Download ON SYSTEMS OF DIFFERENTIAL EQUATIONS IN THE SPACE OF

ON SYSTEMS OF DIFFERENTIAL EQUATIONS IN THE SPACE
OF C  -FUNCTIONS ON SMOOTH MANIFOLDS
G.R.Belitskii* and I.L.Lerman**
* Dept. of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel
** Dept. of Mathematics, Ben-Gurion University of the Negev, Beer-Sheva, Israel
Abstract. Problem of global solvability of the systems of linear differential equations
with singularities is considered. In particular the conditions of surjectivity and of
normal solvability in some special cases are obtained.
1. INTRODUCTION
We consider linear differential operators of the form
( L )( x)  (v )( x)  A( x) ( x)
(1)


in the space of C -functions on a smooth manifold M . Here v is a C - vector
C
M ,
field on
is a
-complex-valued matrix,
A : M  End (C m )

m
 : M  C is an unknown C - vector-function. The differential operator L
acts on a topological space C  -vector-functions, endowed with a standard topology
of convergence with all its derivatives on each compact subset of M .
The properties of the operator L from the point of view of a general theory of
linear operators such as normal solvability, Fredholm or semi-Fredholm have been
researched.
Recall that an operator is called normally solvable if its image is closed. In this case
the equation
( L )( x)   ( x),
where  (x) is any complex-valued C  - function on M , is also called normally
solvable.
A normally solvable operator and the corresponding equation are called Fredholm
if both of subspaces Ker and Co ker are finite-dimensional.
A normally solvable operator and the corresponding equation are called semiFredholm if
min (dim Ker, dim Co ker)  .
The main subject of this work is research of normal solvability, Fredholm and
semi-Fredholm of the operator L for vector fields with non-degenerated singular
points.
Our conjecture is that for a vector field with hyperbolic singularities on a manifold
M the operator L is normally solvable. This was proven in [1] for one-dimensional
manifold M .
Let us recall the notion of glued fields, which was introduced in [2]. A vector field
v on a manifold M is said to be glued from vector fields v on M  if there
exists an open covering M  U  such that v U


are conjugate to v . The latter
means that there exist diffeomorphisms   : U   M  such that v U     v . In

these terms, the main aim of this work is a proving of normal solvability of the
operator L for fields that are glued from hyperbolic linear fields on Rn . Further
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the examples confirm this assertion will be considered. In particular, we will consider
the vector fields on the sphere, the cylinder and the torus. For instance, a vector field
on the cylinder T 1  R1 is presented in the form
(2)
v( x)  (w(u), z ), x  (u, z )  T 1  R1 ,   0,
1
where w is a vector field on T with hyperbolic singular points. It is easy to see
that this field is glued from linear hyperbolic fields. It will be proven in Theorem 10,
that the operator L with the vector field (2) on the cylinder is normally solvable.
Although the considered examples confirm our hypothesis, the question of normal
solvability for such kind of fields still stays open in a general case.
We start this work with research a one-dimensional version of the problem, by
considering, in Section 2, the simplest case of a manifold, M  R1 . Section 3 we
begin with the definition of absorber and nozzle, which were introduced in [2], as
tools of expanding of a local solution to a global one, in a form relevant to our
research. In Section 4, the conditions of normal solvability of the operator L in the
absence of invariant compact subsets for a vector field v is obtained. In Section 5,
normal solvability of the operator with linear hyperbolic field on M  R n is proved.
Further, in Section 6, the case of the operator with two singular hyperbolic points on
the sphere are researched. Section 7 concerns of normal solvability of the operator L
with singular hyperbolic points on the cylinder and the torus.
2. One-dimensional case
We start from the simplest case M  R1 . Then a vector field v has the following
form v( x)  a( x) dxd with a real-valued function a(x) . If a(x) does not vanish then
the operator L is surjective and dim KerL  m . Hence, L is Fredholm. If a point
x0 is singular for v , i.e. a( x0 )  0 , then the problem of formal solvability arises.
Indeed, let  be a solution of the equation
(3)
a( x)´x  A( x) ( x)   ( x).
Substitution of x  x0 into equation (3) yields the condition  ( x0 )  Im A( x0 ) .
Furthermore, by differentiating equation (3) with respect to x and substituting
x  x0 , an infinite system of linear algebraic equations with respect to derivatives
(  ( x0 ), ´( x0 ),...,  ( k ) ( x0 ),... ) is obtained:
 x0   Ax0  x0 

 ´( x )  a´( x )´( x )  A´( x ) x   Ax ´( x )

0
0
0
0
0
0
0
(4)

...


 i  x0   Pi  x0 ,...,  i  x0 
where i  0, 1,..., k ,.. and Pi   x0 ,...,  i   x0  is a polynomial which is a linear




combination of the derivatives { ( k ) ( x0 )} for k  i .
Equation (3) is called formally solvable at x0 if there exists series

ˆ   c k
x  x0 k
,
k!
whose coefficients ck   ( k ) ( x0 ) satisfy all equations of system (4). In this case the
series is the formal Taylor series one of supposed solutions  (x ) .
According to the Leng theorem (see[1]), a sufficient and, evidently, necessary
condition of an existence of a formal solution is compatibility of every finite
k 0
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subsystem of system (4).
Let

ˆ    k
x  x0 k
,
k!
be the formal Taylor series of the given function  (x) and
k 0

