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Singularities of Gradient Vector Fields and Moduli
Singularities & Dynamical Systems S.N. Pnevmatikos (editor) 0 Elsevier Science Publishers B.V. (North-Holland), 1985
81
S I N G U L A R I T I E S OF GRADIENT VECTOR F I E L D S AND MODULI
Floris Takens Rijksuniversiteit Groningen The Netherlands
INTRODUCTION We consider gradient vector fields,i.e.,vector fields X for which there exist a Riemannian metric g and a function V such that g(X,-)=dV. Our considerations are mainly loca1,so we assume all these objectsto be defined on IRn. We assume that X has a singularity in the oriqin, so dV(0) = 0. The analysis of these singularities was motivated by the following considerations. It is known that generic gradient vector fields on compact manifolds are structurally stable [ 2 , 4 ] . This is also true if the vector field depends on one parameter [ 5 1 , while in low dimensions the result even remains valid with more parameters [8].0n the other hand,there is a recent example [61 showing that qeneric k-parameter families of gradients need not be structurally stable if k28. The example is based on a configuration of two saddles with two orbits of non-transverse intersection of stable and unstable manifolds and on the equality of certain eigenvalues at the saddles. This configuration leads to a so called modulus of stability. For the purpose of this paper we can define this as follows. 1.
Let Xq(M) be the space of gradient vector fields on a compact manifold M. Let Wc Xg(M) be a smooth submanifold of finite codimension and let p: W +lR be a smooth function with non-zero derivative.Then we say that (W,u) is a rnvduLub ( 0 6 b t a b i & i t y ) if u(X) # u(X') for X,X'E W implies that X and X'are not topologically equivalent. The codimension of W is also called the c v d i m e n b i a n u s t h e moduLun. The example in [6lleft a number of problemsflike what is the lowest codimension of a modulus of stability of gradient vector fields? or,can a modulus be due to only one (isolated) singularity,or to only one orbit of non-transverse intersection of a stable and an unstable manifold? Here we deal with possible moduli of isolated singularities of gradient vector fields. To state the result we use the
a2
F. Takens
n of k-jets of singularities of gradient vector fields on Wn. space Jk
F O I L n - 5 a n d k b U 6 , j i C 4 & n t k y b i g t h e h e ahe a s m o o t h s u b m a n i dokd WcJt a n d a o m o o t h , j u n c t i o n 1~.: W -+ iR w i t h n o n - z e h o deaiwatiwe b u c h t h a t , w h e n e w m X a n d X ' ahe g h a d i e n t w e c t o h z j i e k d h o n 7Rn w i t h n i n g u k a h i t y i n t h c o h i g - i n a n d s u c h t h u t t h e i h k - j e t s j k l X l and j k ( X ' ) U ~ Ci?n w , . t h e n , i d X a n d X ' U h C t o p o t u g i c a k k y e q u i v a L e n t I f i t a h t h c o h i g i n ) , THEOREM.
V(jklXi) One could call such
u:
=
1J.(jklX'1)
.
n W + l R with W c J K a modulus of stability for
singularities of vector fields. It is not difficult to see that the conclusion of the theorem leads to the existence of a modulus of stability for gradient vector fields in the sens of the above definition. The construction of (W,~J.)is based on the following ideas. Let X be a gradient vector field on IRn with an isolated singularity in the origin.The stable ,unstable set is denoted by Ws, Wu respectively. Both these sets consist of integral curves of X ; we denote by FS,FU
, Wu-0
.
So one may consider Ws , 'W as a cone on FS, FU. Next we define a relation R between if and only if there are X-integral curves passing FS and FU: x R y arbitrarily near both x and y. Then the idea is to mak6 this relathe space of X-integral curves in
Ws-0
tion the union of two diffeomorphisms from FS to FU which we denote by Q + and 5 - . This leads to a diffeomorphism dU = 5 + .511 : FU + FU which is now a topological invariant in the following sense. Let X' be another gradient vector field with singularity which is topologically equivalent with the singularity of X; denote the topological equivalence by h. Fluand dl are defined as above using X' instead of X. Then h induces a homeomorphism from FU to Flu which conjugates d
U
with di or with (d:)-'.
This follows from the "topological chara-
cterization" of dU : if y,y'E FU and if there is some x E FS such that xRy and xRy', then y=y' or y=d (y') or y'=du(y). Finally we make an U example where d has a modulus of stability of the type consideredin [1,31
.
