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Analytic Classification of Resonant Saddles and Foci
109
A N A L Y T I C C L A S S I F I C A T I O N O F RESONANT S A D D L E S A N D F O C I
Jean Martinet
&
Jean-Pierre Ramis
Universite de Strasbourg France
In this work, we apply our results of [17] to the study of resonant analytic saddles or foci in the real plane: we classify exhaustively, up to local analytic transformations of IR 2 , the phase portraits near zero of the differential systems: R = qx ( 2 ) Foci x = y +... (1) Saddles 3 =-py +. . . y = -x +... (p,q are relatively prime positive integers and, in both cases, the dots denote convergent series of order at least two) The I L e h o n a n c e means that, at first order, the phase portraits are 2 defined by polynomials (xpyq for saddles, x +y2 for foci). But, in general, as is well known, ( 1 ) or ( 2 ) are not tineahizabtc; in case (2) for instance, linearizability means that one has a center instead of a focus; in this case, a finite number of tests on the Taylor series provides an o h d e h k at which the system "essentially" departs from the linear one. We deal here only with such systems, which are often called " u e u k " nadd1e.o o h h v c i 0 6 ohdetr k ; the adjective "weak" makes Qbvious sense for foci: the integral curves spiral to the singular point very slowly, in contrast with "strong" foci (non resonant ones), for which they approach it exponentially fast; one can give a similar interpretation for weak saddles, by considering the complexification of the system. To get a better understanding of the analytic structure of such singular points may be an important step in some problems about limit cycles: weak foci and singular cycles (made up of separatrices joining weak saddles) seem to play a prominent role in the global theory of algebraic differential systems in the plane, as "organizing centers" of major bifurcations. Anyway, resonant singular points have been studied for a long time starting with Poincar6, Liapounov, Dulac in particular. Recently, Roussarie [21] and Takens [22] have solved the classification pro-
+...
1
110
J. Martinet and J:P. Ramis
blem in the differentiable case, for saddles and foci respectively: resonant systems which are formally equivalent are differentiably equivalent. The analytic situation is quite different, as was pointed out by Bryuno [5] . We have solved in [17] the classification problem for resonant systems in the c o m p l e x p l a n e C2. We describe the (germ of) "leaf space" C of a resonant system: if kE IN is the order, C is a non-Hausdorff complex analytic manifold of dimension 1 ; it is a "necklace" formed by 2k "beads"(each one a Riemann sphere), each bead sticking to the next one by means of a g u m of analytic diffeomorphism; thus C may be characterized by 2k germs of diffeomorphisms of ( C , O ) , the attaching maps I$i. Resonant systems having the same formal invariants are classified analytically by their leaf spaces. NOW, if we consider the complexification of a h e u L resonant system, the antiholomorphic involution a (a(x,y) = endows the leaf space C with an antiholomorphic involution that we denote also by a . For a heu8 h u d d l e , the involution 0 exchanges two halves of the necklace C , leaving fixed two of the "attaching points"; if one has a (i)=j for two attaching points i and j , then U@ia = $I -1 . where I$ i and 3 $j are the attaching diffeomorphisms ( 0 reverses orientation along the necklace): C is then characterized by k+l attaching maps. The "space of real leaves" corresponds to two germs of real curves, each one at a fixed attaching point. For a ueuk d o c u h , the involution a still exchanges two halves of C; no attaching point is fixed, but two of the beads are globally invariant. We still have a ~ $ ~=a @-' if a ( i ) = j : C is now characterized by j k independant attaching maps. The space of real leaves corresponds to the "equatorial circle" of one of the fixed beads. In this case, an interesting consequence is the following fact: u r ~ yheue u n a P y t i c gehm p (r)= -r+ ,r2k+ +... ' (a# 0 ) is the "Poincar6 map" of a weak focus of order k (i.e the map defined on a ray through 0 by a half-turn along the orbits of the system); its conjugacy class characterizes the weak f o c u s , up to analytic equivalence.
(x,?))
The paper is organized as follows. In the first part, we recall in some details the 6ohmuk classification of resonant diddehentiuL d o h m b : its features are quite different from the analogous theory for vector fields, and interesting by themselves; in particular, the normal forms appear very naturally as simple mehamofiphic c k u n e d 6ofirn.b (see 1.3).
Analytic Classification of Resonant Saddles and Foci
111
The second part is a r&sum& of the main results of [17] ,explaining the nature of the analytic invariants of resonant complex systems. The last two parts are devoted to the study of saddles and foci. We cannot end this presentation without mentioning the papers of Markhashov
[lo]
[ll]
[121 [13] [14] [15]0n
resonant analytic saddles or
foci in the real domain. The results of Markhashov differ completely from ours: he proved in L l O ] Ll2] Ll3] L14] [15] that an analytic focus is always trivial (i.e analytically reducible to its normal f o r m ) ; this, of course, is false: beside the present paper, f o r a critic of Markhashov's proof, see Bryuno [ 4 ] p. 727-728, 131 p .848, [5] appendix (it seems that Basov 121 was the first to prove the divergence of the normalization for certain foci). For saddles, Markhashov claims an analytic classification in LlO][ll]. ned a hOhrrla8 one (Bryuno [3]
It seems that he has only obtai-
p.848). In [11J
he asserts that any real
a i z u t q f i . c saddle is analytically equivalent to an a d g e b h a i c one; in
fact the problem remains open (and we think that this assertion is probably false). 1. CANONICAL NORMALIZATION OF RESONANT DIFFERENTIAL EQUATIONS
TRANSVERSE F I B R A T I O N S 2 We start with a germ of equation at 0 in a: : (1.1) w = py(l+. ..)ax + qx(l+. ..)dy = 0
1.1.
(p and q are relatively prime positive integers); the separatrices of this equation are the coordinate axis. In 1923, Dulac
181
shows that, given arbitrary large positive inte-
gers r , ~ , there exists analytic coordinates in which (1.1) becomes (up to a unity): r s w = py(l+P(u)+x y a(x,y))dx + qxdy = 0 (1.2) where
<
u= xpyq, P is a polynomial of degree n, P(O)=O, r >pn,s> qn, and a is an analytic function. This result can be proved easily,
using an identification process. If, for any r,s, one gets P=O, equation (1.1) is 6ohrnaLBq B i n e a h i In this case, it is well known that (1.1) is a n a l q t i c a l X q R i -
zabLe.
.
n e a h i z a b l e (Bryuno [6] ,Mattel-Moussu [18] ) Thus, we are going to consider the situation in which P# 0 for some r,s. In this case, we
start, using only the first non zero term in P , with an equation: k (1.3) w = (pydx + qxdy) + u ( a +xya)ydx = 0 a # 0 , u = xpyq The integer k 2 1 will be called the vhdeh of w .
112
J. Martinet and J.-P. Rarnis
Let us give a geometrical meaning to ( 1 . 3 ) . Notice first that: k (1.4) w/\ (pydX+qXdy) = qXyU ( ~ 1+xya)dxhdy,
2
This means that the local foliation (by complex curves)% of C -{Ol defined by (1.3) is, outside the separatrices {x=O] and { y = O ] , R h u n 6 v e h h e to the "linear foliation" pydx+qxdy = 0 (u= cst); the integer k represents the "order of contact" of these foliations along their common separatrices. We are going to study the pairs ( w , T I ) , where w is of the form (1.3) and 7~ : C 2 -t (I: is the h i n g u P a h h i b h n t i o n defined by: n (x,y) = u = xpyq
R e m m k I. I. I . The fibration
5
TI, which is transverse to in the sense of ( 1 . 4 ) , is not canonical. Let us look under which condition a function
v = xpyqexph(x,y) will define a fibration having a similar property. We have, in view of (1.3):
+ qdy/y + dh) k =(pydx + qxdy)Adh + u ( a + xya) (q + yah/ay)dx,,dy This equality will be of the form (1.4) if and only if: k (pydx + gxdy)Adh = XYU b(x,y)dx,jdy
w/\dv/v =wn(pdx/x
This means that h(x,y) = Q(u) mod xyuk, where Q is a polynomial in u of degree k. Thus, t h e l j i b h a t i o n n in d e t e h m i n e d m o d u l o xyuk , a n d may be. changed a h b i t h a n i l y a t h i g h e h o t i d e h . 1.2.
