PII. S0161171203206025 http://ijmms.hindawi.com © Hindawi Publishing Corp.
A NOTE ON THE REDUCIBILITY OF LINEAR DIFFERENTIAL
EQUATIONS WITH QUASIPERIODIC COEFFICIENTS
XIAOPING YUAN and ANA NUNES
Received 3 June 2002
The system ˙x=(A+Q(t))x, whereAis a constant matrix whose eigenvalues are not necessarily simple andQis a quasiperiodic analytic matrix, is considered. It is proved that, for most values of the frequencies, the system is reducible.
2000 Mathematics Subject Classification: 37C55, 34A30, 34C20.
1. Introduction and results. Consider the quasiperiodic linear differential equation
˙
x=A+Q(t)x (1.1)
withx ann-dimensional vector,Aa constant square matrix of ordern, and Qa square matrix of ordern, quasiperiodic in timet. We say that a change of variables x =P (t)y is a Lyapunov-Perron (LP) transformation if P (t) is nonsingular andP (t),P−1(t), and ˙P (t)are bounded for allt∈R. Moreover, ifP,
P−1, and ˙Pare quasiperiodic in timet, we refer tox=P (t)yas a quasiperiodic
LP transformation. If there is a quasiperiodic LP transformationx =P (t)y such thatysatisfies the equation
˙
y=By (1.2)
withBa constant matrix, then we say that (1.1) is reducible.
The concept of the reducibility was first considered by Lyapunov (see [5]). There are several authors who investigated the reducibility of (1.1) (see, e.g., [1, 2,6]). The present paper complements the results obtained by Jorba and Simó [2], which we will briefly recall. To this end, we will introduce some notation and definitions that will be used throughout the paper.
We say that a function F is a quasiperiodic function in time t, with the basic frequencies ω= (ω1, . . . , ωr), if there exists a function Ᏺ(θ1, . . . , θr)
which is 2π-periodic in all its argumentsθj,j=1, . . . , r, and such thatF (t)=
Ᏺ(ω1t, . . . , ωrt). We callᏲthe hull ofF (t). The functionF will be called
Letλ=(λ1, . . . , λn)be the eigenvalues ofAandλ0()=(λ01(), . . . , λ0n())the
eigenvalues of ¯A:=A+Q, where ¯¯ Qis the average ofQ(t),
¯ Q=lim
T→∞
1 2T
T
−TQ(t)dt. (1.3)
Assume thatQ(t)is analytic on the strip of widthδ0>0 and that the vector
(λ,√−1ω)satisfies the nonresonance conditions
−1k·ω+l·λ≥|c
k|γ, (1.4)
wherel∈Znwith|l| =0,2 and 0≠k∈Zr. It was shown by Jorba and Simó [2]
that (1.1) is reducible forin some Cantorian setᏱ⊂(0, 0), with0sufficiently
small, provided that
(1) the eigenvaluesλ1, . . . , λnofAare different;
(2) the eigenvaluesλ0
1(), . . . , λ0n()of ¯Asatisfy
dd λ0i()−λ0j()
=0
>2ρ >0 (1.5)
for some constantρand any 1≤i < j≤n.
In [2], the basic idea is to kill the small perturbationQ(t)by KAM iteration. Condition (2) is used to overcome the problem arising from the frequency shift which comes up in this procedure. By a well-known theorem [3, pages 113–115], condition (1) guarantees that the eigenvaluesλ0j()of ¯A=A+Q¯ are differentiable in, and that therefore condition (2) can be imposed.
A natural question is: what happens when condition (1) or (2) is not satisfied? The main result of the present paper is the following theorem which gives an answer to this question.