U
This example suggests that we may expect all complications, known to exist for diffeomorphisms, to show up when studying isolated singularities of gradient vector fields. Though I think is correct,I was not able to prove it:in the present quite special, in fact, d is very close to the time U dient vector field. Finally we should mention that in many cases, like
this expectation examples d is U one map o f a grain [ 6 1 , moduli
of stability were used to show that generic k-parameter families need
83
Singularities of Gradient Vector Fields and Moduli
not be structurally stable if k is greater than or equalthecodimension of the modulus. Such an argument howeveralwaysmakes useof some extra structure. For example in [ 6 1 this conclusion is based on the fact that if we denote the modulus constructed thereby (W',u') and if W' but X' is near W', then X' is not topologically equivalent with
X'f
any X E W'. For our present modulus of stability it isnot clear whether there is such extra structure; hence we cannot conclude tonewtypes of instability of generic parametrised families of gradient vector fields. 2. CONSTRUCTION OF THE MODULUS. A . BLawing up.
First we recall the blowing up construction. Let X be a vector field on lRn with X (0)= O . Then there is an induced vector field "x on Sn-' x R such that Q,(n)=X, where O(w,r)=r.w (we identify S"-'
with the unit
sphere in IR") , e.g. see [ 7 1 . X is said to be obtained by blowingup X. cu If the 1-jet of X is zero in the origin, &=I, then X is zero in the N
points of Sn-lx{0} and we can in fact devide by re; the resulting vector field we denote by x=r-'.Z. We shall apply this method to sinqularities of gradientvector fields. So let Vo be a homogeneous polynomial of degree k. We denote grad Vo
by Xo and the corresponding vector fields on Sn-lx R by To and zO=r-k+%o. Then zOISn-'x{O} is the gradient of VO/Sn-l (again we identify Sn-' with the unit sphere in l R n ) . We shall take Vo so that the critical pints
of Vo I Sn-' are either non-degenerate or are part of critical mani-folds where the second derivative of Vo IS"-' , normal to these critical manifolds is non-degenerate. Also we shall take Vo so that in.the crin- 1 , Vo is nowhere zero. tical points of V o / S In this situation the stable set Ws is a cone on the union of the stable manifolds in Sn-'x{O} of those singularities of 2, IS"-'x{O} where VolSn-' is negative (where we now identify both Sn-I and Sn-'x{O} with the unit sphere in IR"). The unstable set Wu is the union of the unstable manifolds in Sn-lx{O} of those singularities of zolB-'x{O} where Vo ISn-' is positive. If we add to Vo a function V1 with k-jet zero to obtain V=Vo+V1, then the stable and unstable sets of grad Vo and grad V are homeomorphic. A l s o ~o~Sn-'x{O} is equal to flSn-lx{Ol; is the vector field obtained from grad V by blowing up anddeviding
-X
by rk-2.
84
B.
F. Takens
CunhRkucAion
06
Vo.
As announced, V0 will be a homogeneous polynomial. Intheconstruction we first make a polynomial such that FISn-l has the required properties and then choose Vo so that V0l Sn-l = ?ISn-’. For this last construction we need that
N
V(x)
=
v ( -x )
for all x in
Sn-’ or that
make ? so that v(-x) = q ( x ) . A first step in the construction of V is the construction of a p o -
N
V(x)=-v(-x) for all x in S”-’. For this reason we shall
lynomial v :B2+IR such that 1 2 ; - v1 (x,,x2) = v (+xl,fx2) for all (xl,x2)EX? 1
-
we associate to v1 the phase portrait of the following vector field: n extend v1 tot IR by v1 (x,,x2, ,xn) = v (x ,x2) , then take grad ( v1 I Sn-l) 1 and project the integral curves on IR2 (note that the result is inde -
...
pendent of n); we require that this phase portrait has the following form :
in the other quadrants, the phase portrait is determined by symmetry;
-
from the phase portrait it follows that
v, (F) < v1 ( E ) < v 1 (B), v1 (F) < v1 ( D ) < v1 (A) < v 1 ( B )
,
v 1 (C) < v1 (A) ; we alsoassumme that v1 ( B ) is positive and and v I (F) are negative; -
v1 (A), v 1 ( C ) , v,
(D)
,
vl(E)
v1 ISn-’ has critical points corresponding to D and F and has cri-
tical submanifolds corresponding to A, B , C and E, these points and submanifolds in Sn-l will also be denoted by A, . . . , F ; we assume that v1 ISn-’ has nondegenerate 2nd derivative normal to A,B,C and E.