ˆ    k
x  x0 k
,
k!
be the formal Taylor series of a supposed solution  (x ) . Then system (4) may be
written in a recursive form
(5)
 k  kI  Ak  Rk 0 ,..., k 1 , k  0,1,...
where   a´( x0 ) , A  Ax0  and Rk  0 ,..., k 1  are linear functions that are
determined by the given matrix A(x ) and the function  (x) . Hence, if it holds that
(6)
 k  specAx0 , k  0,1,...
then equation (3) has a solution. Condition (6) is one of versions of so-called the
absence of resonances from the theory of vector fields. Note that if   0 a singular
point x0 is called hyperbolic. In this case condition (6) holds for every large enough
k.
Thus, in the absence of resonances, equation (3) is formally solvable at x0 for an
k 0
arbitrary C  -function  (x) . Conversely, condition (6) is necessary for equation (3)
to be formally solvable at x0 for every C  -function  (x) . In other words, the
following assertion is true.
Proposition 1. Let x0 be a singular point of the vector field v . Then equation (3)
is formally solvable at x0 for every vector-function  (x) if and only if condition (6)
holds.
Proof. The part "only if" remains to be proven. Assume k  N to be the smallest
number such that  k  specAx0  .
Consider a vector u  Im Ax0   kI  and set  ( x)  x  x0  u . Show that (3)
for the given  (x) is not formally solvable at x0 . Indeed, the first k equations
of system (4) have the form:
0  Ax0  x0 


0  a´( x0 )´( x0 )  A´( x0 ) x0   Ax0 ´( x0 )

...

 k -1

0  a´( x0 ) ( x0 )  Ax0  k -1 ( x0 )  ...

uk! kI  Ax0  k  ( x0 )  Rk  0  x0 ,...,  k 1 x0 
k
Since  j  specAx0  for

j  k 1

all equations have only trivial solutions
 x0   0 ,...,  ( k 1) ( x0 )  0 . Hence, the first non-trivial in the left side equation is
uk! kI  Ax0  k  x0 .
This equation has no solution since u  ImkI  Ax0  .
However, an existence of a formal solution may not imply local solvability.
Example 1. Consider the simplest equation
a( x) ´( x)  b( x),
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(7)
where
exp 1 / x , x  0
a( x)  
 0, x  0
and
2
exp 1 / .2 x , x  0
b( x )  
.

0, x  0
2
Equation (7) is formally solvable at x  0 and an arbitrary C  - vector-function 
is its formal solution. But for x  0 it holds that ba(( xx))   . Hence the equation has
no local smooth solutions in a neighborhood of x  0 .
In this example the point x0 is a singular point of "an infinite type" of the vector
field v( x)  a( x) dxd .
Note that the point x0 is a singular point of a finite type if the function a(x) has
at the point a root of a finite order. This means that the function a(x) may be
presented as a( x)  ( x  x0 ) r b( x) with r  0 and b( x0 )  0 .
Theorem 1. (Kyznezhov,[4])Let x0 be a singular point of a finite type for the vector
field v (x ) . If equation (3) has a formal solution ̂ then there exists its C  - local
solution whose formal Taylor series at x0 coincides with ̂ .
It is clear that if equation (3) has only a singular point then local solvability implies
a global one. But this does not occur if the equation has more than one singularity.
Example 2. Consider the equation:
x( x  1) ´( x)  t ( x)   ( x)
(8)
where  (x) is a given C  - smooth complex-valued function on R1 and
t  const . The equation has two hyperbolic singular points x1  0 and x2  1 . The
corresponding open intervals are U1  (, x2 ) and U 2  ( x1 , ) . In this case the
condition of the absence of resonances is fulfilled at x1 iff t Z  and at x2 iff
t  2Z  . Choosing t {Z   2Z  } it is obtained that the equation is formally
solvable at the points x  0 and x  1 and besides Ker( L U i )  {0}, i  1, 2 .
A solution of equation (8) may be written in the form:
 x  Cx z( x), x  0,1,
where
t
x
z ( x) 
, x  0,1
1 x
satisfies the corresponding homogeneous equation. For the function C (x) is
obtained:
 ( x)
C´( x) 
, x  0,1.
z ( x) x( x  1)
Let the function  (x) have the form:
 ( x)  z ( x) x( x  1) ( x),  (0)   (1)  0,
where  (x ) is flat at the points x  0 and x  1 . Show that a global solution exists
if and only if:
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1
  (s)ds  0.
(9)
0
On each of the intervals U i for the given function  (x) there exists a local solution
i .
Indeed, since the function  (x ) is flat at x  0 , the function
x
1 ( x)  z1 ( x)   (s)ds, 1 (0)  0,
0
where
z1 
xz
x
, x  U1
is a C  -local solution on U1 , while the function
x
2 ( x)  z2 ( x)   (s)ds,  2 (1)  0
1
with
z2 ( x)  z( x), x U 2
is a C  -local solution on U 2 , since  (x ) is flat at x  1.
Assume that there exists a global solution  of equation (7). Then in the absence
of resonances on each of intervals
Ui
it holds that
  i  KerL U i  {0} .
Consequently the local solutions 1 and  2 have to coincide on the intersection
U1 U 2  (0,1) , if a global solution does exist:
1 ( x)  2 ( x) , x  (0,1),
implying
x
  (s)ds  0.
0
The latter is a necessary and sufficient condition for global solvability. Hence, if 
is positive on (0,1) then (8) has no C  -solution. For example, if


