Lectures on analytic differential equations

349 Pages·2007·2.568 MB·English

by  Yulij Ilyashenko, Sergei Yakovenko

LECTURES ON ANALYTIC
DIFFERENTIAL EQUATIONS 1
Yulij Ilyashenko2
Sergei Yakovenko3
Moscow State University,
Steklov Institute of Mathematics, Moscow
Independent University of Moscow, Russia,
Cornell University, Ithaca, U.S.A.
E-mail address: [email protected]
Weizmann Institute of Science, Rehovot, Israel
E-mail address: [email protected]
WWW page: http://www.wisdom.weizmann.ac.il/~yakov
1Version of June 22, 2003
2supportedbythegrantNSFno.0100404
3TheGershonKekstprofessorofMathematics,supportedbytheIsraeliScienceFoundation
grantno.18-00/1
1991 Mathematics Subject Classification. Primary 34A26, 34C10;
Secondary 14Q20, 32S65, 13E05
Henri Poincar´e (April 29, 1854–July 17, 1912)
David Hilbert (January 23, 1862–February 14, 1943)

Contents
Preface iii
Notations vii
0. Functions of complex variables vii
Chapter 1. Normal forms and desingularization 1
1. Holomorphic differential equations 2
2. Holomorphic foliations with singularities 12
3. Formal flows and embedding theorem 21
4. Formal normal forms 32
5. Holomorphic normal forms 48
6. Holomorphic invariant manifolds 62
7. Topological classification of holomorphic foliations 68
8. Desingularization in the plane 91
9. Complex separatrices of holomorphic line fields 116
Chapter 2. Singular points of planar analytic vector fields 125
10. Singularities of planar vector fields with characteristic
trajectories 126
11. Algebraic decidability of local problems. Center–focus
alternative 141
12. Holonomy and first integrals 163
13. Zeros of analytic functions depending on parameters and small
amplitude limit cycles 180
Chapter 3. Linear systems: local and global theory 215
iii
iv Contents
20. General facts about linear systems 216
21. Local theory of regular singular points 226
22. Analytic and rational matrix functions.
Matrix factorization theorems 237
Appendix: meromorphic solvability of cocycles 253
23. The Riemann–Hilbert problem: positive results 260
24. Negative answer for the Riemann–Hilbert problem in the
reducible case 269
25. Riemann–Hilbert problem on holomorphic vector bundles 276
26. Linear nth order differential equations 295
27. Irregular singularities and the Stokes phenomenon 313
Appendix: demonstration of Sibuya theorem 325
Bibliography 333
Index 341
28. To Do List 344
Chapter 1
Normal forms and
desingularization
1
2 1. Normal forms and desingularization
1. Holomorphic differential equations
1.1. Differential equations, solutions, initial value problems. Let
U ⊆ C × Cn be an open domain and F = (F ,...,F ): U → Cn a holo-
1 n
morphic map (vector function). An analytic ordinary differential equation
definedbyF onU isthevectorequation(orthesystem ofnscalarequations)
dx
= F(t,x), (t,x) ∈ U ⊆ C×Cn, F ∈ On(U). (1.1)
dt
Solution of this equation is a parameterized holomorphic curve, the holo-
morphic map ϕ = (f ,...,f ): V → Cn, defined in an open subset V ⊆ C,
1 n
whose graph {(t,ϕ(t)): t ∈ V} belongs to U and whose complex “velocity
vector” dϕ = (cid:0)df1,..., dfn(cid:1) ∈ Cn at each point t coincides with the vector
dt dt dt
F(t,ϕ(t)) ∈ Cn.
The graph of ϕ in U is called the integral curve. From the real point
of view it is a 2-dimensional smooth surface in R2n+2. Note that from the
beginning we consider only holomorphic solutions which may be, however,
defined on domains of different size.
The equation is autonomous, if F is independent of t. In this case the
image ϕ(V) ⊆ Cn is called the phase curve. Any differential equation (1.1)
canbemadeautonomousbyintroducingafictitiousvariablez ∈ Cgoverned
by the equation z˙ = 1.