Singularities of Gradient Vector Fields and Moduli W e o b s e r v e t h a t f o r each
.
A3
,... , A n ,
...
not a l l zero,
85
3 -sphere
the
: A 1 i s an i n v a r i a n t s u b m a n i f o l d I ( x , , . . , X n ) E sn-1 I x 3 : . . : x n = A 3 : n n-1) A c o n s e q u e n c e of t h i s i s t h a t f o r e a c h p o i n t aEA f o r grad (v, I S
.
( a s s u b s e t of Sn-’)
t h e two b r a n c h e s of t h e o n e - d i m e n s i o n a l u n s t a b l e
m a n i f o l d of a a p p r o a c h B i n t h e same p o i n t . W e
shall
distroy
this
l a s t p r o p e r t y by a d d i n g t o v1 a p e r t u r b a t i o n of t h e f o l l o w i n g form V(Xl,. - . , x n ) = v 1 ( X I
N
,X2)+E.X1
3
. v 2 , r l ( x 3 , .. . , X n ) ,
where v 2 , n ( ~ 3 , . . . , ~ n )i s a homogeneous p o l y n o m i a l , d e p e n d i n g on qClR and s a t i s f y i n g v of v as
2&-1
vllS
vl Sn-’ .
(-x)=-v ( x ) ; w e s h a l l come b a c k t o t h e d e f i n i t i o n 217 2r q h a s t h e same c r i t i c a l p o i n t s and c r i t i c a l submanifolds
For
s h a l l w e s t i l l have t h a t f o r e a c h p o i n t
E
E
s m a l l , two d i f f e o m o r p h i s m s
~ + , ~ , ~, (o-,El,(a) a ) x1
2
0
,
.
x1 5 0
is the l i m i t point
Of
t h e n w e see from t h e f o r m u l a f o r
o+rE,r7
’ %,n
Sn-3
using
the
I f w e i d e n t i f y A and B w i t h
:
branch
of
the
aEA
two b r a n c h e s of WU(a) a p p r o a c h B , b u t now i n d i f f e r e n t p o i n t s . defines, for
2
This
A -B:
Wu(a) i n
that
f o r some c o n s t a n t c . From t h i s it a l s o f o l l o w s t h a t
o
-
B e f o r e w e a n a l y s e t h e s i t u a t i o n f u r t h e r w e want t o show t h a t FU c a n b e i n t e r p r e t e d a s t h e map d U : FU . o -1
+IE,r7
-,€In
mentioned i n
C. A n a L y h h
the
06
introduction.
ghad V o
.
From t h e p r e c e d i n g c o n s t r u c t i o n s and r e m a r k s i t f o l l o w s t h a t u n s t a b l e s e t o f GadV
0
the
is the cone on B o r , more p r e i s e l y , on two copiesof t h e
(n-3)- s p h e r e , o n e i n {x 2 > 0 1 and one i n I x 2 < O l . The s t a b l e s e t i s more c o m p l i c a t e d : w e h a v e t o t a k e t h e u n i o n of t h e s t a b l e m a n i f o l d s o f t h e n- 1 i s n e g a t i v e . T h i s means, i n t h e ( x l , x 2 ) s i n g u l a r i t i e s where VolS p l a n e d e s c r i p t i o n : t h e c u r v e C-A-D-F-E, including t h e endpoints. In sn- 1 t h i s c o r r e s p o n d s w i t h t h e t w o c l o s e d ( n - 2 ) - d i s c s f o r t h e l i n e C-A-D
(one i n { x
2
>ol
and o n e i n {x < O ? ) , f o u r c u r v e s f o r D-F 2
each q u a d r a n t ) , and an (n-2)-sphere
f o r E-F.
