.)
c03
d21
Figu e 12
Theo em 5.10 ([LL]).The s a a o in eg able saddles a e gi en by
(A)d21 +3c30 =c21 +d12 =c12 +3d03 =0,
(B)c21 =d12 =c30 −3d21 =3c12 −d03
=3c03d30 −4d21d03 =0,
(C)c21 +d12 =c30c12 −d21d03 =c2
30c03 +d30d2
03
=c2
12d30 +c03d2
21 =0.
(5.15)
The s a um (A)consis s o Hamil onian sys ems.
The s a um (B)consis s o sys ems ha ing a Malkin fi s in eg al
cons uc ed gene ically om an in a ian qua ic and an in a ian sex ic
which gi e a a ional fi s in eg al.
The s a um (C)consis s o e e sible sys ems (called symme ic in
[LL]), possibly in he gene alized sense o Theo em 5.6 o in he sense
below.