I N T E G R A T Z N G FACTORS
From this point, we shall need to use ~ o h m a L Idiwehgentl
heti-Leh
i n
t u o v a t i i a b l e h . They will be of a particular type that we describe
first.
6~
as the k i n g o h hotimal b e k i t h = c a xryS r,s> o buch .that a l l t h e b e h i e n i n o n e v a t i i a b l e C a xr and C s a r r,s X eZhve.age o n ( x ( < p(yI
o ( p may d e p e n d o n $ 1 . We define
@[[x,y]]
i
I S
ys
Now, if we fix two relatively prime, positive integers p,q, it is easy to check that any
:=
E:
2
can be written as: fn(x,y)u" (u = xpyq)
1 n >O where the functions fn are analytic on a common polydisk in C 2 . (The coefficients fn are not unique, but there is a canonical choice, if one factors out the powers of u = xpyq in
?.)
Analytic Classification of Resonant Saddles and Foci
P R O P O S l T l O N 1 . 2 . I . T h e u n u l y t i c 6uhm (1.3) w = [ p y d x + q x d y ] + u k la + x y a ) y d x k hu6 u iuhmu& i n t e g h u t i n g 3uctu4 xyu e x p ( - i ; ) ,
i=
h
phuvided t h a t
6
113
# 0)
2E
i.
It
i b
unique
B(OJ-0,u n d o n e h u h :
A
=
P l u ) m u d wgu
k
w h e f i e P i n u p u l y n u r n i u l ud d e g 4 e e k i n u . h
Phuud.
Recall that F is an integrating factor of a differential A
form w if
w/F is a closed form (Cerveau-Mattei [17] )
. From
(1.3) we
get:
w/xyuk= du/uk+l + cxdx/x + w ' where w ' has analytic coefficients vanishing at 0. Hence F=xyuke is an integratiQg factor if; drefdu/uk+l + efadx/x
-;
n
+ e fw l ] =
0
which is equivalent to:
k h k (1.5) dffi(pydx+qxdy) + ~ Y Udfr\dx +XYU (dw'+df,jw') = 0 r s Setting f = r,s C br,sx y , ( 1 . 5 ) gives a system of linear equah
A
h
.
tions in the coefficients b
-
Notice that: r,s If (r,s) is not a multiple of (p,q), (1.5) determines b
from r,s bi . with i
These remarks suffice to prove that f is unique and that: k f = P(u) mod xyu A
r To show that fEB, it is enough to notice that the series r C br,sx and brrsys are solutions of 4 e g u L a k diddehential Q Q u a t i o n b h
A
u d u h d e h o n e ; an easy induction proves that these series have a common disk of convergence. Now, consider again a differential form (1.3), and write its integrating factor F as: G = xyuk/(l+Q(u)) (l+xyuk;)
A
h
where Q is a polynomial of degree ,
A
i'
is a c L o b e d r j u h m , w i t h c o e r j d i c i e n t b i n 8 ( w ' has no polar part);thus n h n w ' = dH HEB, H ( O ) = O h
h
We reformulate this, with obvious changes of notations, in the following:
J. Martinet and .I.-P. Ramis
114
COROLLARY I . ~ u c . ~ u . '?. I
w/f
(1.6)
with
1 I . 3 1 , w i l l 7 ii.ite.ghuting
L c R w b e u n u i i u L y t . i c 4ohm
2. 2.
Then: il+Q(u) J d u / u k t ' + adxlx
=
Q(U) 'CiIU+
. . . t" b - I u h - l ;
a,B
Et;
Bdy/y
+
+
p B - q a # 0,
d;
M, Aio,=
0.
I . 3 . NORMALIZATION
We are going to put (1.6) into a normal form, keeping the fibration TI
h
unchanged. From now on
we shall denote by Dn the g k u u p
I
04
dUhmUe
ttiunh (iokrnutiollb : A
$1
XI=
x exp q$(x,y)
i(O)= 0
$Eii,
A
Y'= Y exp-p@(x,y)
h
These transformations are characterized by the properties n,O and D$(O)= Id. We shall denote by u HULy t i c d i6 3 (10vi p h i n m b
.
Now, consider the form ( 1 . 6 )
=
n,
D T c E n the subgroup consisting
of
and the formal transformation: h
A
: (x,Y)
h
(x exp @,(x,Y) l~ exp @2(x,y)) where $ and 42 are the elements of 6 defined by: 1 and p&l + q;, = 0 a i l + B i , = fi It is obvious that: O
+
, w/F ,
L
A
with wo= (l+Q(u))du/uk+l+ndx/x+Bdy/y. Summarizing the results we have obtained up to now, we have the following: TO;
= IT
TIfEOREIM I . 3 . I .
=O
wo
L e t w be a
henorlUi.lt
w i R h u thuYi6wehAC 4ibkution (41
w huh
(ii)T h e w0=
Cl
UvlUeytAc
dohm 0 3 o h d e h h ,
togotlleh
T .
Unique inttghating
A
dUCtOh
h
FEE.
h
l w / F , ? r ) muy b e viohrnalize.d i n t o : i I + Q l u l J d L ( / u k + l +adxlx + B d y / y , I T l x , y l - u
pUih
(lJ@-qCi # 0 )
b y rne.u116
0 4 u unique nohrnalizing
=
x P yY
@ED . A
h
fhUnb~O,ltTIU.t~On
,.
The unicity in (ii) is obvious: the identity mapping is the unique element of D which preserves the above normal form4 The meromorphic normal form o may be further simplified, in the 0 following way: 1 ) The differential equation:
( l + Q ( u ))du/uk+l= dv/v k+1
defines a unique holomorphic function u (i.e v = u exp +(u), tes in c 2 : Y
:
(x,Y)
$lO)=O).
+
+
v(u) , v(O)=O, v' ( 0 ) = 1
Then, the analytic change of coordina-
ix exp i l ( u ) ,Y exp $ , ( u ) )
Analytic Classification of Resonant Saddles and Foci
where aQl+ 09, = 0 transforms obviously w
0
PQ1+ 49.2
and into:
115
= @
dv/vk+l + adx/x + f$dy/y and preserves the fibration n (but moves the fibers: n o Y = 2) The complex numbers
ci
and
a
V
O
~ )
.
are the t i e h i d u e h of the closed form
on the separatrices {x=O} and {y=O}. Actually, only the ratio wo B/a is geometrically interesting: through a linear change of coordinates (x,y) (ax,by), one may always reduce (in the complex domain)
-
to the case where
PO-qa = 1.
In the sequel, we shall be interested in the pairs ( F u , n ) , where the local foliation of ( C 2 , 0 ) defined by a resonant equation
7, is
In view of the above results, w e 4 h U l e c o n b i d e n o n k y t h e a n a k y ,tie p U 4 h b ( 7 0 . n ) w h i c h [email protected] ~ J L I ~ I u Pt rP e~d u c i b l e , thfiiuugh a u n i q u e e k e o
=O.
vn' A
I1leH.t 0 5
" . f )
RO
I
Olle
04
Rke
1(04t?14:
riohnlut
: wo = p q d x t q x d y n :nix,y)=u=xr~yq.
+
u b [ a y d x + Bxdy)
pa-qa # 0
Remunkh . 1) If the integrating factor
of a resonant form is analytic, then
the normalizing transformation $ is itself analytic: this is obvious from our computations. This fact was first noticed by Cerveau-Mattei ( [ 7 ] ,p.129-134), with a different approach.
2) It is interesting to compare the formal normalization of resonant forms w and resonant vector fiels X. In the latter case, the key is the (formal) Jordan decomposition:
x = $ + i h
[&i]
= 0 h
where S is a semi-simple (diagonalizable) vector field, and N a nilpotent one. In the former case, as we have seen, the key is the (formal) decomposition : _
A
w = F.w
1 ,. where F is the integrating factor, and w1 a meromorphic closed form. h
1.4. A P P L l C A T l O N T O R E A L R E S O N A N T V l F F E R E N T l A L E O , U A T l O N S We consider a re'al analytic differential equation, defined in a neighborhood of 0 in R 2 (x,y), as a compEe.x e q u a t i o n w h i c h in i n u a h i a n t unden t h e (antihokornofiphic) iviuukutioii u: ( x , g ) * If w =O is a real resonant equation, the involution a must transform each separatrix into a separatrix. Therefore, two cases are to be considered. S a d d l e h : e a c h h ~ . p u h u t t h i x i h a-inwahiant ( i . e in a h e a L c u h u e ) .