The s t a b l e
(one in
set i s
the
c o n e o n a l l t h i s . Because of t h e symmetry w e may r e s t r i c t o u r a t t e n -
86
F. Takens
tior, to {x 2 0 1 . 1Now we consider the relation R between FS
and FU defined in
the
introduction. h'e shall make use of both vector fields Xo-gradVo and
-
Xo, obtained from Xo by blowing up and deviding by
the appropriate power of r. We shall often identify X integral curves with the corresponding
2
integral curves in Snxtr
be identified with the corresponding subsets in Sn-'x{ 01. Restricting to {x2>Ol, FU is an (n-31-sphere which we can identify with B:each integral curve in FU approaches, for t-l-m I a unique point in BcSn-'x{O). F S , also restricted to (x2>Ol consists of an
x0
(n-2)-disc I two curves and an (n-2)-sphere. We restrict to the part of FS which consists of the interior of the minus the point where the two curves are attached; this we denote by Fs:'l'here is a canonicril projectionn: ys+
our attention (n-2)-disc , part of FS Ac Sn- 1 x{Oi: has a unique limit point in A for t -+ + m.
integral curve in Fs each 0 Using the fact that A and B, as submanifolds of Sn-'x{O1, are normally hyperbolic invariant manifolds of go (this time not restricted to S"-lx{Ol) we see that an orbit x in Ps is related with an orbit y in FU if and only if 0 (n(x))=y or o - r E l ,(TI (x)) =y This means that
.
+,E,rl
-1
is indeed dUFU+FU as defined in the introduction + I E r r l *o- I E rn (except that we had here the further complications that P" F FS and 0
that dim.Ps=dim FU+l) in the sense that for each yly'in FU such that for some xEPS we have x Ry and x Ry',
D. T h e c h o i c e 0 4
v
2,rl
and c o n n - t h u c t i o n
y=y-, or y=du(y') 04
,
or y'=dU(y).
-the m o d u L u s .
Ke choose v so that grad (v2 is a two-parameter family 2r r l lrl of gradient vector fields on Sn-3 wich satisfies the following specifications:
-
for gency ties; sense
some line ! in the 0-plane, qrad(v2,nlSn-3) has an orbit of tanof a stable and an unstable manifold of hyperbolic singularitransverse to this line the tangency unfolds generically in the of [ 5 ] ;
-
along L the modulus (of topological conjugacy) defined in [2] isnot constant. 2 For example in the case n=5 (so Sn-3=S ) one might take v2 suchthat 2 ,TI grad(v2,n/S ) has the following local phase portraits:
Singularities of Gradient Vector Fields and Moduli
87
J
/--
If h and h 2 are the contracting and expanding eigenvalues as indica1 ted, then the above mentioned modulus is h,/A,. I L -1 If we now take E sufficiently smal1,thenalso d = a U,E,17 +,E,T) @-,EJl has two hyperbolic fixed points which have a tangency for T) on some line L near 1. Along this line L the modulus, associated with this tangency, is again not constant. P7e fix E and only consider n E L . This gives a one - parameter family of homogeneous polynomials V 0,s
(SEX?
is the parameter) say of degree k. Now we take WcJf: as the lR we take set of k-jets of {Vo,s}sEIK, so dim W = 1 , and a s map u : F: the modulus associated with the tangency. -f
This is indeed a modulus as announced because if V is any smooth function with the same k-jet as V then, as we mentioned in the 0,s
section on "blowing up", the unstable sets of gradV
and
are homeomorphic and, by construction the maps dU for
gradV
gradV
0,s
and
gradV are conjugate. This means that the modulus of the tangency 0,s is also the same in both cases. REFERENCES
[ l ] S . Newhouse
& J. Palis & F. Takens, Stable families of diffeomorphisms. Publ. Math. I . H . E . S . 5 8 ( 1 9 8 3 ) p. 5 - 7 1 .
[ 2 1 J . Palis, On Morse 385
- 403.
- Smale
dynamical systems. Topology 8 ( 1 9 6 9 ) p.
[ 3 1 J . Palis, A differentiable invariant of topological conjugacies.
Asterisque 5 1 ( 1 9 7 8 ) p. 3 3 5
- 346.
[ 4 1 J. Palis & S. Smale, Structural stability theorems. Proc. A.M.S.
F. Takens
88
Symp. in Pure Math. 1 4 ( 1 9 7 0 ) p. 2 2 3 - 2 3 2 .
F. Takens, Stability of parametrized families of gradient vector fields. Ann. of Math. 1 1 8 ( 1 9 8 3 ) , 3 8 3 - 4 2 1 .
[51 J. Palis
&
[ 6 1 F. Takens, Moduli of stability for gradients:generic k-parameter
families of gradient vector fields are not always structurally stable. In this volume. [ 7 ] F. Takens, Singularities of vector fields. Publ. Math. I.H.E.S. 4 3 ( 1 9 7 4 ) p. 4 7 - 1 0 0 .