,  0   1  0,
2 
2


x
1

x



then C -solutions are absent.
Assume that the vector field v has an arbitrary number of isolated singular points
under the condition of local solvability of the equation at each one of them. Denote
the collection of all singular points and consider open intervals
{xi }
{U i  ( xi 1 , xi 1 )} . The criterion of global solvability of the equation is presented by
the following theorem.
Theorem 2. Assume that equation (3) is locally solvable in each of open intervals
U i . If the equation has a global solution then for every collection of local solutions
 ( x)  exp  
1
{ i }  C  (U i ) there exist vector-functions  i  KerL U i such that
i 1 ( x)  i ( x)   i1 ( x)  i ( x), x U i U i 1.
(10)
Conversely, if condition (10) is fulfilled for a collection of local solutions { i } then
the equation has a global solution.
Proof. Assume that equation (3) has a global C  -solution  . Let { i } be a set
of C  - local solutions on U i . Then the differences  i    i
homogeneous equation
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satisfy the
L i x   0 , x  U i .
Thus the functions  i satisfy condition (10).
Conversely, suppose that condition (10) is fulfilled for some of C  -local
solutions { i } and some of functions  i  KerL U i . Consider the vector function
 (x ) which
is defined by
 x    i x   i x  , x  U i .
The function  is well defined since the differences  i  i and i1   i 1
coincide on the intersection U i  U i 1 .
The function  (x ) on every interval U i satisfies the equation:
L x   L i x   L i x    x  .
Consequently  (x ) is a global solution of equation (3).
Corollary 1. Let all singularities of the vector field v be of finite orders. Then the
operator L is normally solvable.
Proof. Let {xi } be the collection of all singular points of the operator L . Suppose
that all the singularities of the operator are of finite types. Then these singular points
are isolated.
Consider intervals {U i  ( xi 1 , xi 1 )} forming an open covering of R 1 . To prove
the statement of Corollary 1 it is sufficient to prove normal solvability of the operator
on each of intervals U i and finite-dimensionality of Ker ( L U i ) (see Corollary 3.2
in [8]).
For an interval U i consider a sequence { n ( x)}  Im( L U i ) which converges to a
vector function  ( x)  C  (U i ) . For the limit function  (x) it needs to be proven
that  ( x)  Im( L U i ) .
Since { n ( x)}  Im( L U i ) equation (3) is formally solvable at
xi for each of
functions  n (x) . The formal solvability of equation (3) for the function  n (x)
means that every finite subsystem of system (4) has a solution.
It follows from
 ni  xi    i  xi  ,
that it holds also for the limit function  (x) that every finite subsystem of system
(4) has a solution. According to the mentioned above the Leng Theorem, there exists
a general solution which satisfies all the equations of the system. Thus equation (3) at
xi with the limit function  (x) has a formal solution. Further, by Theorem 1 for
 (x) there exists a local solution which may be extended dynamically on the whole
interval U i .
Hence,  U i  Im L U i and the restrictions L U i are normally solvable.
It remains to ensure that dim Ker( L U i )   for every i . In every open interval
( xi , xi 1 )
Since
the operator has no singular points. Therefore dim Ker( L U i  U i 1 )  n .
Ker( L U i )  Ker( L U i U i 1 )
it
holds
dim Ker( L U i )  dim Ker( L U i  U i 1 ) . Consequently dim Ker( L U i )  n .
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that
Hence the operator L is normally solvable.
In fact, the situation of Example 2 is general in the following sense.
Theorem 3. For every collection of vector-functions {ci } KerL U i U i 1  there
exist a vector-function  (x) and local solutions { i }  C  (U i ) of equation (3) such
that ci  i 1  i .
Proof. Consider vector-functions
{ci } KerL U i U i 1  . There exist vector-
functions  i C  that are defined on open intervals U i and satisfy the condition:
ci  i 1  i .
The vector-function
 x   L i x  , x  U i
is well defined since
L i 1 x   L i x  , x  U i  U i 1 .
It is evident that  i are local solutions of equation (3).
Summarizing the previous facts we obtained the following criterion of surjectivity of
the operator L .
Corollary 2. Let all singularities {xi } of the vector field v be of finite types. Then
the operator L is surjective if and only if the following conditions are fulfilled:
(a)  i k  specAxi , k  0,1,... where i  a( xi ) ;
(b) each collection of vector-functions {ci } KerL U i U i 1  permits of a
representation ci   i 1  i where  i  KerL U i .
If v has a unique singular point then the condition of surjectivity has been reduced
to (a). But if the vector field has even two singular points condition (b) may not be
fulfilled (see Example 2).
The criterion of surjectivity of L with two singular points has the following form:
Corollary 3. Let the vector field v have two singular points x1 and x2 of finite
types. Then the operator L is surjective if and only if condition (a) is fulfilled and it
holds that
(11)
dim Ker( L U1 )  dim Ker( L U 2 )  n.
Indeed, if condition (11) holds then each of vector-functions
c  KerL U1 U 2 
may be written in the form: c   1  2 , where  i  KerL U i , i  1, 2 . Thus
condition (b) of Corollary 2 is also fulfilled.
Conversely, if every vector function c  KerL U1 U 2  may be represented in the
mentioned form, then
KerL U1 U 2   Ker( L U1 )  dim Ker( L U 2 ),
implying (11).
For instance, in Example 2 condition (a) holds at both of the singular points x1  0
and
x2  1 . However, condition (b) is not satisfied. Indeed, in this case
Ker( L U i )  {0}, i  1, 2 while dim KerL U1 U 2   1 .
Example 3. Consider the equation:
x 3 (1  x) 3 ´( x)  2t (2 x  1) ( x)   ( x).
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(12)
The equation has two non-hyperbolic singular points x1  0 and x2  1 .
Therefore, if it is assumed that t  0 then condition (a) is satisfied at both of the
singular points.
Furthermore, a general solution of the corresponding homogeneous equation has
the form:


t
, x  0,1.
 x   C exp   2
2 
 x 1  x  
Therefore if
then
while
dim Ker( L U i )  1, i  1, 2
Re t  0
dim KerL U1 U 2   1 . Hence, condition (b) is also satisfied, implying that for
Re t  0 the equation has a global solution for every C  -function  (x) .
For Re t  0 condition (b) fails and operator (12) is not surjective.
Of course, all previous considerations may be immediately carried out on the unit
circle T .
In order to formulate conditions of Fredholm for the operator L consider system
(4) at a singular point xi of the vector field v . This is an infinite linear system with
respect to an infinite vector w  (w0 ,..., wi ,...) . The system may be rewritten in the
form
Ti w  
where Ti is the corresponding linear operator in the space of infinite sequences w .
Denote d i  dim Co ker Ti in an algebraic sense. For example, if condition (6) holds
then d i  0 . Denote by S the space of all collections c  {ci } such that
ci  KerL U i U i 1  . Let
representation
ci   i 1  i ,
S0  S
consists of elements of
S
permits of the
with  i  Ker( L U i ) . It is obvious that S 0 is subspace.
Denote b  co dim S 0 and s  dim KerL . Note that s  n .
Theorem 4. Let all the singularities of the vector field v be of finite types and such
that i d i   , b   are satisfied. Then the operator L is Fredholm of
indL  b   d i  s.
i
In particular, if all singularities are hyperbolic (a´( xi )  0) , then d i   , and we
obtain
Corollary 4. Let the vector field v on R1 have a finite number of hyperbolic
singularities. Then equation (3) is Fredholm.
Since the number of hyperbolic singularities on the unit circle T is finite, the
operator L on T is Fredholm as well.
3. Multi-dimensional case. Absorber and Nozzle
It is clear that a multi-dimension case of M is more complicated. We start this part
of our work with some additional relevant definitions.
Let F t (x) be a flow of the vector field v via a point x . Denote by
(t  ( x), t  ( x)) a maximal open interval of an existence of the flow. For a compact
manifold M such interval for all x coincides with the real line. Recall that a subset
S is called v -invariant if t    and for all x  S and t  0 it holds that
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F t ( x)  S .
Let W be a subset of the manifold M and consider the closed interval
t  (W ), t  W  : t  x , t  x .
xW
In particular, it is possible that the interval is reduced to zero. But if it has inner
points, then the corresponding open interval (t  (W ), t  (W )) is a maximal interval of
an existence of the trajectory F t (x) for all x W .
Definition 1. A v -invariant subset M   M is called an absorber if for every
x  M there exists a neighborhood W and a number t 0  0 such that
t  t 0 , F t u   M  , u W .
Subset M   M is called a nozzle if it becomes an absorber after the substitution
t  t . In other words there exists t 0  0 such that
t  t 0 , F t u   M  , u W .
Example 4. Let us consider an example of a constant vector field

xe
n
where e is a vector in R . Its flow is F t ( x)  x  te . Let f (x) be a linear
functional such that f (e)  0 . Then the subset { A  {x : f ( x)  1}} is an absorber
and the subset {N  {x : f ( x)  1}} is a nozzle.
Example 5. Consider the vector field
x  x m , m  0
on the real line.
If m is odd then the point is a node. Then every closed neighborhood of this point
is a nozzle and R1 is an absorber.
If m is even then the point is a point of overflow. In this case, every interval
  , is an absorber and  ,   is a nozzle.
Theorem 5. Let M  , M  be an absorber and a nozzle for a vector field v ,
respectively. If the equation
( L )( x)   ( x)
(13)
has a solution in a neighborhood of the intersection M   M  then it has a global
solution. In particular, if M   M  is empty set, then the equation is solvable for
every vector-function  (x) .
Proof. Assume that  0 : M  C p is a local C  -solution of equation (13) in a
neighborhood U of the intersection M   M  , i.e.  0 is defined on M and
satisfies the equation
L 0 x   x , x U .
A supposed global solution of (13) will be searched in the form
  0 
where the vector-function  is a solution of the equation
(14)
v   A  ~
~
Here     L 0 and does vanish in U .
It remains to prove that equation (14) has a solution for every C  -vector-function
~x that equals identically to zero in a neighborhood of the intersection
7-44
M   M  . For this aim will be used Lemma 1 [see below]. The lemma claims the
existence of a decomposition of the vector-function ~x in the form
~       .
Here  
M
0
and  
 0 . Since equation (13) is linear, it follows that suffice
M
it to find solutions   of the equations
(15)
L  x    x .
The vector-functions   are obtained in the same way. Therefore we will describe
in detail only a solution of the equation for the function   .
Consider the following system of differential equations:


x  v x 
.
(16)

 y  A x  y     x 
Let ( F t ( x), G t ( x, y)) be the corresponding flow. The projection G t ( x, y ) may
be presented in the form:
t


G t x, y   t , x  y    1 s, x   F s x ds  ,
(17)
0


where  (t , x) is a fundamental matrix of the corresponding linear homogeneous
system. It is easy to check that if a function   is a solution of the equation
(18)
  F t x  G t x,  x  , t  R1 ,
then   satisfies the equation (15).
 