Theorem1.1. LetΩ0⊂Rr+be a compact set with positive Lebesgue measure
and assume thatQ(t)is quasiperiodic with frequencyω∈Ω0and analytic in
the strip of widthδ0>0. Then, for a sufficiently small positive constantγ, there exist a subsetΩ⊂Ω0withMeas(Ω0\Ω)=Meas(Ω0)(1−O(γ1/n
2
))and a
suffi-ciently small constant = (δ0, γ) >0such that for any∈(0, ), system (1.1) is reducible. More exactly, there is an analytic quasiperiodic transformation x=P (t)ysuch that (1.1) is changed into
˙
y=By, (1.6)
whereBis a constant matrix withA−B =O().
of small divisors, certain frequencies must be removed from the original fre-quency setΩ0at each iteration step. Showing that the remaining frequencies
form a big subset ofΩ0through the estimates ofSection 3concludes the proof.
Remark1.2. When one ofλjis not simple, the functionsλ0j()are not
nec-essarily differentiable in. Therefore, in the hypothesis ofTheorem 1.1, we have to regard the tangent frequenciesω, instead of, as the parameters used to overcome the frequency shift in KAM iterative steps. Thus, we cannot find explicitly a tangent frequency vector ωsatisfying some Diophantine condi-tions such thatTheorem 1.1holds true. On the other hand, inTheorem 1.1it is not necessary to excise a subset of small measure from(0, ∗). In this sense, Theorem 1.1complements the results of [2]. Yet another complementary ap-proach is that of [1], where ωis fixed and reducibility is proved for “most” matricesA.
2. Proof of Theorem 1.1. The proof ofTheorem 1.1 is based on Newton iteration. Before we state the main iterative lemma, we need to introduce some notation.
In the following, we denote byC,C1,C2, . . .positive constants which arise
in the estimates, byᏽthe hull of a quasiperiodic functionQ(t), and by ˜ᏽthe average ofᏽon ther-torus. For a matrix-valued functionQ(t), define
QD:=sup t∈D
Q(t), (2.1)
where·is the sup-norm of the matrix.
Denote bymthe number of the iterative step, and let
(1) mbe the sequence that bounds the size of the perturbation before the
mth iteration step withm=(1+ρ)m−1andρ=1/3, for example;
(2) δmbe the sequence that measures the size of the analyticity domain in
the angular variables aftermiteration steps with
δm=δ0−
δ0
2
1+···+m−2 ∞
j=1
j−2 form≥1; (2.2)
(3) Um=U (δm)= {θ∈(C/2πZ)N:|Imθ|< δm};
(4) qmbe the sequence that measures the width of the analyticity domain
in the frequency space aftermiteration steps withqm=1/4n
2
m+1 , where
nis the dimensional number of system (1.1); (5) C(m)be a constant of the formC1mC2.
LetΩ0=Π0⊃Π1⊃ ··· ⊃Πm−1be the closed sets inRr+and letΠm⊂Πm−1
be as defined inductively inSection 3. Letᏻlbe the complexql-neighborhood
get a quasiperiodic linear differential equation
˙
x=Am−1+mQm(t;ω)
x, (2.3)
where the following conditions are satisfied:
(H1)m Am−1=A+1ᏽ˜1(ω)+···+m−1ᏽ˜m−1(ω),m≥2, A0=Awith ˜ᏽl(ω)
analytic inᏻl, andᏽ˜lᏻl≤1 forl=1, . . . , m−1;
(H2)m the hullᏽmofQm(t;ω)is analytic inUm−1×ᏻm−1and
ᏽm(θ, ω)Um−1×ᏻm−1≤1. (2.4)
Let
Am=Am−1+mᏽ˜m. (2.5)
Then (2.3) can be rewritten as
˙
x=Am+mQm∗(t)
x, (2.6)
whereQ∗m(t)=Qm(t)−ᏽ˜m. Following [2], we will find a change of variables
x=E+mPm(t)
y, (2.7)
whereEis the unit matrix such that (2.3) is changed into
˙
x=Am+O
m+1
x (2.8)
verifying conditions (H1)m+1and (H2)m+1. This change of variables is given by
the following lemma.