Floris TAKENS University of Groningen Department of Mathematics P.O.B. 800 9 7 0 0 AV Groningen THE NETHERLANDS
*
*
*
81
S I N G U L A R I T I E S OF GRADIENT VECTOR F I E L D S AND MODULI
Floris Takens Rijksuniversiteit Groningen The Netherlands
INTRODUCTION We consider gradient vector fields,i.e.,vector fields X for which there exist a Riemannian metric g and a function V such that g(X,-)=dV. Our considerations are mainly loca1,so we assume all these objectsto be defined on IRn. We assume that X has a singularity in the oriqin, so dV(0) = 0. The analysis of these singularities was motivated by the following considerations. It is known that generic gradient vector fields on compact manifolds are structurally stable [ 2 , 4 ] . This is also true if the vector field depends on one parameter [ 5 1 , while in low dimensions the result even remains valid with more parameters [8].0n the other hand,there is a recent example [61 showing that qeneric k-parameter families of gradients need not be structurally stable if k28. The example is based on a configuration of two saddles with two orbits of non-transverse intersection of stable and unstable manifolds and on the equality of certain eigenvalues at the saddles. This configuration leads to a so called modulus of stability. For the purpose of this paper we can define this as follows. 1.
Let Xq(M) be the space of gradient vector fields on a compact manifold M. Let Wc Xg(M) be a smooth submanifold of finite codimension and let p: W +lR be a smooth function with non-zero derivative.Then we say that (W,u) is a rnvduLub ( 0 6 b t a b i & i t y ) if u(X) # u(X') for X,X'E W implies that X and X'are not topologically equivalent. The codimension of W is also called the c v d i m e n b i a n u s t h e moduLun. The example in [6lleft a number of problemsflike what is the lowest codimension of a modulus of stability of gradient vector fields? or,can a modulus be due to only one (isolated) singularity,or to only one orbit of non-transverse intersection of a stable and an unstable manifold? Here we deal with possible moduli of isolated singularities of gradient vector fields. To state the result we use the
a2
F. Takens
n of k-jets of singularities of gradient vector fields on Wn. space Jk
F O I L n - 5 a n d k b U 6 , j i C 4 & n t k y b i g t h e h e ahe a s m o o t h s u b m a n i dokd WcJt a n d a o m o o t h , j u n c t i o n 1~.: W -+ iR w i t h n o n - z e h o deaiwatiwe b u c h t h a t , w h e n e w m X a n d X ' ahe g h a d i e n t w e c t o h z j i e k d h o n 7Rn w i t h n i n g u k a h i t y i n t h c o h i g - i n a n d s u c h t h u t t h e i h k - j e t s j k l X l and j k ( X ' ) U ~ Ci?n w , . t h e n , i d X a n d X ' U h C t o p o t u g i c a k k y e q u i v a L e n t I f i t a h t h c o h i g i n ) , THEOREM.
V(jklXi) One could call such
u:
=
1J.(jklX'1)
.
n W + l R with W c J K a modulus of stability for
singularities of vector fields. It is not difficult to see that the conclusion of the theorem leads to the existence of a modulus of stability for gradient vector fields in the sens of the above definition. The construction of (W,~J.)is based on the following ideas. Let X be a gradient vector field on IRn with an isolated singularity in the origin.The stable ,unstable set is denoted by Ws, Wu respectively. Both these sets consist of integral curves of X ; we denote by FS,FU
, Wu-0
.
So one may consider Ws , 'W as a cone on FS, FU. Next we define a relation R between if and only if there are X-integral curves passing FS and FU: x R y arbitrarily near both x and y. Then the idea is to mak6 this relathe space of X-integral curves in
Ws-0
tion the union of two diffeomorphisms from FS to FU which we denote by Q + and 5 - . This leads to a diffeomorphism dU = 5 + .511 : FU + FU which is now a topological invariant in the following sense. Let X' be another gradient vector field with singularity which is topologically equivalent with the singularity of X; denote the topological equivalence by h. Fluand dl are defined as above using X' instead of X. Then h induces a homeomorphism from FU to Flu which conjugates d
U
with di or with (d:)-'.
This follows from the "topological chara-
cterization" of dU : if y,y'E FU and if there is some x E FS such that xRy and xRy', then y=y' or y=d (y') or y'=du(y). Finally we make an U example where d has a modulus of stability of the type consideredin [1,31
.