(x,;].
J. Martinet and J.-P. Ramis
116
The linear part of o may be written, in real coordinates, as PYdX + V d Y (there is no restriction on the values of the integers p,q) We consider only saddles having a finite order k (i.e which are h not linearizable). The integrating factor F, being unique, is h e a l . Moreover, we may choose a beak t i r U n b W e h A t dibhution IT: Dulac’s theorem quoted in 1.1 shows the existence of a complex fibration; it is real modulo ~ y ( x ’ y ~ )(being ~ unique modulo this ideal, following Remark 1.1.1); as it may be modified arbitrarily inside this ideal, it can be choosen real. We then proceed as in 1 . 3 , and obtain: E a c h a u z a k y t i c p n i h ($o,n)
dibhatiouz
1% b
wo=
w i t h a thannwehbe
t h h o u g h a u n i q u e h e a l ;cCIT, i n t o a h e a k n o k -
IT) t h U V I n d O h m b ,
:
( a b a d d k c o 6 o h d e h h,
i l + O , l ~ l l( p y d x + q x d y l + u k [ a y d x + a x d y ) = 0
lTJx,yl= u
a polynomiak
=
xpyq
C J ~d e g h e e
as in 1.3, a (real) change of coordinate in the basis of n , and (real) linear transformations (x,y)--f (ax,by), we may assume that, in the normal form (1.8): Q = 0 , pf3-qCr‘l (k odd), pa-qa=+l (k even)
F o c i : t h e n e p a h a t h i c e b ahe e x c h a n g e d by u . The symetry u implies that p=q= 1; the linear part of o is: xd:: + ydy in a convenient real coordinate system: the singularity is either a c e n t e h o h a d o c u b ; infinite order corresponds to a center (the result of Bryuno quoted in 1.1 is a generalization of the famous theorem of Poincare-Liapounov); we consider only equations of finite order k, that is weah a o c i 0 6 o h d e h k . It is of course convenient to use complex coordinates (E,,q) such that the (complex) separatrices are c = O , q = O , the involution being defined by u ( < , n ) = ( t , t ) :the real plane is then q = E . Using the same argument as in the previous case, one proves that a weak focus admits a real transverse fibration r(it is a Liapounov function). The integrating factor being real too, we obtain as in 1.3 Each a n a l y t i c p a i h
(Fu,~J ( a weab
docun
04
u h d e t l h, w i t h a h e a l
thclMbVQhne 6 i b h a t i o n ) t h a n b d o h m n , t h h o u g h a u n i q u e
n u h m a C b,ohrn:
Z E ~ i~n t,o
a heal
117
Analytic Classification of Resonant Saddles and Foci
As in the previous case, we loose no generality in assuming that Q=O. Moreover, by using linear transformations ( E r n ) -+ (at, a n ) , and possibly (C,n)- ( q , E , ) (these are real transformations), we may assume that: a-a = i With these simplifications, the normal form ( 1 . 9 ) reads, in real coordinates (x,y) (E, = x+iy) : k k o0= (1+ AU ) (xdx+ydy) + (xdy-ydx) = 0 2 2 2 TI(x,y)= u = 5 : = x +y h = Rea .
70:
2. ANALYTIC INVARIANTS (COMPLEX DOMAIN) 2. I .
INTRODUCTION
In part one, we have described the 60htnaL i n w a h i a n t b (p,q,k,a,B) of a pair (Tw,.rr), where w is a resonant form of finite order k, and n is a transverse fibration. Here, we consider the set of analytic pairs (Fo,a) with given formal invariants, and we intend to classify these pairs up to analytic diffeomorphisms of ( C 2 , 0 ) ; that is, we look for their a n a e y t i c inwahinntb . Using the results of section one, we may formulate this problem as follows. Fix a normal form: k : wo= pydx+qxdy + u (aydx+Bxdy) = 0 (pB-qay= 1 ) (1.7) O' n(x,y)=u= xPyq
1
rv
Consider the analytic pairs (C ,TI) (a being the same as in ( 1 . 7 ) ) w which are D -equivalent to ( 1 . 7 ) ; each of these corresponds to a uTI A h nique normalizing transformation 0 ED : h
WAQ, A
.
0
0
= 0
o
n
(caution:Q o is not, in general, an analytic form, but only the w o product of an analytic form by a formal series h, with h(O)# 0 ; see 1171 . ) and (Fw,,.rr) are analytically equivalent Obviously, the pairs if and only if: (the group of a n a l y t i c d i d d e o m o h p h i b m b ) $ i l o i w , E Da Thus, our problem will be solved if we are able to c h a h a c t e h i z e t h e clacldeb i . V , ( i . e e l e m e n t n 0 6 t h e q u o t i e n t bpUCe ~ , f D n l w h i c h have t h e p h o p e h t y : ;-'(To] i b an a n a l y t i c d o l i a t i o n . The solution we give to this problem rests upon a very natural h
(5,~)
F=
h
118
J. Martinet and J.-P. Ramis h
"geometrical" interpretation of the elements of D I which we are going to sketch now; the following ideas generalize a viewpoint which is classical in the analytic theory of differential equations; for more details, see [l6] and [ 1 7 ] . 2 . 2.
F O R M A L DTFFEOMORPHTSMS A N D C O C Y C L E S A
~n
n
We recall that O E D ~is defined by (x,y) + (x exp q$,y exp-p$),$eB, i.e $ =C @,(x,y) (xPyqIn =C $ un , where the $n are analytic on a com11
mon polydisk A c C 2 ( A will be fixed, for simplicity, throughout this paragraph). h
We shall consider systematically the elements of B as the h e b t h i c t i o n n t o t h e A U I L ~ ; U C C X ~ t ( u J x C ~ ( x , y d)e,d i . n e d b y u = x r J q q , 0 6 rjotimuL 2 b e h i e 4 . i n u w i . t h c o e d d i c i c n t s u n u Y y R i c o n Act ( x , y ) ; in this context, IT will denote the projection (u,x,y) -+ u from C3 to C 2 . In particular every @ E D will be thought of as the restriction to C of the formal diffeomorpEism (u,x,y) + (u,@(u,x,y)) of C3 along { O i x A . NOW, we sketch the main ideas. 1)Sectorial diffeomorphisms. Let u = { U E C ~Iu/
h
-
4
A
-4
B(U) -P B 9 'L where B ( U ) is the ring of functions which are analytic inside U, and 21 are Cm along {OIxA (a part of the boundary of U).
Example I . The interested reader should do the following fundamental exercise; show that the integral:
(L # R+ is a half-line going from 0 to
a)
defines an element of B(U) (U = half-plane with bissector L), and its Taylor series at 0 is $ =C n!un (this integral is the Borel's sum of the series A
n
4)
We have $= r$' if $ - $ I is L n d L n L t e L y ddut a l o n g {OIxA: it vanishes, with all its partial derivatives with respect to u , for u=O. We denote by
B m ( U ) C B(U)
the ring of BPuL d u n c Z L o n 6 .
Now, we define DT(U) as the g h o u p (under composition of mappings) of transformations:
o
:
(u,x,y)E~ + (u,x eq',y
e-p')
9 EB(U)
Analytic Classification o f Resonant Saddles and Foci
119
(actually, to consider Dn(U) as a transformation group, one must allow the radius and aperture of U to vary slightly, but this is an easily overcomed technical difficulty) Similarly, D " ( U ) C D , ( U ) will be the subgroup defined by the condition @ E B ~ ( U ) ,i.e 0 is i i i d i n i t e L y 6Put w i t h h e s p e c t t o t h e identity i i i a p p i n g a t u big { 01 x n .