Lemma 2.1(iterative lemma). Assume that (H1)m and (H2)m are fulfilled. Then there is a quasiperiodic LP transformation
x=E+Pm(t)
y, (2.9)
wherePm(t)is quasiperiodic with frequencyωand its hullᏼm(θ;ω)is analytic
inUm×ᏻmsuch that (2.3) is changed into
˙
y=Am+m+1Qm+1(t)
y, (2.10)
whereAmandQm+1satisfy the conditions (H1)m+1and (H2)m+1.
Proof. Rewrite (2.3) as
˙
x=Am+mQm∗(t)
x, (2.11)
whereQ∗m(t)=Qm(t)−ᏽ˜mandAm=Am−1+ᏽ˜m. Hence, we can write
ᏽ∗
m(θ, ω)=
0≠k∈Zr
ˆ ᏽ∗
m(k;ω)e √
where ˆᏽ∗m(k;ω) is the k Fourier coefficient of ᏽm∗(θ, ω) in θ. Let Mm =
(1/bm)|lnm|, wherebm=δm−1−δm, and let
ᏽ∗1
m(θ, ω)=
0≠|k|≤Mm
ˆ ᏽ∗
m(k;ω)e √
−1k·θ,
ᏽ∗2
m(θ, ω)=
|k|>Mm
ˆ ᏽ∗
m(k;ω)e √
−1k·θ, (2.13)
so that
ᏽ∗
m(θ, ω)=ᏽ∗m1(θ, ω)+ᏽ∗m2(θ, ω). (2.14)
We claim that
ᏽ∗2
m(θ, ω)
Um×ᏻm≤
C(m)m, (2.15)
whereC(m)is a constant of the formC1mC2. In fact, ᏽ∗2
m(θ, ω)
Um×ᏻm≤
|k|>Mm ᏽˆ∗
m(k;ω)ᏻme √
−1k·θUm
≤
|k|>Mm ᏽˆ∗
m(θ;ω)
Um−1×ᏻme−|k|δm−1e|k|δm
≤
|k|>Mm
e−|k|(δm−1−δm)= m
|k|>0
e−|k|(δm−1−δm)
≤C(m)m.
(2.16)
Next, we perform the change of variables as in (2.7), where E is the unit matrix inRn, to transform (2.10) into
˙
y=E+mPm
−1A m+m
AmPm−P˙m+Q∗m1
+m+1Qm+1 y, (2.17)
where
m+1Qm+1=mQ∗m2+2m
E+mPm
−1
Q∗mPm. (2.18)
We would like to have
E+mPm−1Am+mAmPm−P˙m+Q∗m1
=Am (2.19)
and this implies that
˙
Pm=AmPm−PmAm+Q∗m1. (2.20)
In order to solve this equation, we consider
ω·∂ᏼm
∂θ =Amᏼm−ᏼmAm+ᏽ
∗1
whereᏼis the hull ofP (t). Write
ᏼm(θ, ω)=
0≠|k|≤Mm
ˆ
ᏼm(k;ω)e √
−1k·θ. (2.22)
Then we get
−1(k·ω)ᏼˆm(k)=Amᏼˆm(k)−ᏼˆm(k)Am+ᏽˆ∗m1(k), 0<|k| ≤Mm, (2.23)
where we omit the dependence onωto simplify the notation. That is,
−1(k·ω)E−Am
ˆ
ᏼm(k)+ᏼˆm(k)Am=ᏽˆm∗1(k), 0<|k| ≤Mm. (2.24)
By LemmasA.2and3.1, (2.24) is solvable forω∈ᏻmand
ᏼˆm(k;ω)ᏻm≤−1k·ωEn2−En⊗Am+AmT ⊗En−1
ᏻm
ˆ ᏽ∗
m(k)
ᏻm
≤Cγ|k|τ
m
ᏽ∗ m
Um−1×ᏻm−1e−|k|δm−1
≤Cγ|k|τ
m
ᏽmUm−1×ᏻm−1e−|k|δm−1
≤C|k|τ
γm
e−|k|δm−1,
(2.25)
where in the last inequality we have used (H2)m. Therefore,
ᏼm(θ, ω)Um×ᏻm≤ 0<|k|≤Mm
ᏼˆm(k;ω)ᏻme√−1k·θUm
≤
k∈Zr
C|k|τ
γm
e−|k|(δm−1−δm)
≤ C
γm
k∈Zr
|k|τe−C3|k|(δ0/m2)≤C(m),
(2.26)
where the last inequality follows fromLemma A.1. Then, the function
Pm(t)=ᏼm
ω1t, . . . , ωrt;ω
(2.27)
solves (2.20). By (2.26) and (2.18), it is easy to show thatᏽm+1Um×ᏻm≤1. We
omit the details.