U
This example suggests that we may expect all complications, known to exist for diffeomorphisms, to show up when studying isolated singularities of gradient vector fields. Though I think is correct,I was not able to prove it:in the present quite special, in fact, d is very close to the time U dient vector field. Finally we should mention that in many cases, like
this expectation examples d is U one map o f a grain [ 6 1 , moduli
of stability were used to show that generic k-parameter families need
83
Singularities of Gradient Vector Fields and Moduli
not be structurally stable if k is greater than or equalthecodimension of the modulus. Such an argument howeveralwaysmakes useof some extra structure. For example in [ 6 1 this conclusion is based on the fact that if we denote the modulus constructed thereby (W',u') and if W' but X' is near W', then X' is not topologically equivalent with
X'f
any X E W'. For our present modulus of stability it isnot clear whether there is such extra structure; hence we cannot conclude tonewtypes of instability of generic parametrised families of gradient vector fields. 2. CONSTRUCTION OF THE MODULUS. A . BLawing up.
First we recall the blowing up construction. Let X be a vector field on lRn with X (0)= O . Then there is an induced vector field "x on Sn-' x R such that Q,(n)=X, where O(w,r)=r.w (we identify S"-'
with the unit
sphere in IR") , e.g. see [ 7 1 . X is said to be obtained by blowingup X. cu If the 1-jet of X is zero in the origin, &=I, then X is zero in the N
points of Sn-lx{0} and we can in fact devide by re; the resulting vector field we denote by x=r-'.Z. We shall apply this method to sinqularities of gradientvector fields. So let Vo be a homogeneous polynomial of degree k. We denote grad Vo
by Xo and the corresponding vector fields on Sn-lx R by To and zO=r-k+%o. Then zOISn-'x{O} is the gradient of VO/Sn-l (again we identify Sn-' with the unit sphere in l R n ) . We shall take Vo so that the critical pints
of Vo I Sn-' are either non-degenerate or are part of critical mani-folds where the second derivative of Vo IS"-' , normal to these critical manifolds is non-degenerate. Also we shall take Vo so that in.the crin- 1 , Vo is nowhere zero. tical points of V o / S In this situation the stable set Ws is a cone on the union of the stable manifolds in Sn-'x{O} of those singularities of 2, IS"-'x{O} where VolSn-' is negative (where we now identify both Sn-I and Sn-'x{O} with the unit sphere in IR"). The unstable set Wu is the union of the unstable manifolds in Sn-lx{O} of those singularities of zolB-'x{O} where Vo ISn-' is positive. If we add to Vo a function V1 with k-jet zero to obtain V=Vo+V1, then the stable and unstable sets of grad Vo and grad V are homeomorphic. A l s o ~o~Sn-'x{O} is equal to flSn-lx{Ol; is the vector field obtained from grad V by blowing up anddeviding
-X
by rk-2.
84
B.
F. Takens
CunhRkucAion
06
Vo.
As announced, V0 will be a homogeneous polynomial. Intheconstruction we first make a polynomial such that FISn-l has the required properties and then choose Vo so that V0l Sn-l = ?ISn-’. For this last construction we need that
N
V(x)
=
v ( -x )
for all x in
Sn-’ or that
make ? so that v(-x) = q ( x ) . A first step in the construction of V is the construction of a p o -
N
V(x)=-v(-x) for all x in S”-’. For this reason we shall
lynomial v :B2+IR such that 1 2 ; - v1 (x,,x2) = v (+xl,fx2) for all (xl,x2)EX? 1
-
we associate to v1 the phase portrait of the following vector field: n extend v1 tot IR by v1 (x,,x2, ,xn) = v (x ,x2) , then take grad ( v1 I Sn-l) 1 and project the integral curves on IR2 (note that the result is inde -
...
pendent of n); we require that this phase portrait has the following form :
in the other quadrants, the phase portrait is determined by symmetry;
-
from the phase portrait it follows that
v, (F) < v1 ( E ) < v 1 (B), v1 (F) < v1 ( D ) < v1 (A) < v 1 ( B )
,
v 1 (C) < v1 (A) ; we alsoassumme that v1 ( B ) is positive and and v I (F) are negative; -
v1 (A), v 1 ( C ) , v,
(D)
,
vl(E)
v1 ISn-’ has critical points corresponding to D and F and has cri-
tical submanifolds corresponding to A, B , C and E, these points and submanifolds in Sn-l will also be denoted by A, . . . , F ; we assume that v1 ISn-’ has nondegenerate 2nd derivative normal to A,B,C and E.