If (t ,x ) = (t ,x ,...,x ) ∈ U is a specified point, the initial value
0 0 0 0,1 0,n
problem, sometimes also called the Cauchy problem, is to find an integral
curve of the differential equation (1.1) passing through the point (t ,x ),
0 0
i.e., a solution satisfying the condition
ϕ: V → Cn, ϕ(t ) = x ∈ Cn. (1.2)
0 0
In what follows we will often denote by dot the derivative with respect
to the complex variable t, x˙(t) = dx(t).
dt
The first fundamental result is the local existence and uniqueness theo-
rem.
Theorem 1.1. For any holomorphic differential equation (1.1) every point
(t ,x ) ∈ U there exists a sufficiently small polydisk D = {|t−t | < ε, |x −
0 0 ε 0 j
x | < ε, j = 1,...,n} ⊆ U, such that solution of the initial value problem
0,j
(1.2) exists and is unique in this polydisk.
This solution depends holomorphically on the initial value x ∈ Cn and
0
on any additional parameters, provided that the vector function F depends
holomorphically on these parameters.
Fromtherealpointofview,Theorem1.1assertsexistenceof2nfunctions
oftwo independentrealvariableswhose graphis asurface inCn+1 ’ R2n+2,
1. Holomorphic differential equations 3
with the tangent plane spanned by two real vectors ReF,ImF. To derive
this theorem from the standard results on existence, uniqueness and dif-
ferentiability of solutions of smooth ordinary differential equations in the
real domain, one should use rather deep results on integrability of distri-
butions, see Remark 2.10 below. Rather unexpectedly, the direct proof is
simpler than in the real case in the part concerning dependence on initial
conditions. This proof is given in the next section.
1.2. Contracting map principle. Consider the linear space A(D ) of
ρ
functions holomorphic in the polydisk D and continuous on its closure,
ρ
A(D ) = {f: D → C holomorphic in D and continuous on D }. (1.3)
ρ ρ ρ ρ
This space is naturally equipped with the supremum-norm,
kfk = max|f(z)|, z = (z ,...,z ) ∈ Cn, (1.4)
ρ 1 n
z∈Dρ
andthusnaturallyasubspaceofthecomplete normed (Banach)spaceC(D )
ρ
ofcontinuouscomplex-valuedfunctions. Thoughholomorphicfunctionsmay
have very complicated boundary behavior and thus A(U) ( O(U), they are
continuous and therefore for any smaller domain U0 relatively compact in U
(i.e., when U0 b U), there is an obvious inclusion A(U0) ⊂ O(U).
Theorem 1.2. The space A(D ) and its vector counterparts Am(D ) are
ρ ρ
complete (Banach) spaces.
Proof. Any fundamental sequence in A(D ) is by definition fundamental
ρ
in the Banach space C(D ) and has a uniform limit in the latter space. By
ρ
the Compactness principle (Theorem 0.5), this limit is again holomorphic
in D , i.e., belongs to A(D ). (cid:3)
ρ ρ
A map F of a metric space M into itself is called contracting, if
dist(F(u),F(v)) 6 λ dist(u,v) for some positive real number λ < 1 and
all u,v ∈ M.
Theorem 1.3 (Contracting map principle). Any contracting map F: M →
M of a complete metric space M has a unique fixed point w ∈ M such that
F(w) = w.
This fixed point is the limit of any sequence of iterations u = F(u ),
k+1 k
k = 0,1,2,... beginning with an arbitrary initial point u ∈ M.
0
Proof. For any initial point u ∈ M, the sequence u , k = 1,2,... is
0 k
fundamental, since dist(u ,u ) 6 λkdist(u ,u ) and by the triangle in-
k k+1 0 1
equality dist(u ,u ) 6 dist(u ,u )λk/(1−λ) for any k < l. By completeness
k l 0 1
assumption, the sequence u converges to a limit w ∈ M. Since F is con-
k
tinuous, passing to the limit in the identity u = F(u ) yields w = F(w).
k+1 k
4 1. Normal forms and desingularization
If w ,w are two fixed points, then dist(w ,w ) 6 λdist(F(w ),F(w )) =
1 2 1 2 1 2
λdist(w ,w ) which is possible only if dist(w ,w ) = 0, i.e., when w =
1 2 1 2 1
w . (cid:3)
2
1.3. Picard operators and their contractivity. Denote by D = {|z−
ε
z | < ε, |t−t | < ε} ⊂ Cn+1 a polydisk centered at the point (t ,x ) ∈ U
0 0 0 0
and sufficiently small to belong to U.