Proof ofTheorem1.1. Obviously, (1.1) satisfies the conditions (H)mwith
m=1. In fact, condition (H2)1may always be fulfilled by a suitable rescaling
of. Thus, byLemma 2.1, there exists a sequence of transformationsx=(E+
mPm(t))y, m=1,2, . . ., such that the hullsᏼm of thePm(t)are analytic in
the domainsUm×ᏻm, and
ᏼmUm×ᏻm
Let
U∞×ᏻ∞= ∞
m=1
Um×ᏻm. (2.29)
Then, all theᏼm,m=1,2, . . . ,are well defined in the domainU∞×ᏻ∞. Set
Φ(θ, ω)= ···◦E+mᏼm(θ, ω)◦···◦E+2ᏼ2(θ, ω)◦E+1ᏼ1(θ, ω),
Φ(t;ω)= ···◦E+mPm(t;ω)◦···◦E+2P2(t;ω)◦E+1P1(t;ω).
(2.30)
Note thatE+mᏼm(θ;ω)U∞×ᏻ∞≤1+mC(m)≤1+2−m. We see thatΦ, and
thusΦ, are well defined. Let
x=Φ(t)y. (2.31)
Since mᏽmU∞×ᏻ∞ ≤ m → 0 as m → ∞, the transformation x = Φ(t)y
changes (1.1) into
˙
y=By, (2.32)
whereB=A+∞j=1m˜ᏽm. This, together withLemma 3.3, completes the proof
ofTheorem 1.1.
3. Estimates on the allowed frequencies set. LetΠl(0≤l≤m−1)be a
sequence of compact subsets ofRr
+withΩ=Π0⊃Π1⊃ ··· ⊃Πm−1and denote
byᏻlthe complexql-neighborhood ofΠl,l=0, . . . , m−1. Recall that
Am(ω)=A+1ᏽ˜1(ω)+···+mᏽ˜m(ω), (3.1)
where, forl=1, . . . , m−1, the ˜ᏽl(ω)are analytic, and real for real arguments
in the domainᏻl, andᏽ˜l(ω)ᏻl≤1; and ˜ᏽm(ω)is analytic, and real for real
arguments in the domainᏻm−1, and˜ᏽm(ω)ᏻm−1 ≤1. Denote by| · |d the
determinant of a matrix. Let
k(m):=
ω∈Πm−1:−1k·ωEn2−En⊗Am+AmT ⊗End<
γm
|k|τ1
,
(3.2)
whereγm=γ/m2n
2
andτ1=(r+1)n2,
Πm=Πm−1\
0<|k|≤Mm
k(m), (3.3)
whereMm= |lnm|/(δm−1−δm)is the number of Fourier coefficients we must
consider at themth step of the iteration, and denote byᏻmthe complexqm
Lemma3.1. Letτ=τ1+n2−1and
G(ω)=−1k·ωEn2−En⊗Am(ω)+ATm(ω)⊗En. (3.4)
Then, forω∈ᏻmand0<|k| ≤Mm, the inverse ofG(ω)exists and it is analytic
in the domainᏻmwith