Since  
M
 0 it is true that for every x there exists a neighborhood W such


that for s  t0 it holds   F s u  0 for all u W . Hence, the integral

  x     1 s, x   F s x ds
(19)
0

converges and presents a C -mapping on M .
Immediate verification shows that   (x) is a solution of equation (18).
Indeed,

  F x     1 s, x   F s t x ds 
t
0


 

   1 u  t , F t x   F u x du 
t
t
  1

u
t , x    u, x   F x du    1 u, x   F u x du   G t  x,   x ,
0
 0




Since


t , x  1 u, x    1 u  t , F t x
The latter follows from the following flow property
G t  s x, y   G t F t x, G s x, y  .
Setting in (18)    0 , we arrive at the following equality




u, x    u  t , F t x t , x 
7-45

(20)
that is equivalent to (20). Thus, a solution   of equation (15) is obtained.
Similarly, the vector-function   is obtained in the form

  x     1 s, x   F s x ds .
(21)
0
A vector-function       is a C  -solution of equation (14).
Lemma 1. Let us consider two closed subsets S1 and S 2 of a manifold M . If a
C  -function  equals identically to zero in a neighborhood of the intersection
S1  S 2 then there exists the decomposition
      ,
where  
S1
 0 and  
S2
0 .
Proof. Consider the complements U1  M \ S1 and U 2  M \ S 2 . The intersection
S1  S 2 is closed while the subsets U1 ,U 2 are open on M . It is obvious that the
intersection
U1  U 2  M \ (S1  S 2 )
is an open subset as well. The submanifold U1 U 2 is a para-compact set. Therefore
there exists a decomposition of the unity as a sum of C  -functions  1 ( x) and  2 ( x) .
The functions  1 ( x) and  2 ( x) must satisfy the following conditions: they are
determined on U1 U 2 , supp  i  U i and  1 ( x)   2 ( x)  1 .
Let us set
 x  x  , x  U 1
  x    1
0 , x  U1

and
  x       .
It is easy to see that the functions   and   are well defined everywhere on the
manifold M and it holds that  
S1
 0 , 
S2
 0 and         0 .
Let us return to Examples 4 and 5. In Example 4 A  N is empty set. Hence,
according to Theorem 1 the operator L is surjective. In Example 5, the intersection
of an absorber and a nozzle contains a neighborhood of the origin. As a consequence,
local solvability implies a global one.
4. Equations in the absence of invariant compact subsets
The typical example of the vector field without invariant compact subsets in R n is a
constant vector field v ( x )  e , where e is some vector.
It is well known that on the real line in the absence of invariant compact subsets a
vector field is conjugate with a constant one and the corresponding operator has the
form
(22)
( L )( x)  ´x  A( x) ( x).
The operator L on R1 is surjective and dim KerL  n .
In contrast to the real line, in the plane, even in the absence of invariant compact
subsets, the vector field may be not conjugate with a constant field. As a consequence,
the operator may be not surjective and even not normally solvable. The following
example demonstrates this in the case A( x)  0 .
7-46
Example 6. Let us consider the manifold
M  R 2 \  , 0
that is evidently diffeomorphic to the plane. Consider the vector field


v 

, x   ,  M .


It is obvious that the flow F t x  e t x where
1 0 

  
 0  1
has no invariant compact subsets. However, for instance, the Abelian equation
v x  1
has no continuous solutions. Indeed, if  is a solution then
 et  ,    ( , et )  t.
If  ,   0 , then we arrive at a contradiction as t   . Thus the operator
L  v is not surjective. Moreover, the operator is not normally solvable. In order
to prove that it suffices to check that Im L is dense in the space of C  - functions on
M . That means that for every C  -function  (x) there exists a sequence
{ n ( x)}  Im L that converges to  (x) . To show that consider a sequence of
functions { n ( x)} such that each of them equals to zero inside of a circle of the
radius 1 / n and equals to 1 outside of the circle of the radius 2 / n . Easy to show
that the sequence { n ( x)} converges to the unity in C  -topology on M . Hence the
sequence { n }  { n ( x) ( x)} converges to the function  (x) .
It remains to prove that  n ( x)  Im L . For any n  0 , let us consider the subsets
1

M  n    x  ( , ) :   
2n 

and
1

M  n    x  ( , ) :   .
2n 

The subset M  (n) is an absorber for the vector field v , and M  (n) is a nozzle.
Their intersection is contained in the neighborhood of the radius 2 / n of the point
x0  0 , where  n ( x)  0 . Therefore, according to Theorem 1 the equation
( L )( x)   n ( x),
has a solution. Thus  n ( x)  Im L .
The same arguments show that Im L is dense in the space of C  - functions on
M for every matrix A(x ) .
In the considered example the field was not conjugate to a constant one because of
an existence of so-called non-wandering compact subsets.
Definition 2. A subset K  M is called wandering for a vector field v if
t  (K )   and t  (K )   and there exists a number t 0  0 such that
 t  t 0 , F t K   K is empty set .
This means that every trajectory F t (x) leaves the subset K in a finite period of
time. If K is not wandering, then it is called non-wandering subset.
7-47
Theorem 6. If all compact subsets K  M are wandering for a vector field v ,
then the operator L is surjective.
Proof. Note that if all compact subsets are wandering then the field is full, i.e. its
flow F t (x) is defined for every t  R and x  M . It was proven in [3] that in this
case the flow is conjugative with a flow of shifts. This means that there exists a
manifold B such that dim B  dim M  1 . There also exists a diffeomorphism
 : M  R1  B such that
F t (s, z )  (s  t , z) where (s, z)  R1  B . Hence
the operator L , in an obvious sense, is equivalent to the operator
 ~
~
L  s, z  
 As, z  s, z .
s
that is surjective.
5. Operator with only a hyperbolic singular point in R n
The main result of this section is the following statement.
Theorem 7. Let v (x ) be a linear hyperbolic field in R n . Then equation (3) has a
global solution iff it is formally solvable.
 