Obviously, we have: A
Dn = D n ( U ) / D Y ( U ) h
Thus, we may consider a formal 0 as the Taylor series along {OlxA of a "true transformation" of a substantial domain: we have given some flesh to a g h o s t ! 2 ) The characteristic cocycle
of E; Dn. Cover a neighborhood of O E C ( U ) with a finite number of sectors U o l
... (UonU,# 0 , U , n U 2 # 0 , ..., but the intersection of any three sec tors is empty, putting aside 0).We fix such a covering Given Q E D ~ ,write O= Oo= O 1 =..., where Oi€DT(Ui) is defined up -1 to composition by an element of D;(Ui). The transformations O 0 o O 1 ,
U1,
,.r.
h
u.
h
.. .
. -1 , @ 1 ~ ~ 2
belong obviously to D ; ( U 0 q U 1 ) , ne u ? e c ~ ~ - c o c y c t e :
, . . . : they defi-
D ;( U 1P U 2)
c ( 0 )E C1 (u;DT) If one chooses other representatives 0; of @ , the cocycle c(0') will be cuhumuPoguub to c(@), because they are related by an element h
de.temnineo u u i i i q u e e L e m e n t 0 6 t h e
of C o ( u ; D ; ) . This means that c o h u m a ~ o gy d p u c c :
pi
E
H~ (U;D;)
Notice that this is ~ i o n - u b e P i u n cohomology: this construction is similar to those of bundle theory.
It is easy to see that [i]=[i'] if and only if @ ' = O o Y , Y E D n ; n this means that each class @ . D n C 6 n is c I i u k u c t e t i z c d by an element A
1
of H (Il;D;), which we shall call the c h u h u c t e h i d t i c
h
cLc(dd
(or cocy-
h
cle, with a slight abuse) of O . D n . We have proved in [ 1 7 ] the fundamental result:
Euch ePement
06 H 1 (U;U;) id
the chntactexistic
c l u ~ d0
4
un
eLe.ment
A
06
QTIDTT.
ExumpPe 2. Consider the covering consisting of two sectors Uo=C-R U
1
=C-!R-.i, and F E D T ( U o n U l )
(n(x,y)=xy) defined by:
F: (u,x,y) -t (u,x exp f (u),Y exp-f (u)) f (u)=exp-l/u if Re u > O , f (u)=O if Re u
+ .i
J. Martinet and J.-P. Ramis
120
( L . = OA. is a segment, Re A . > O is very small, Im AoZO and Im A l < O ) 3 3 7 defines 0 .ED,- (U.): (u,x,y) + (u,xexp $ . (u),y exp-$] (u)) such that: 3
3
Show that
1
Oo~Q;l
=C
=
n!unW
F
I
u
Finally, we have, for each covering of ( C , O ) by a finite number J of sectors, a uiatuhal i d e n t i a i c a t i o n 0 6 D , / D n w i P h H (‘?,(;U;). This is the main tool for solving our problem about foliations. 2 . 3 . ANALYTIC
h
INVARIANTS
We go back to the data ( 3 , , n ) considered in 2.1. Our main result can be stated (loosely) as follows: A-1 Given Q E D ~ , 0 (To) is an analytic foliation if and only if, for a convenient covering ‘L((which we are going to describe), 0 is charac1 terized by a cocycle in C (I(:DY) which preserves the foliation (this means that the cocycle preserves the restriction of yo to each domain UonUl, U1r)u2 , . . . ) A
h
h
yo
1) T h e
6oliutiun
%.
.
2. -1 2 We describe it on a domain U =IT ( U ) C C (x,y), where U C C ( u ) is any ’L sector with vertex 0: each leaf of in U (we put aside the separatrices x=O, y=O, which lie in n -1 (0))is a one-to-one covering of U through ,-, as ‘To andn are transverse. More precisely, write: ojo/xyuk = du/uk+l+adx/x+6dy/y = ( 1+Xuk ) du/uk+’+mdx/x+ndy/y
yo
(where m,n are i n t e g e h h , pn-qm=l,)i =na-mB : these choices are consistent because we assume (see 1.3) pB-qa=l;X mod 2 does not depend on the choice of m,n) It follows that is defined by the aihnt i n i e g h u l : H(u,x,y) = xmynK(u) with K(u) = u x exp-l/ku k These definitions depend on U in general, because ux requires the choice of a determination of Log u on U (we often use the principal determination). 2. The l e a 4 H = C E ~has then, for each UEU, a unique point [x,y)sU such that n(x,y)=u: it is determined by xpyq=u, xmyn=c/K(u) (recall that pn-qm=l ) A transformation F: (x,y) + (x exp q+,y exp-p+) preserves if and only if HoF is a function of H (this means that $ is a function of H).
TO1;
.
Fo1G
Analytic Classification of Resonant Saddles and Foci
D F ( U ; Z ) C DG(U) consisting of transformations which preserve
121
yo;each
one is unique-
ly defined by a relation: f(z)= z+a2 z
where
2
+...
gent to the identity. k Similarly, if Re u
-1 .
II,F
= f,H
is any local diffeomorphisrn of C at 0, tan-
on U , we have an analogous result, using
instead of H. k If U contains a line on which Re u =0, it is easily seen that Y I U e l ' e n i e u R 06 D (U) ( 5 t a - t 0 4 I I O ~ ] p h e o e h w e n b u t t h e h d c n t i t y rnapphiig. k The lines ; e u =O will be called the S t O i . , e b L i n e n 0 4
3,.
To.
2 ) Co~lcluniOn5
%(('To)as
the covering o f C(u) such that the sectors U0flu1 , are lim ted by two consecutive Stokes lines o f 7;. For ins+. tance, if k=l, it consists of the two sectors [email protected] and U 1=C\R.i; in general, it is made up of 2k sectors of aperture 2n/k. Define
U1"U2,.
.-
It is now possi le to state precisely our result about the analytic classification of the pairs
(FW,71 see );
[17].
notrrncie 6ohm ( 1 . ? ) , 1 ( = 1 { () F t h e T H E O R E M 2.4. L e t ( f o , r ] b e O p o n d i v l g couefiing o h C ( u ) . ( 4 ) one hun: H~ i24;~~i';F~I I = C' 171;~;i3~i
CUhhCb-
Each c o c y c t e i n d e 6 i n e d b y 2 h t o c u P a n a l y t i c d i 6 6 e u m u t r p h i h r n ~ 0 6
(z,~)
C a t 0, tungciq-t t o . t h e i d e n t i t y . ( i h l T h e h e c o c y c C e h c L a h h i d y ahe p a i t r n up t w a n a e y t i c cquiwal e n c e : g i v e v i '3, t h e c h a f i a c t e f i i b t i c cLu6h 0 4 i.tn n u f i m a L i z i n g wn h t h a n h A o n r n a t i u n 0~ DT i 6 h e p f i e b e n t e d b y a u n i q u e c o c y c l e in C 1 (V;D:(Fo)) ; e v e h y c o c y c l ~d e d i n c n U M u n d y t i c 4oPiation Ju, up t o a n a P y t i c e q u i v a l e n c e .
x(c2,. ..
This foliation is obtained, geometrically, by gluing
To1 -Uo
to
%l?ll, T I C l to : the transition mappings are the terms of the cocycle. This viewpoint enables us to describe the "leaf space"
as a "necklace" of 2k Riemann spheres (see [17] 2.5.
SUMMABILITY
(To,n) ; let
and the introduction)
..
(IA;D;(F~))
F = ( F ~ , F ~ -,1 E c l be such that[$]= be one of the corresponding cocycles; let
We still fix a normal form
A
h
F,
J. Martinet and J.-P. Ramis
122
’f= Qic.Dn (Ui),
is a u n i q u e family O = ( a O , Q l ,... ) ,
(To). Then, there
A-1
and set
@
such that: A
h
and Moreover, [@]= F means that:
o I ~ ( F ~ ~ G ~ )
@ , =@
A
1
-1-
-1
- Fo,
OOoa1
=
F1,
...,
%
%
%
%
On u o / l ~ 1 t ~ l ~ ~ 2 t ~ . .