Singularities of Gradient Vector Fields and Moduli W e o b s e r v e t h a t f o r each
.
A3
,... , A n ,
...
not a l l zero,
85
3 -sphere
the
: A 1 i s an i n v a r i a n t s u b m a n i f o l d I ( x , , . . , X n ) E sn-1 I x 3 : . . : x n = A 3 : n n-1) A c o n s e q u e n c e of t h i s i s t h a t f o r e a c h p o i n t aEA f o r grad (v, I S
.
( a s s u b s e t of Sn-’)
t h e two b r a n c h e s of t h e o n e - d i m e n s i o n a l u n s t a b l e
m a n i f o l d of a a p p r o a c h B i n t h e same p o i n t . W e
shall
distroy
this
l a s t p r o p e r t y by a d d i n g t o v1 a p e r t u r b a t i o n of t h e f o l l o w i n g form V(Xl,. - . , x n ) = v 1 ( X I
N
,X2)+E.X1
3
. v 2 , r l ( x 3 , .. . , X n ) ,
where v 2 , n ( ~ 3 , . . . , ~ n )i s a homogeneous p o l y n o m i a l , d e p e n d i n g on qClR and s a t i s f y i n g v of v as
2&-1
vllS
vl Sn-’ .
(-x)=-v ( x ) ; w e s h a l l come b a c k t o t h e d e f i n i t i o n 217 2r q h a s t h e same c r i t i c a l p o i n t s and c r i t i c a l submanifolds
For
s h a l l w e s t i l l have t h a t f o r e a c h p o i n t
E
E
s m a l l , two d i f f e o m o r p h i s m s
~ + , ~ , ~, (o-,El,(a) a ) x1
2
0
,
.
x1 5 0
is the l i m i t point
Of
t h e n w e see from t h e f o r m u l a f o r
o+rE,r7
’ %,n
Sn-3
using
the
I f w e i d e n t i f y A and B w i t h
:
branch
of
the
aEA
two b r a n c h e s of WU(a) a p p r o a c h B , b u t now i n d i f f e r e n t p o i n t s . defines, for
2
This
A -B:
Wu(a) i n
that
f o r some c o n s t a n t c . From t h i s it a l s o f o l l o w s t h a t
o
-
B e f o r e w e a n a l y s e t h e s i t u a t i o n f u r t h e r w e want t o show t h a t FU c a n b e i n t e r p r e t e d a s t h e map d U : FU . o -1
+IE,r7
-,€In
mentioned i n
C. A n a L y h h
the
06
introduction.
ghad V o
.
From t h e p r e c e d i n g c o n s t r u c t i o n s and r e m a r k s i t f o l l o w s t h a t u n s t a b l e s e t o f GadV
0
the
is the cone on B o r , more p r e i s e l y , on two copiesof t h e
(n-3)- s p h e r e , o n e i n {x 2 > 0 1 and one i n I x 2 < O l . The s t a b l e s e t i s more c o m p l i c a t e d : w e h a v e t o t a k e t h e u n i o n of t h e s t a b l e m a n i f o l d s o f t h e n- 1 i s n e g a t i v e . T h i s means, i n t h e ( x l , x 2 ) s i n g u l a r i t i e s where VolS p l a n e d e s c r i p t i o n : t h e c u r v e C-A-D-F-E, including t h e endpoints. In sn- 1 t h i s c o r r e s p o n d s w i t h t h e t w o c l o s e d ( n - 2 ) - d i s c s f o r t h e l i n e C-A-D
(one i n { x
2
>ol
and o n e i n {x < O ? ) , f o u r c u r v e s f o r D-F 2
each q u a d r a n t ) , and an (n-2)-sphere
f o r E-F.
The s t a b l e
(one in
set i s
the
c o n e o n a l l t h i s . Because of t h e symmetry w e may r e s t r i c t o u r a t t e n -
86
F. Takens
tior, to {x 2 0 1 . 1Now we consider the relation R between FS
and FU defined in
the
introduction. h'e shall make use of both vector fields Xo-gradVo and
-
Xo, obtained from Xo by blowing up and deviding by
the appropriate power of r. We shall often identify X integral curves with the corresponding
2
integral curves in Snxtr
be identified with the corresponding subsets in Sn-'x{ 01. Restricting to {x2>Ol, FU is an (n-31-sphere which we can identify with B:each integral curve in FU approaches, for t-l-m I a unique point in BcSn-'x{O). F S , also restricted to (x2>Ol consists of an
x0
(n-2)-disc I two curves and an (n-2)-sphere. We restrict to the part of FS which consists of the interior of the minus the point where the two curves are attached; this we denote by Fs:'l'here is a canonicril projectionn: ys+
our attention (n-2)-disc , part of FS Ac Sn- 1 x{Oi: has a unique limit point in A for t -+ + m.