First, let us discuss the notion of formal solvability for the multi-dimensional case.
Let x0 be a singular point and  : R n  C m -a supposed solution of the equation
(v )( x)  A( x) ( x)   ( x), x  M
(23)
Substituting x  x0 in (23) we get the condition  ( x0 )  Im A( x0 ) .
Furthermore, differentiating equation (23) with respect to x and substituting x  x0
, the infinite system of linear algebraic equations with respect to derivatives (
 ( x0 ), 1 ( x0 ),...,  k ( x0 )   ( k ) ( x0 ),... ) is obtained:
(v )( x0 )  A( x0 ) ( x0 )   ( x0 )


(v1 )( x0 )  A( x0 )1 ( x0 )   1 ( x0 )  R1[ ( x0 )]

(24)

...

 (v k )( x0 )  A( x0 ) k ( x0 )   k ( x0 )  Rk  ( x0 ), 1 ( x0 ),...,  k 1 ( x0 )
where  k ( x0 )   ( k ) ( x0 ) and Ri  x0 ,...,i x0  are linear combinations of the
derivatives { x0 ,...,  k 1 ( x0 )} .
As in one-dimensional case equation (23) is called formally solvable at x0 if there
exists formal series
I

x  x0 
ˆ   C I
,
I!
I
whose coefficients CI  C m satisfy all equations of system (24). Here I  I1 ,..., I n 
I!  I1!...  I n !
is
a
integer-valued
multi-index,
and
x  x0 I


I1


In
 x1  x10  ...  xn  xn0 . In this case the series is the formal Taylor series
of a supposed solution  (x ) .
By the Leng Theorem [5] a necessary and sufficient condition for an existence of a
formal solution is compatibility of every finite subsystem of system (24).
By 1 ,..., n denote the eigenvalues of   v( x0 ) and 1 ,...,  m the eigenvalues of
A( x0 ) . It is well known that the numbers
7-48
n
 j   I i i
i 1
are the obstacles to solvability of the system.
The condition
n
 j   I i i
(25)
i 1
guarantees formal solvability at x0 for every  (x) .
Condition (25) is one of versions of a so-called the absence of resonances the
vector fields theory.
[Proof of Theorem 7] . Remind that if equation (31) with hyperbolic point x0 is
formally solvable at x0 then it has a local C  solution  0 [6]. Without loss of
generality we assume that x0  0 . By S  denote the invariant subspaces that
corresponding to   spec with Re   0 and Re   0 , respectively. Then
R n  S   S  , and for all x  R n it holds that x  x  x where x  S  .
The subset
M     x : x    ,   0
is an absorber for the vector field v , while the subset
M     x : x    ,   0
is a nozzle.
Let a local solution  0 be determined in a neighborhood U of x0 . For small
enough  it holds that M     M     U . Hence, according to Theorem 5, the
equation has a global solution.
Corollary 6. Let v be a linear hyperbolic field in R n . Then the operator L is
normally solvable.
Proof. Consider a sequence { n ( x)  Im L} that converges to a vector-function
 ( x) C  . For the limit function  (x) we have to prove that  ( x)  Im L .
Since  n ( x)  Im L , it follows that equation (23) is formally solvable at x 0 for
each of functions  n (x) . Formal solvability of equation (23) for a function  n (x)
means that each of finite subsystems of system (24) has a solution.
From
 ni  x0    i  x0 
Follows that every finite subsystem of system (24) has a solution for the limit
function as well. According to above mentioned Leng`s Theorem, there exists a
general solution which satisfies all equations of the system. Thus equation (23) at x0
with the limit function  (x) has a formal solution. Then, by Theorem 7, for  (x)
there exists a global solution. Hence,   Im L and the operator L is normally
solvable.
If a singular point of the operator is a hyperbolic saddle, then dim KerL   [7].
If the number of resonances is finite then dim CoKerL   and the operator is semiFredholm. But if the singular point is a node, then both of spaces KerL and CoKerL
are finite-dimensional implying
Corollary 7. If a linear hyperbolic field in R n is a node, then the operator L is
Fredholm.
7-49
6. Operator on the sphere with two nodes
Consider a vector field on the n-dimensional sphere S n , having two fixed points.
Theorem 8. The operator