The main features of F are: (i) The Fils are defined over sectors of aperture n / k . (ii) Writing Fi: (u,x,y) (u,x expqfioH,y exp-pf i (see 2 . 3 ) , k fioH is e x p o r t e n t i a l l y @ a t 0 6 utzdeh k: 1 I f i o H l I = o(exp-l/l uI ) -f
T h e s e . p h o p e h t i e . s e n n u h e that :lu,x,gl J u , x e x p q $ , g e x p - p $ l .in iz-summctb&e., as a consequence of Ramis’ theory (see 1201). Let us ex-f
plain the meaning of this fundamental property, for k=l (Borel-summability)
.
Essentially, knowing the formal transformation a , one can c o m p u t ~ % % (i.e the normalizing transformations on Uo,U1) as follows. 1 ) Write, as in 2 . 2 :
i
=c @,(X,Y)U” (X,Y)E A Define 6 (t,x,y) =I gn(x,y)tn/n! (inverse Borel-transform of 4 ) Then, this series has a n o r i - z e t z o tzad-iuh o d c u n u ~ t z y ~ i t c (in e t) and -
$ which is holomorphic
defines, by analytic continuation, a function on RxA,where:
111
&
~~
- ~ . .... . .
.
.
~
R
-
-D+o
0
I)-
=
+ -
~ - ( D U D)
(the half lines D+ and D- lie on the + bissectors R and !R- of U o n U 1 )
f.1
Moreover, $ has e , x p o n e n t i a l g t z o w t h (at most) when
t
+ m
.
R e m u h k . Ecalle’s theory of resurgent functions gives much more information on
5:
the cocycle F defines precisely the location and the na-
ture of the singularities of
6 with
respect to t: in general,
analytic on the universal covering of @ h Z (see [9] ) . 2 ) For each half line L starting from 0 in C(t), Lf R+ or I R - , @,(u,x,y)
=
JL
$(t,x,y)exp-t/u.dt/u
6
is
set:
(Bore1 transform of
5)
and aL: (u,x,y) (u,x exp q$,y exp-p$) It is not hard to check that (see example 2, in 2 . 2 ) : - $L is analytic on UL x A (UL = open half plane with bissector L), and $,=I$ ; thus 0 ED (U), and OL=@ +
h
-
A
h
L
n
A
.
When L moves in the lower (resp. upper) half plane, the QL glue together and define Oo~DT(Uo) (resp. O1~Dn(U1)),with O = Q 1 = @ -
A
0
T h e .t,tann 6 O h m U t h O n b O o ,
Zo
~tzaklb6OhmUti~~Mb,ohl
and
@
U ~ L p Qt i C C h b e e q
c,,
06
.the
h
.
( u n i q u e ) notzmaLizing
y = G-J(ro).
123
Analytic Classification of Resonant Saddles and Foci
They are called the (Borel) sums of the b ~ c i g u ! u h
I l t i ~ ~oil b ,f
To
Stokes lines.
:
n
@
;
the lines R
+
and R- are
they are the bissectors of two consecutive
In the general case, when k>l, the procedure which leads to (Oo,O n I' ) knowing O is quite similar: one replaces in 1) the factorial
...
by r(l+n/k) (inverse Leroy-transform): in 2 ) , exp-t/u.dt/u by ktk-'exp(-t k/u k ) .dt/u k (For more details, see [ 1 6 ] . )
.
3. REAL DOMAIN: RESONANT (OR WEAK) SADDLES In this chapter, we use the previous results to obtain the classification of h c u L u n u e y t i c b u d d l e b which are equivalent, through a hcu.8 O E D ~ ,to the h e a t i.iunrnu! d u f i m : k wo= pydx+qxdy+u (aydx+Bxdy) = 0 a , BER (1.8) IT (XIy ) =u=xpyq A
1 3,:
A
pB-qa=l (k odd) pB-qa= *l (k even) As in 2.3, the cunipeex d i r l ? i u t i o n To is defined by the first integral : (3.1)
FI =
xmyn7, n(u)
The Stokes lines of
'To,
x k K(u) = u exp-l/ku x = +(na-mB) in C (u), are defined by Re uk =0,and the where
singular lines by Im uk=O. Notice that the real plane IR 2 C C 2 lies in TI
-1
(R), and IR consists of two singular lines.
The analytic weak saddles ( 7 , n ) of order k and invariants a,B are in one-to-one correspondance with their normalizing transformations, from the results of 1.4. Thus, they are classified, up to real analytic equivalence, by the cocycles: h
such that
F =[O]
F
E
C1(?(;D;(Fo))
for a ncaL O E
sT.
We are going to study these cocycles in details, assuming k=l for simplicity. In this case, the c o v e r i n g u o f C(u) consists of two sectors U oru1 +. with bissectors RTi, R . 1 , and aper. set: ture 2 ~ We
y@ -L
--
_1; _ R~
~
\.
uoqul
=
u+uu-
1
with Re u>O on U+, Re u
(U;D;(F~))
A cocycle F = (F+,F-)EC is canonically defined by: (3.2) H+oF = f+oB+ and H - o F - = f-oH%+ -1 % -1 where H+ = H on U+ = n (U,) , H-= 1 / H on U= TI (U-), and f+,fare local diffeomorphisms of ( @ , O ) , tangent to the identity:
124
J. Martinet and J. -P. Ramis
(3.3)
f+(z
= z
+ n>c 2 anzn
f-(z)= z
+ n>z 2 bnzn
In what follows, we shall always denote by u the antiholomorphic involution defined by conjugation ( u ( x , y ) = ( X , y ) in C2, u(u)=; in C(u), etc . . . ) . Notice that JI exchanges the sectors U0 and U1 in C(u), ?, % Uo and U1 in C 2 , but it leaves invariant each sector U+,U-. PROPOSZTZON 3 . 1 .
A cocycle F=
t i c u s a teaL tkanbdo.tmation i E -I 13.4) OOFOO = F (i.e
J
[ ' L l ; P y ( F o ) )i b c h a a a c t e h i n i d a n d u n L y id: -I U O F + O U = F + , U ~ F - ~ C= I F : ' )
[F+,F-)
C
E
an
P h o o d . a) Necessity.
Let i E 6n be a real transformation, with characteristic class 1 % % F=(F+,F-) E C (?(;D;(Fo)); let O O r O 1 be the "sums" of 6 on Uo, U1. We have obviously: uoOooo = @ i.e O ~ ( X , Y )= o l ( x , y ) $1 % since 5 transforms U 0 into U1, and @ is real (see the summation formulas in 2.5). From this follows immediately: ?, CioF o a = u o @ O ~ - ' o o = o 0 0 - l = F ; ' on U+ + 0 1 1 0% and, similarly: u o F - 0 ~= FI1 on U-. b) Sufficiency . 1 Consider an F E C ( ~ / ; D ~ ( ~ o satisfying ) ) condition ( 3 . 4 ) ; take any with characteristic class F: then o @ u E DT has obviously the @E D same characteristic class, in view of ( 3 . 4 ) ; therefore: h
A
h
h
h
A
h
U O O O U = @oG
-1
where G E D T is a n u k y t i c ; this equality gives Goo = O o o o O , which shows that Goa is an a n t i h o l o m o h p h i c i n v o l u t i o n 0 6 C 2 at 0. Assume that we have shown: (3.5) G O D = G'oooGI-' with G'E Dn h It follows that 0 ' = OoG' E D n is real: indeed: h
I S O Q ' O ~
=
h
A
h
h
i~o@oUoUoG'o~= OoGoUoG'oU = OoG' = 0 '
Thus, we have found a real diffeomorphism with characteristic class F. PhLood 0 6 ( 3 . 5 ) . It is well known that any antiholomorphic involution of Cn at 0 is conjugate to the canonical involution u(x)=x through a holomorphic diffeomorphism ( [ l ] ,p.16; see too [19] ,p.263-264). From this we may write here, in C2: G ~ = U
~
~
0
0
0
~
;
~
where G 1 is holomorphic, with DG 1 ( O ) = I (because DG(O)=I), and defined up to composition on the right by a real analytic diffeomorphism Having chosen such a G1, notice that the function n ' = n a G 1 is h e a l ;
Analytic Classification of Resonant Saddles and Foci
indeed noGouoG1 CToTToGl = TOG1 0 0
= noG OU
.