integral curve in Fs each 0 Using the fact that A and B, as submanifolds of Sn-'x{O1, are normally hyperbolic invariant manifolds of go (this time not restricted to S"-lx{Ol) we see that an orbit x in Ps is related with an orbit y in FU if and only if 0 (n(x))=y or o - r E l ,(TI (x)) =y This means that
.
+,E,rl
-1
is indeed dUFU+FU as defined in the introduction + I E r r l *o- I E rn (except that we had here the further complications that P" F FS and 0
that dim.Ps=dim FU+l) in the sense that for each yly'in FU such that for some xEPS we have x Ry and x Ry',
D. T h e c h o i c e 0 4
v
2,rl
and c o n n - t h u c t i o n
y=y-, or y=du(y') 04
,
or y'=dU(y).
-the m o d u L u s .
Ke choose v so that grad (v2 is a two-parameter family 2r r l lrl of gradient vector fields on Sn-3 wich satisfies the following specifications:
-
for gency ties; sense
some line ! in the 0-plane, qrad(v2,nlSn-3) has an orbit of tanof a stable and an unstable manifold of hyperbolic singularitransverse to this line the tangency unfolds generically in the of [ 5 ] ;
-
along L the modulus (of topological conjugacy) defined in [2] isnot constant. 2 For example in the case n=5 (so Sn-3=S ) one might take v2 suchthat 2 ,TI grad(v2,n/S ) has the following local phase portraits:
Singularities of Gradient Vector Fields and Moduli
87
J
/--
If h and h 2 are the contracting and expanding eigenvalues as indica1 ted, then the above mentioned modulus is h,/A,. I L -1 If we now take E sufficiently smal1,thenalso d = a U,E,17 +,E,T) @-,EJl has two hyperbolic fixed points which have a tangency for T) on some line L near 1. Along this line L the modulus, associated with this tangency, is again not constant. P7e fix E and only consider n E L . This gives a one - parameter family of homogeneous polynomials V 0,s
(SEX?
is the parameter) say of degree k. Now we take WcJf: as the lR we take set of k-jets of {Vo,s}sEIK, so dim W = 1 , and a s map u : F: the modulus associated with the tangency. -f
This is indeed a modulus as announced because if V is any smooth function with the same k-jet as V then, as we mentioned in the 0,s
section on "blowing up", the unstable sets of gradV
and
are homeomorphic and, by construction the maps dU for
gradV
gradV
0,s
and
gradV are conjugate. This means that the modulus of the tangency 0,s is also the same in both cases. REFERENCES
[ l ] S . Newhouse
& J. Palis & F. Takens, Stable families of diffeomorphisms. Publ. Math. I . H . E . S . 5 8 ( 1 9 8 3 ) p. 5 - 7 1 .
[ 2 1 J . Palis, On Morse 385
- 403.
- Smale
dynamical systems. Topology 8 ( 1 9 6 9 ) p.
[ 3 1 J . Palis, A differentiable invariant of topological conjugacies.
Asterisque 5 1 ( 1 9 7 8 ) p. 3 3 5
- 346.
[ 4 1 J. Palis & S. Smale, Structural stability theorems. Proc. A.M.S.
F. Takens
88
Symp. in Pure Math. 1 4 ( 1 9 7 0 ) p. 2 2 3 - 2 3 2 .
F. Takens, Stability of parametrized families of gradient vector fields. Ann. of Math. 1 1 8 ( 1 9 8 3 ) , 3 8 3 - 4 2 1 .
[51 J. Palis
&
[ 6 1 F. Takens, Moduli of stability for gradients:generic k-parameter
families of gradient vector fields are not always structurally stable. In this volume. [ 7 ] F. Takens, Singularities of vector fields. Publ. Math. I.H.E.S. 4 3 ( 1 9 7 4 ) p. 4 7 - 1 0 0 .
Floris TAKENS University of Groningen Department of Mathematics P.O.B. 800 9 7 0 0 AV Groningen THE NETHERLANDS
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