L

x
v 

xA
x

x
on the sphere with the vector field that is glued from two linear hyperbolic nodes is
normally solvable.
Proof. Denote by L U i : C (U , C p )  C (U , C p ) the operator L that acting on
any open subset U . Furthermore, we assume that the field v is glued from linear
nodes at points- x1 , x2 . Let us show that the subspace Im L is closed. Indeed, the
field v has the invariant open covering U1  Sn \ x1  , U 2  Sn \ x2  . The
operator L U i , i  1, 2 is normally solvable and it holds that Ker ( L U i ) are finitedimensional. Then, according to Corollary 3.2 [8] the operator L is normally
solvable.
Note that in contrast to the one-dimensional case [1], for n  2 the operator L on
the sphere is semi-Fredholm, even if the vector field is glued from two linear
hyperbolic nodes. Indeed, in spite of the fact that Ker ( L U i ) are finite-dimensional,
dim KerL U1  U 2    . In order to show that, recall that all compact subsets of the
vector field v on the intersection U1 U 2 are wandering. Therefore the vector field
v on U1 U 2 is conjugate to a shift on R1  Sn1 . It means that there exists a
diffeomorphism T : Sn \ x1 , x2   R1  Sn1 such that (Tv)( s, z)  ( / (s), 0) ,
where s  R 1 and z  Sn1 . Hence the homogeneous equation (23) is equivalent to
the equation
 s, z  ~
 As, z  s, z  .
s
The general solution of this equation has the for
 ( s , z )   ( s , z )c ( z )
where ( s, z ) is the fundamental matrix of the system that depends on variable z as
a parameter while c : Sn1  C p is an arbitrary C  vector-function. Therefore
dim CoKerL   , implying dim CoKerL   .
Now we state a criterion of global solvability.
Theorem 9. Assume that equation (23) has a C  -solution on each of open subsets
U1 ,U 2 , where U i  Sn \ xi , i  1, 2 .
a) If the equation has a global solution then for every collection of "local"
solutions
 i  C  (U i ), i  1, 2 there exist vector-functions  i  KerL U i such that
2 ( x)  1 ( x)  2 ( x)  1 ( x), x U1 U 2 .
(26)
Conversely, if condition (26) is fulfilled for a collection of local solutions { i } , then
the equation has a global solution.
b) For every vector function c  KerL U1 U 2  there exist a vector-function
 (x)
7-50
and local solutions { i }  C  (U i ) of equation (23) such that c  2  1 .
Proof. Assume that equation (23) has a global C  -solution  . Let 1 and  2
be C  - local solutions on U1 and U 2 , respectively. Then the differences
 i    i satisfy the homogeneous equation
L i x   0 , x  U i .
Thus the functions  i , i  1, 2 satisfy condition (26).
Conversely, suppose that condition (26) is fulfilled for some of C  -local solutions
{ i } and some of functions  i  KerL U i . Consider a vector-function  (x ) that is
defined by
 x    i x   i x , x  U i , i  1,2.
Since the differences 1  1 and  2  2 coincide on the intersection U1 U 2
, the function  is well defined.
The function  (x ) on every set U i satisfies the equation:
L x   L i x   L i x    x .
Consequently  (x ) is a global solution of equation (23).
To prove the second part of the theorem, consider vector-function
c  KerL U1 U 2  . There exist vector functions  i  C  , i  1, 2 which are defined
on the open subsets U i and satisfy the condition: c  2  1 (see [9]). Define a
vector-function  (x) by the formula:
 x   L i x  , x  U i , i  1,2.
Since
L 2 x  L1 x , x U1  U 2 .
the function  (x) is well defined.
Thus, this is evident that  i are local solutions of equation (23).
7. Examples on the Cylinder and the Torus
Recall that we consider a vector field v(u, z )  ( w(u ), z ) where u  T 1 and z  R1
, on the standard two-dimensional cylinder M  T 1  R1 . Here w is a vector field
on the circle
with "non-degenerated singularities", i.e. has the form
T1
1
d
w(u )  a(u ) du , u  T where a(u ) is a C  -function with roots of finite order.
For such fields on T 1 normal solvability was proven in [1]. Moreover, conditions of
surjectivity and of Fredholm were obtained. Let, as above, A : M  End (C m ) be a
C  -complex-valued matrix.
Theorem 10. If v is glued from vector fields of the form (2) on M , then the
operator
( L )( x)  (v )( x)  A( x) ( x), x  M
is semi-Fredholm with dim KerL   .
Before the proving of the theorem, let us state conditions of global solvability of the
equation L   on M  T1  R1 . At first, note that if   0 , then T 1 is an
absorber for the vector field v , while for   0 it is a nozzle. Therefore, to prove
solvability of the equation, suffice it to prove its solvability in a neighborhood of the
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circle T 1 .
Let us assume that   0 . By differentiating the equation
(v )(u.z )  A(u, z ) (u, z )   (u, z )
(27)
with respect to z and substituting z  0 the infinite system of linear equations with
respect to derivatives ( (u ), 1 (u ),..., k u  z 0   (zuk , z )
k
z 0
,...) , which is similar to the
system (24), is obtained:
(v )(u )  A(u ) (u )   (u )


(v1 )(u )  A(u )1 (u )   1 (u )  R1[ (u )]


...