1
: but
naGo5 =
o c n since G
125 E
D
71'
hence
Set
G1(xty) = (x+ S,(x,y) ,y+ nl(x,y)); then we have: IT' (XrY) = (x+Sl)p(y+nl~q A s T ' is real the curves x+E1 =O and y+ql =O are real and analytic; thus we have: x+S1 = a X,Y) (x+S) y+nl = b(x,Y) (y+n) where 5 and r\ are real analytic, of order -22 at 0, and a,b are analytic, with a(O)=b(O)=l, aPbq t e a l . It is then easy to define a real analytic G2 such that: n ' = noGZ ~ the diffeomorphism G I = G10Gi1 belongs We have then I T ~ =G noG2; to D n and gives the expected relation (3.5) This proposition, added to the results of sections 1 and 2, proves the following: THEOREM 3 . 2 .
Thc h e a l a n a l y t i c weah b a d d 4 e b
06
ondek one, w i t h 604-
mul i n u u h i a n t h ( p , q , k = l , a , B ) , a h c c h u h u c t e k z e d , u p . t o h e a l a n a l y t i c 1 e q u i v u l c n c c , b y f h e e f e m e n t b F E C (1i;Dy(30 1 h u c h Rhctf u o F o u = F - 1 , whete :
The "reality condition" ( 3 . 4 ) may be interpreted easily in terms of the local diffeomorphisms introduced in (3.2) and (3.3). Assuming that H+ and H- are defined by the principal determination of Log u on U+ and U-, one has: -2i71XH 5H+ u = H, and uH-5 = e It follows that F=(F+,F-) will satisfy condition ( 3 . 4 ) if : -1 e2inX5f-a e -2inX = and af+o = f+ The first of these relations means that uof+ is an antiholomorphic involution of C at 0; thus, from the result cited in the proof of Prop.3.1, we have: -1 uof+ = g+ouog+ where g+ is a local diffeomorphism of ( C , O ) , tangent to the identity; finally we obtain: (commutator of u and 9), = [u,g+] f+ = ug+ug;' Similarly, one proves that:
f?
I
f- = e 2inXco19where g - is any local diffeornorphism of
(C,O),
with linear part e
.
inX
J. Martinet and J.-I? Ramis
126
The "explicit" computation of the analytic invariants of a weak saddle is a difficult problem (Ecalle's theory of resurgent functions gives, in principle, a way of computing them: see [9]). We shall present here a two-parameters family for which we can compute the invariants, using one of our previous results ([16],p.154-158). The idea is to construct non trivial resonant saddles by blowing up Riccati equations with an irregular singularity. We consider the family: w = (l+ax+Xu-buy)du-xudy = 0 (u=n(x,y)= xy) arb where a,b,A are real parameters. The equation w =O is formally equivalent to the normal form: a,b w = (l+Au)du-xudy = 0 (p=q=l; k=l) 0
A
through a unique formal transformation
,.Qi
1
@E
h
Dn:
h
x' = x exp @
y ' = y exp-; These differential equations can be obtained from the Riccati equations : 2 €, dri/dS = ( l + X g ) r i + a€, - b€,n2
by means of the Hopf blowing up: 5 = x y , ~ = YThe computations we have made in the above reference show that the invariants of the saddle w ~ , ~ =are, O with the notations of 3 . 2 : f + ( z ) = z/(l+pz)
li = -2ina/r (l+a)r (I+@)
a ,B
being the rooes of
f - ( z ) = z/(l+vz)
and
iTX - 2 i ~ b e /r(i-a)r(i-@) a'+Xa-ah = 0.
3.4. R E M A R K S O N TffE G E N E R A L C A S E ( k
v
=
>
2 ) ; T H E L t A F SPACE.
The coveringu of C(u) relative to the normal forms (1.8) of order k consist of 2k sectors with aperture 2n/k, and bissectors the S R v k e h f i n e n Re uk = 0. The reality condition on the characteristic classes F E C 1 (2(;D'' ('fo) is still: (3.4)
UFU =
F
)
-1
In the plane C(u), u is the reflexion with axis IR, a n i n g u l a k L i n e
127
Analytic Classification of Resonant Saddles and Foci
o f T0 which bissccts two of the intersections UinUi+l: these sectors are invariant through 0 ; the other 2k-2 sectors UinUi+l are associated in pairs by u . Extending the procedure used for k=l, we may conclude:
C J [ ~ ~ ; D ~ i )3 0u e) h i 6 q i . n g ( 3 . 4 ) [ a n d Zhun t h e e q u i wufcvicc c t u ~ , e n 0 6 w c a k n u d d t e n 0 6 o h d ~ hk , w i z h 6 i x e . d dotmu.! i n w u h i u n t n ) a h e d i . d i r i c d b y t h e c h o i c e 0 4 k + l u n a L y t i c d i 6 6 e o m o h p h i n m n u6 in x z, t h e o t h e h o u h e R u n g e n t [ C , O ) ; o i n c 06 t i i c m l i u ~ t i n e a h p u f i z z + e T ~ cocycVcin P
F
t
% o . t h e identity, und t h e y ( t h e o t h c f i w i b e u h b i t h a h y .
In our paper [ 1 7 ] , we have shown that the resonant complex foliations are characterized by their "leaf spaces", which we describe as "necklaces of Riemann spheres" (see the Introduction). In particular, the complex foliation defined by a saddle of order k gives rise to a necklace C of 2k spheres. The reality condition means that C is equipped with an antiholomorphic involution u : this involution exchanges two halves of the necklace, and fixes two of the 2k "attaching points". The submanifold of real leaves consists of the fixed points of u (two "small arcs"). 3.5.
Cm-NORMALlZA710N
Let w = 0 be a resonant analytic saddle, formally equivalent to a normal form w = 0. Roussarie has shown [21], in the more general cao w se where w has C coefficients, that this equation is equivalent to wo=
0 through a Cm diffeomorphism at 0 in R2. Actually, he proves
still more: he classifies, up to conjugacy, germs of C"
vector fields
( i - e ,he takes ir,to account the parametrization of the inteqral curves by the time). This is a difficult result; one has mainly to show that any i n 6 i n i t e L y 6 k u t pehtuhbation
06
w
0
is equivalent to w o ,
by
means of a Cm diffeomorphism which is infinitely flat with respect to the identity map of R 2
.
We shall show how, in the case of anaeykic equations, our theory
-
z.
allows to define a local Cm diffeomorphism Y of R2 at 0, which transforms the foliation -T- into We shall do this, for simplicity, when the order k is one. A h Let O E DTIbe the h e a l 6 o h m u k normalizing transformation, which ta~ )its ) characteristic class, kes Tu to To, and let F E C 1 ( ' ~ / ; D ~ (be which satisfies (3.4):
uFu = F -1
.
? , %
yo;
The sums Oo and (D1 of on the domains U ,U1 transformJw into 0 the real plane R 2 is contained in each of these domains; but it is n o t i n w a h i a n t throuqh any of these transformations (otherwise, we
128
J. Martinet and J.-P. Rarnis h
would have 5@ u = Qo= Ql,i.e 0 would be analytic, and there is no0 thing to prove). There are two ways of overcoming this difficulty. 1) Set x' = x exp q$i (i = 0,l) *i (y' = y exp-p$i The "reality" of i means that @1 = 0, is then defined, in UonU1, by any one of the two The foliation
.
