Lectures on analytic diff. equations (web draft, March 2006)

545 Pages·3.909 MB·English

by  Ilyashenko Yu., Yakovenko S.

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 March 6, 2006
2supportedbythegrantNSFno.0100404
3TheGershonKekstProfessorofMathematics
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
Chapter 1. Normal forms and desingularization 1
1. Holomorphic differential equations 2
2. Holomorphic foliations and their singularities 12
3. Formal flows and embedding theorem 22
4. Formal normal forms 32
5. Holomorphic normal forms 49
6. Holomorphic invariant manifolds 64
7. Topological classification of holomorphic foliations 70
8. Desingularization in the plane 92
9. Complex separatrices of holomorphic line fields 118
Chapter 2. Singular points of planar analytic vector fields 127
10. Singularities of planar vector fields with characteristic
trajectories 128
11. Algebraic decidability of local problems. Center–focus
alternative 143
12. Holonomy and first integrals 165
13. Zeros of analytic functions depending on parameters and small
amplitude limit cycles 182
Chapter 3. Linear systems: local and global theory 217
14. General facts about linear systems 218
15. Local theory of regular singular points 227
16. Analytic and rational matrix functions.
Matrix factorization theorems 238
iii
iv Contents
Appendix: meromorphic solvability of cocycles 254
17. The Riemann–Hilbert problem: positive results 261
18. Negative answer for the Riemann–Hilbert problem in the
reducible case 269
19. Riemann–Hilbert problem on holomorphic vector bundles 277
20. Linear nth order differential equations 296
21. Irregular singularities and the Stokes phenomenon 314
Appendix: Demonstration of Sibuya theorem 326
Chapter 4. Advanced analytic normal form theory and nonlinear
Stokes phenomena 335
22. Nonlinear Stokes phenomenon for parabolic and resonant germs336
23. Complex saddles 364
24. Nonlinear Riemann–Hilbert problem 381
25. Nonaccumulation theorem for hyperbolic polycycles 392
Chapter 5. Global properties of planar polynomial foliations 415
26. Algebraic leaves of polynomial foliations on the complex
projective plane CP2 416
Appendix: Foliations with invariant lines and algebraic leaves of
foliations from the class A 452
r
27. Perturbations of Hamiltonian vector fields and zeros of Abelian
integrals 456
28. Generic global properties of analytic foliations of complex
projective plane 492
Appendix. First steps of formal classification of finitely generated
groups of conformal germs 518
Bibliography 525
List of Figures 535
Index 537
30. Finitely generated groups of holomorphic germs 533
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 = (ϕ ,...,ϕ ): 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)dϕ1,..., dϕn(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) and every
point (t ,x ) ∈ U there exists a sufficiently small polydisk D = {|t−t | <
0 0 ε 0
ε, |x −x | < ε, j = 1,...,n} ⊆ U, such that solution of the initial value
j 0,j
problem (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 ??), 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
for some positive real number λ < 1 and all u,v ∈ M the inequality
dist(F(u),F(v)) 6 λ dist(u,v) holds. A point w ∈ M is fixed (by F), if
F(w) = w.
Theorem 1.3 (Contracting map principle). Any contracting map F: M →
M of a complete metric space M has a unique fixed point in M.
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
F Awfully written!
until the end of the 1.3. Picard operators and their contractivity. Denote by D = {|z−
ε
subsection... z | < ε, |t−t | < ε} ⊂ Cn+1 a polydisk centered at the point (t ,z ) ∈ 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 ,z ) ∈ U, is the operator
0 0
P: An(D ) → An(D ),
ε ε
Z t (1.5)
Pf(s,v) = v+ F(s,f(z,v))ds.
t0
DenotebyL andL theboundsforthemagnitudeofF anditsLipschitz
0 1
constant in U: for any (t,x),(t,x0) ∈ U,
|F(t,z)| 6 L , |F(t,z)−F(t,z0)| 6 L |z−z0|. (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 |ds| 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(s,v)−f0(s,v)||ds| 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
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
1. Holomorphic differential equations 5
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-
pendent of t,x,y) right hand side F = const ∈ Cn the Picard approxima-
tions stabilize immediately: if f (t,v) = v, then f (t,v) = f (t,v) = ··· =
0 1 2
v+(t−t )F.
0
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 (independent of t and x) (n × n)-matrix
ij
with complex entries. Reasoning by induction, one can see that the Picard
approximations 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.