 (v k )(u )  A(u ) k (u )   k (u )  Rk  (u ), 1 (u ),...,  k 1 (u )
where  k u  z 0   (zuk , z )
k
z 0
(28)
.
Obviously, if   Im L , then the system has C  -solutions
 k (u) 
 k
z k
, k  0,1,...
z 0
Note that from the Fredholm property of the operator L on T 1 follows that for
every finite subsystem, the space of solutions is finite-dimensional. Hence the system
is solvable iff every its finite subsystem is solvable.
Lemma 2. For any C  - complex-valued function 
on the manifold
1
1
M  T  R with a vector field of the form (2), the equation L   has a global
C  -solution if and only if system (28) is solvable.
Proof. The part "if" remains to be proven. Let us assume that a set of the functions
{ k (u ), u  T 1} forms solutions of system (28). Using the Borel Lemma, may be
constructed a function ~ on the manifold M such that
 k ~
 k u   k z 0 , k  0,1,...
z
and consequently it holds that
L~     ,
where  is a flat function on T 1 .
A supposed global solution of the equation L   will be searched in the form
  ~  
where  is a solution of the equation
(29)
L ( x)   x
that is flat on the circle. Repeating the above considered arguments regarding to
equations (15)-(18) we can represent a solution of (29) in the form

 x     1 s, x  F s x ds ,
(30)
0
where  is the fundamental matrix and F t is the flow of v . It remains to check
that this formula presents a C  -vector-function on M .
Indeed, by fixing a compact neighborhood U  {(u, z), z  a}  M of the circle
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T 1 , an estimate  1 ( x, s)  Ce Ns , x  U , s  R1 is valid.
Since the function  is flat on the circle, it holds that
p
 k  F1 s u , es z  C kp esp z , z  a , k , p  0,1,2...
(31)
Therefore, for under the integral sign expression takes place the estimate
p
 1 x, s  F s x   C0 p Ce s p N  z .
(32)
 




We obtain convergence of the integral by choosing p  N /  . Thus it holds that
  C 0 . In the same may be obtained an estimate for under the integral expression
sign of the derivative of expression (30)
 1 x, s  F s x k  C~e s p N  z p , x  (u, z).
Convergence of the corresponding integrals implies that  C  .
Proof of Theorem 10. Consider a sequence of vector-functions { n ( x)} such that
 n  Im L and that converges to a vector-function  C  . For the limit function 
we have to prove that   Im L as well. In order to prove this we will show that
system (28) is solvable for the limit vector function  . Each of the equations of the
system is a linear equation on the circle with a vector field whose singularities are
hyperbolic of finite order. The Fredholm property of equations of this type was
proven in [1].
Since  n  Im L and
 n k    k  ,
it holds also for the limit function  that for every finite index k system (28) has a
solution. Therefore, by the Lemma 2,   Im L .
It remains to ensure that the space of solutions of the corresponding homogeneous
equation
(33)
v x  Ax x , x  M
is finite-dimensional.
Indeed, every C  -vector-function  may be written in the form
p0
1
f k (u) z k   (u, z ),
k
!
k 0
 ( x)  
with an arbitrary index p0 , where   o( z
p0
) and f k (u ) 
(34)
 k ( u , z )
z k
z 0
.
Furthermore, by applying the operator L to (34) we arrive to the equality
L ( x)   L f k (u) z k   L (u, z).
p0
For   KerL it is obtained that
(35)
k 0
L ( x)   L 1

(36)
.
z 0

k 0  k!
Equation (36) for every fixed choice of f k has a unique solution   o( z p0 ) .
Indeed, for every two solutions 1 ,  2 ,   1   2 it holds that L( 2  1 )  0 .
It follows from (29)-(32) that the function  satisfies the equality
 x    1 (t , x) F t x .
p0
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f k (u) z k
and the estimate
 u, z   C (u, z )C p e t  N p  z ,
0
0
p0
(37)
where index p0 satisfies the inequality N  p0  0 , and consequently, as t  
yields   0 .
Furthermore, backing to (32), we see that a definite choosing f k (u) yields a
unique representation for the vector-function  . The functions f k satisfy system
(28) for k  p0 and   0 . The system is homogeneous system of linear equations
on the circle, with the vector field whose singularities are of hyperbolic type. It was
proven that the space of solutions of such system has a finite dimension [1]. For the
given index p0 , the space of solutions of the restricted to k  p0 system (28)
coincides with the dimension of the KerL . Hence dim KerL   .
Under conditions of Theorem 10, the operator L may be not Fredholm because a
dimension of CoKerL may be is infinite.
Example 7. Let a vector field on the cylinder T 1  R1 have the form

v( x)  ( w(u ), z ), x  (u, z )  T 1  R1 ,
(38)
z
where w(u )  sin u u . Consider the equation:
(v )( x)   ( x)   ( x)
(39)
where  is an irrational constant.
System (28) for the operator L in this case has the form
(40)
(w k )(u)  (k   ) k (u)   k (u), k  0,1,2,...
with  k (u )   (zuk , z )
k
z 0
. Here the field w is glued from two hyperbolic fields. The
corresponding covering of
is U1 U 2
where U1  T 1 \ {0}
and
T1
1
U 2  T \ { } . Since  is irrational, i.e. in the absence of resonances, it holds that
dim Ker( L U i )  0 ,
i  1,2,... .
A solution of every equation of (40) may be written as:
 u   Ck u  sk (u), u  0, ,
where
 k   
u
sk (u )  tan
, u  0, 
2
satisfies the corresponding homogeneous equation. For the function C k (u ) is
obtained:
 k (u )
C k ´(u ) 
, u  0,  .
s k (u ) sin u
Assuming  k (u ) to be flat at 0 and  , we arrive to the following criterion of
solvability of equation L   :

0
 k (u )
 k (u )
(41)
0 sk (u) sin u du  0,  sk (u) sin u du  0, k  0,1, 2,..
From an infinity of a number of conditions follows that dim CoKerL   .
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