h
closed meromorphic forms (because w A @Tu = 0, i=O,l;see 1.2.2) : 1 0
du/u2 + adx/x + @dy/y + d$ Therefore, it is defined as well, in lR du/u2
+
adx/x
+
@dy/y
+
= 0
i , by
d$
(i=O,l) the real equation:
= 0
where )I = (@o+ @ 1 ) / 2 = Re $o = Re $l. 2 The fuction J, is C" in a neighborhood of 0 in IR : it is analytic , ,
(as Qo and Q ) outside the coordinate axis, and it admits $I as its 1 Taylor series at each point of these axis. Then the real C" diffeomorphism: Y
x' = x exp q$
IY' = Y exP-PvJ answers obviously the question. 2) Using 3.1 and 3.2, one may write the characteristic cocycle of 0
0 as:
F = 5GuG-l = [u,G] where G = (G+,G-), and G+,G- are elements of D;(U+) which p h e A t h w e the foliation To.A straightforward computation shows then that -1 G, o O 1 preserves the two quadrants of !R2 defined by xpyq=u > 0, and -1 o Q 0 preserves the two quadrants where xpyq=u 0. The pair G-1 (G;~oO,,G- "ao) thus defines a Cm local diffeomorphism of R2 at 0: it is still analytic outside the coordinate axis, and admits 0 as its Taylor series at each point of these axis. This diffeomorphism answers the question. Notice, moreover, that the Cm normalizing transformation obtained here belongs obviously to the Gevrey class of order 2. h
4 . REAL DOMAIN: WEAK FOCI
We consider a normal form: w0=
k (1+ Xu ) (xdx+ydy)
+
k !! (xdy-ydx) = 0 2
(XER)
2
which represents a weak focus of order k in IR (x,y), with a transver
3-
se fibration T , made up of concentric circles (each leaf of 0 is a spiral which crosses each circle). In C 2 , with complex coordinates:
129
Analytic Classification of Resonant Saddles and Foci
5
=
q = x-iy
x+iy
this normal form reads: k w = du + u (aqdc + i S d r \ ) = 0 0
c1
sn
=X -i/2
.rr(E,rl)= u = The foliation To is defined by the first integral: (4.1) H = nu-(iX+1/2)exp-l/ikuk k+l(iX+1/2)du/u + dq/q) (write wo/icnuk = du/u Therefore, the Stokes lines, in C(u), of the complex foliation k k are defined by Re iu = 0, and the singular lines by Im iu = 0. 2 -1 + Notice that IR C C 2 ( n = c ) lies in 71 (R ) , that is in the inverse 2 image of a Stokes line; notice to that II has modulus one i n R
'5
.
The analytic weak foci (T,T)of order k and invariant X are in oneto-one correspondance with their normalizing transformations, from the result of 1.4. Thus, they are classified, up to analytic equiva1 E C fl/;D;(To)) such that F for a h e r d
=[$I
lence, by the cocycles F A
h
@
E
D7.
We are going to describe these cocycles in some details for k=l.
Of course, the "reality condition" will be stated, as in the saddle case, by means of the involution of C2 which corresponds to our real
({,c),
plane, that is o ( c , q ) = a(u) = G. For k=l, the covering %(=%((To) of C(u) consists of the two sectors Uo,U1 with aperture 217 and bis-
y-) -+
-+
sectors R-,R . We denote by U+,Uthe components of UonUl on which
..~
1,
R-
\
.-./
Im u > 0 and Im u < 0. Notice that o leaves invariant each
R+
r7-
of the Ui's, but exchanges U+ and U - . The elements of C 1 ( ~ ! ; D ~ ( ~ o )are ) the
pairs F = (F+,F-) defined on (4.2)
where
I-:+oF+=
H+ = 1/H on U+,
U+,U-
H-oF-= f- O H -
f+ O H +
H-
=
€-
by:
H on U- (we use flat integrals), and f+ tangent to the iden-
are local analytic diffeomorphisms of ( C , O ) , tity. PROPOSITION 4 . 1 . T h e c u c y c l e F
h e a l tfianndukmatiun (4.3)
E ;
G17 i
o ~ F + o o= F I J
C J ( ~ ( ; € J ~ ( ~ i o b) ) c h a h a c t e f i i n t i c
E
d and u n L q id: ( i . e odFou
=
06
u
F-I)
The proof is analogous to the one of Proposition 3.1 and we shall A h omit it. Notice only that if @ E D17 is real (that is u @ u = O ) and sumA
h
J. Martinet and J.-P. Ramis
130 mable with sums deed
0
%
%
on Uo,U1, then we have
0 Q . o = Oi
(i=O,l); in-
preserves Ui, and the definition of the sum (see 2.5) gives
immediately these equalities. Condition (4.3) follows obviouslyfi Let us now describe more precisely the cocycles having property (4.3), in terms of the one dimensional diffeomorphisms f+,f-. To do this, we assume that H+ and H- are defined by the principal determination of Log u ( 0 < Arg u < IT on U+, - - 7 ~ < Arg u < 0 on U - ) . Then, we have:
OH+5 = Hand it follows that F will satisfy (4.3) if and only if: -1 o f + o = fn This means that, having chosen f ( z ) = z + a z (any conver+ n>2 n gent series), f- will be defined by its inverse: ~
-1
= f+(z). f- ( 2 ) = 2 + n>2 1 anz Using these natural conventions, we have proved: THEOREM 4 . 2 .
T h e u n a t q t i c c t a n h e n o i weuk 6oc.i w i t h dohmUk? i n w a h i u n . t b
k - i ,A ~f?,U h Q i n o n e - t o - o n e c o t 4 e n p o n d a n c e W i t h t h e t o c u t u n n t g t i c diddeomohphinmn o d ( C , O ) :
As in the saddle case, our computations of [16]allow us to give explicitely the "characteristic diffeomorphism" of a family of weak foci. P R O P O S l T l O N 4.3. T h e weak
dOCUb
o h okdeh one:
[ f + u [ x 2 -2y) + 2 b x y ) [ x d x + y d y ) + ~ ( x d y - y d x 0 ) = ( u = x2 + q2 ) ljohrnul inwuhian.tn k = J , A = O ; it.) u n u t y t i c i n w a h i u n t i n t h e l o c a l diddeomohphibm u 6 [C,O) d t 6 i n t . d b y : ,a = -2ei6innh%', 6 = Ahg(u+-bi) I ; , [ z l = z/ / -f-+TU Z hub
P h o o a . The "blowing down"p(a two sheeted covering) defined by:
v= u/2i
(r
t= 2i (x-iy)/(x+iy) = 2 i ~ / c
, n ) into the Riccati equation: arb 2 2 o f = v dt-(t-40v+ Bvt )dv = 0 ( B = (a-bi)/2)
transforms the pair
and the projection (v,t) -t v. The foliation associated to this Riccati equation is formally equivalent to the normal form: v 2dt-tdv = 0 whose inverse image under p is: (xdx+ydy) + :(xdy-ydx)=
0
For these normal forms, we use the flat first integrals:
Analytic Classification of Resonant Saddles and Foci
K+(v,t) = t
-1 -l/v
e (Re v>O) H+(x,y) = rl -1u1/2e1/1u (Imu>O) with the relations: 1 2 z H + =
K-(v,t)
=
131
teliv
(Re v
~ - ( x , y )= qu -1/2e-l/iu (Imu
2 i ~ I= ~ - o p
K+o P
Denote by G = (G+,G-) the c h a h u c t e h c b t i c c v c y c L e 06 Rhe R i c c u i ~ and set KoG = g0K. The results of ([16],p.154-57)
e y u a t ~ u w'=O, ~
show that g = ( g + , g - )
g+(z) with:
is given by:
= z/(l+llz)
g-(z)
=
Z/(l+VZ)
=
, 2+b i ieiesinnJa
_.
=
~
' 2+b3 ' 4 i e - isinnv'a ~.
v
7-
The characteristic cocycle F of w = 0 is obtained by lifting G through p ; a straightforward computation gives the result@ These foci are defined too by the following algebraic vector field of degree three: x = -y -xu/2 - ay(x2-y2 ) - 2bxy2 9 = x - Y U / ~+ ~ X ( 2S- y 2) + 2bx2y 2 (notice that xi + y$ = -u /2)
1-
Bryuno has given in [5] examples of analytic weak foci with a d i V Q I L ~ Q M Rn v h m a L i z i n g than&6ohmation; they are constructed, too, from the study of Riccati equations. He obtains, for each order k and invariant X , a d i n c h e t e d c t m i l y of examples, which correspond probably (for k=1) to characteristic diffeomorphisms: f+(z) = z
+
zs
+...
( s >/ 2) But the simplest numerical example he gives is a vector field of
degree five.