Definition 1.4. The Picard operator P associated with the differential
equation (1.1) and the initial value (t ,x ) ∈ U, is the operator
0 0
P: An(D ) → An(D ),
ε ε
Z t (1.5)
Pf(t,v) = v+ F(t,f(z,v))dz.
t0
DenotebyL andL theboundsforthemagnitudeofF anditsLipschitz
0 1
constant in U: for any (t,x),(t,x0) ∈ U,
|F(t,x)| 6 L , |F(t,x)−F(t,x0)| 6 L |x−x0|. (1.6)
0 1
Lemma 1.5. If the polydisk D is sufficiently small, the Picard operator P
ε
(1.5) restricted on An(D ) is well defined and contracting. More precisely,
ε
for sufficiently small ε its contraction factor λ does not exceed εL , where
1
L is the Lipschitz constant for F in U.
1
Proof. Explicit majorizing of the integral shows that
Z |t−t0|
|Pf(t,v)−v| 6 L |dz| 6 L ε,
0 0
0
so if ε is chosen sufficiently small, the operator P is well defined on An(D )
ε
and maps this space into itself. For any two vector functions f,f0 defined
on such small polydisk D , we have by virtue of the same estimate
ε
Z |t−t0|
kPf −Pf0k = sup L |f(z,v)−f0(z,v)||dz| 6 εL kf −f0k.
1 1
|t−t0|<ε 0
If εL < 1, the operator P is contracting. (cid:3)
1
Proof of Theorem 1.1. Assume ε be so small that the εL < 1 so that by
1
Lemma 1.5, the Picard operator P is contracting. By Theorem 1.2 the fixed
point of this operator (which exists by Theorem 1.3 and Lemma 1.5) is a
holomorphic vector function f: D → Cn that satisfies the integral equation
ε
Z t
f(t,v) = v+ F(t,f(z,v))dz. (1.7)
t0
1. Holomorphic differential equations 5
For each fixed x , the function ϕ(t) = f(t,x ) clearly satisfies both the
0 0
initial condition (1.2) and the differential equation (1.1). By construction,
it depends holomorphically on the initial condition x .
0
To prove holomorphic dependence on additional parameters, one can
treatthemasfictitiousdependentvariables. Assumethatthevectorfunction
F = F(t,x,y) depends holomorphically on additional parameters y ∈ Cm,
and consider the initial value problem (recall that the dot means the deriv-
ative d)
dt
(
x˙ = F(t,x,y), x(t ) = x ,
0 0
(1.8)
y˙ = 0, y(t ) = y .
0 0
Solution of this initial value problem is a function f(t,x,y,x ,y ) holomor-
0 0
phically depending on all variables. (cid:3)
Remark 1.6. ForadifferentialequationwiththerighthandsideF(t,x)the
shifted solution x0(t) = x(t−y), y ∈ C1, satisfies the shifted equation x˙0 =
F(t−y,x0)whichanalyticallydependsontheparametery. ByTheorem1.1,
thisshowsthatsolutionsoftheinitialvalueproblemdependholomorphically
also on the t-component of the initial point (t ,x ) ∈ U.
0 0
1.4. Principal example: exponential formula for linear systems.
The proof of the existence theorem is constructive: solution of a differential
equation is the uniform limit of its Picard approximations, iterations of the
Picard operator.
In the simplest case of a differential equation with constant (i.e., inde-
pendentoft,x,y)righthandsideF = const ∈ Cn thePicardapproximations
stabilize immediately: if f (t,v) = v, then f (t,v) = f (t,v) = ··· = v+tF.
0 1 2
A linear system with constant coefficients is the system of equations
x˙ = Ax, x ∈ Cn, A ∈ Mat(n,C) (1.9)
where A = ka k is a constant (n × n)-matrix with complex entries inde-
ij
pendent of t. Reasoning by induction, one can see that the Picard approx-
imations for solution of (1.9) which start with the constant initial term
f (t,v) = v, have the form
0
(cid:16) (cid:17)
f (t,v) = E +tA+ t2A2+···+ tkAk v. (1.10)
k 2! k!
Indeed,
Z t
Pf (t,v) = v+ A·(cid:0)E +sA+···+ skAk(cid:1)vds
k k!
0
= Ev+(cid:0)tA+···+ tk+1 Ak+1(cid:1)v = f (t,v).
(k+1)! k+1
These formulas motivate the following fundamental object.