4.4. R E M A R K S O N THE G E N E R A L C A S E ( b 3 2 ) The covering u o f E(u) relative to the normal forms of order k consists of 2k sectors with aperture 2 ~ / k ,and bissectors the Stokes lines Re iuk = 0. The "reality condition" for a cocycle F is still: -1
UFO = F But now, in contrast with the case of saddles, the involution o in C ( u ) is the reflexion with respect to a S t o k e 6 L i n e ; therefore, no sector UinUi+l is preserved under U , and the reality conditioh means that F consists of k pairs of related diffeomorphisms. Thus, we may conclude :
The
COCyCkeb
F w h i c h c l a ~ ~ i d yu ,p t o anuLytic e q u i v u L e n c e , t h e
132
J. Martinet and J.-P. Ramis
ohdeh k [ a n d i n w a h i a n t X E W 6,Lxed) a h @ d e 6 i n e d b y t h e (C,O), tan5~n.tt o t h e i d e n t i t y , and o t h e h w i h e a h b i t h a k y . weak 6 0 t h choice
06
0 6 k Local a n a l y t i c di6~eomohphihmh 0 6
As for the (complex) leaf space of a weak focus of order k , it con-
sists of a necklace of 2k Riemann spheres, equipped with an antiholomorphic involution 0 ; the fixed points of u are the equatorial circles of 2 of these spheres. 4.5. C" N O R M A L l Z A T I O N Takens has shown [22] that every weak focus may be transformed, through a Cm diffeomorphism of (R2,0), into its formal normal form. For analytic foci, this result is an obvious consequence of our theory (more than in the saddle case); the point is that, as we have already noticed, the real plane now lies over a S X u k e h P i n e of C i u ) ; A A this means that the normalizing transformation O E D ~of a weak focus has a h e a l hum O M R 2 (see 2.5); this sum defines obviously a Cm diffeomorphism of (IR',O) which normalizes the focus. Moreover, this normalizing transformation is canonical, analytic outside 0, and belongs to the Gevrey class of order s = l+l/k. 4.6. W E A K FOCI A N D T H E I R P O I N C A R E M A P .
Consider an analytic weak focus of order k , and a piece of analy2 tic line L going through the origin 0 E IR , parametrized by s coordinate r (r=O at the origin). Take any point meL (close to O ) , and follow the integral curve through m (in counterclockwise direction for instance) until you hit again L, at a point m'= P(m); the mapping P is an analytic germ, and: 2k+l+. P(r) = -r + ar (a # 0 ) The square of P (under composition) is the Poincari. map of the focus, but it is more convenient to use P for what we have in mind. The map P is defined up to analytic conjugacy, in the group of analytic diffeomorphisms of ( R , O ) ; its conjugacy class is an invariant of the focus. As a final remark, we are going to deduce from our theory the following :
..
0 6 w e a k 6 o c i 0 6 vtrdetr k aae i n o n e - t o - o n e cuhtreopvndance w i t h t h e c o n j u g a c y ceahheh 0 6 a n a 2ktl+. L y t i c h e a e gehrnh P ( k 1 =-tr+ah ..
T H E O R E M 4.7. T h e a n a l y t i c e q u i u a L e n c e Clahheh
We emphasize the main fact: each P is the "Poincar6 map" of a weak focus.
Analytic Ctassification of Resonant Saddles and Foci
Sketch
phood.
133
Here, the key word is "holonomy". We have shown in
[17] that resonant differential equations have the same moduli space as the resonant germs of diffeomorphisms of ( C , O ) , the relation between the two problems being defined by the holonomy of the separatrices. [low, the mapping P relative to a weak focus is a holonomy map'L ping. To see this, transform the complex foliation into J , 2. through a Hopf blowing up of C 2 at 0 ; the foliationT has two singular points O,m on the singular divisor P (C), and . t h h ~6 Q~ p U h U t h i C c 3 : the "origina1"ones S , S ' , and the divisor P , ( @ ) (taking out the singular points). It is clear that P is the holonomy map of the leaf P l ( C ) ~ { O , ~ ] The . singular points 0,- are resonant with eigenvalues p=1 (on S , S ' ) and q=2 (on P1 ( C ) ) . Let C be the leaf space of 3 : it is a necklace of 2k spheres, endowed with an involution a. Let C ' be the "orbit space" of the map P: it is also a necklace of 2k spheres, endowed with an involution a'. Using the above description, we get a n a t u h a L a n a b y t i c i b O m O h p h i b f f l between the two necklaces C and X', compatible with u and 0 ' . One checks easily that, when C describes the space of all necklaces representing the moduli space of weak foci, X' describes the w h u l c set of necklaces corresponding to the moduli space of real analytic diffeomorphisms with linear part r * -r. (The two moduli spaces are a priori isomorphic.) I
3
REFERENCES A. Andreotti, P. Holm: Quasianalytic and Parametric Spaces, in Real and Complex Singularities, O s l o 1976, Sijthoff and Noordhoff Intern. Publishers. V.V. Basov: Divergence of Transformations of Real systems to the normal form in the case of nonrough focus (in russian), deposited in the All-Union Institute of Scient. and Technical information at no 1206-78. A.D. Bryuno: On local invariants of differential equations, Mat. Zanetki,l4,4 1973),499-507= Math.Notes, 844-48. A . D . Bryuno: The normal form of real differential equations, Mat. Zametki 18,2(1975),227-41=Math.Notes,722-731. A.D. Bryuno: Divergence of a real normalizing transformation, Mat. Zametki 31,3(1982),403-410=Math.NotesI 207-211.
J. Martinet and J. -P. Ramis
A.D. Bryuno: Analytical form of differential equations, Trudy Moskov.Mat. Obsc.25(1971),120-262= Trans. Moscow Math. SOC. 25 (1971),131-288 (1972).
D. Cerveau, J.F. Mattei: Formes integrables holomorphes singuliiires, Astgrisque 97 (1982). I:. Dulac: Sur les cycles limites, Bull. SOC. Math. F r m c e , 5 l (1923), 45-188. J. Ecalle: Les champs de vecteurs locaux resonnants de C v : classification analytique, to appear in: Publication R.C.P 25, no 32 , Strasbourg (1984). L.M. Markhashov: The analytic equivalence and stability of systems of second order under the resonance 1:l (Russian), Preprint no 14,Institute of Problems in Mech.,Moscow (1972). L.M. Markhashov: Analytic equivalence of second order systems with arbitrary resonance, Prikl. Mat. Mekh. 36,6(1972),1030-42. L.M. Markhashov:On analytic equivalence of systems of ordinary differential equations with resonances, Preprint n036, Inst. of problems in Mech., Moscow (1974). L.M. Markhashov: Invariants of multidimensional systems with a resonance relation, Prikl.Mat.Mekh.38,2(1974),233-239. L.M. Markhashov: On the analytic equivalence of systems of ordinary differential equations under resonance, in: Problems of analytical mech., Theory of stability and control (Russian), Nauka, Moscow (19751, 189-195. L.M. Markhashov: The method of invariants in problems about equivalence of ordinary equations, in: Cybernetics and Computational techniques (Russian),no 39, Naukova Dumka, Kiev (1978), 45-53. J. Martinet, J.P. Ramis: Problgmes de nodules pour des equations differentielles non lingaires du premier ordre, Publ. Math. 1.H.E.S 55 (1982), 63-164. 3 . Martinet, J.P. Ramis: Classification analytique des equations diffgrentielles non lineaires rgsonnantes du premier ordre, to appear in Ann. Ec. Norm. Sup. (1984). J.F. Mattei, R. Moussu: Holonomie et Integrales premisres, Ann. Sc. E c . Norm. Sup. 13 (1980), 469-523. J.K. Moser, S.M. Webster: Normal forms for real surfaces in Q:2 near complex tangents and hyperbolic surface transformations, Acta Math. 150:3-4 (1983), 255-296.
Analytic Classification of Resonant Saddles and Foci
[20i J . P .
Ramis: Les series k-sommables et leurs applications, Springer Lect. idotes in Physics 126 (1980).
[21] R . Roussarie: ModSles locaux de champs et de formes, Asterisque 30 (1975). [22] F. Takens: Normal forms for certain singularities of vector fields, Ann. Inst. Fourier, XXIII,2(1973),113-195. Jean Martinet
Jean-Pierre Ramis
Institut de Mathematiques 7 , Rue Ren6 Descartes
67084 Strasbourg Cedex France
135