G−1(ω)ᏻm≤Cγ−1
m|k|τ. (3.5)
Proof. By the definition ofΠm, we get that forω∈Πmand 0<|k| ≤Mm,
ᏹk(ω)≥ γm
|k|τ1. (3.6)
It is easy to see that
G(ω)ᏻm≤C5|k|, k≠0, (3.7)
whereC5=2(max{|ω|:ω∈Π} + A +1). Since detG(ω)=ᏹk(ω),G−1(ω)
exists forω∈Πmand
G−1(ω)=adjG(ω)ᏹ
k(ω)
, (3.8)
where adj is the adjoint of a matrix. Thus, for 0<|k| ≤Mm,
G−1(ω)Πm≤C6 |
k|n2−1
γm/|k|τ1 =
C6γm−1|k|τ. (3.9)
Now, we assume thatω∈ᏻm. Then there is anω0∈Πmsuch that|ω−ω0|<
qm. Thus,
G−1ω
0G(ω)−G
ω0
≤G−1(ω)Πm∇
ωG(ω)ᏻm·ω−ω0
≤C6γm−1|k|τG(ω)ᏻm−1
qm
qm−1−qm
≤C6γm−1|k|τ+1
qm
qm−1−qm
≤C6Mmτ+1γ−m1
qm
qm−1−qm
≤C6m(6+2r )n 2
lnmτ+11/4n
2
m+1
γδ0 −1
m1/4n2−1/4n
2
m+1
<1 2.
Therefore,E+G−1(ω
0)(G(ω)−G(ω0))has its inverse which is analytic in
ᏻmsince
E+G−1ω0
G(ω)−Gω0 −1
= ∞
j=0
−G−1ω0
G(ω)−Gω0 j
. (3.11)
So,G(ω)has its inverse forω∈ᏻmand
G−1(ω)=E+G−1ω 0
G(ω)−Gω0 −1
·G−1ω 0
≤E+G−1ω 0
G(ω)−Gω0 −1
·G−1ω 0
≤Cγ−1
m|k|τ.
(3.12)
Lemma3.2. LetK=n2(n+1)2(diamΠ0)r−1. Then the Lebesgue measure of
k(m)verifies
Meask(m)≤
Kγ1/n2 |k|r+1
1
m2. (3.13)
Proof. Recall thatql=1/4n2
l+1 , let q 1
l =(5/6)ql+(1/6)ql+1, and denote
by ᏻ1l the complex ql1-neighborhood of Πl. Obviously, ᏻl+1 ⊂ ᏻ1l ⊂ ᏻl and
dist(∂ᏻ1
l, ∂ᏻl)=(1/6)(ql−ql+1) > (1/12)ql. Noting thatᏽ˜lᏻl≤1 and using
Cauchy’s theorem, we get for 1≤s≤n2and 0≤l≤m−1,
l∂ωsᏽ˜lᏻ
1 l ≤
l
12q−l1
s˜
ᏽlᏻl≤1l/2. (3.14)
The combination of (3.1) and (3.14) leads to
∂s
ωAm(ω)ᏻ
1
m−1≤1/2. (3.15)
Let
B(ω):= −En⊗Am(ω)+ATm⊗En. (3.16)
Then
∂s
ωB(ω)ᏻ
1
m−1≤1/2. (3.17)
Set
ᏹk(ω)=−1k·ωEn2+B(ω)d. (3.18)
We are now in a position to estimate∂s
ωᏹk. To this end, writeB(ω)=(bij).
Then
ᏹk(ω)=
−1n2(k·ω)n2+
1≤l≤n2−1
φl(ω)(k·ω)n
2−l
where
φl(ω)=
1≤ji≤n2
σj1···jlb1j1···bljl (3.20)
andσj1···jl∈ {−1,+1,− √
−1,+√−1}. Observe that for 1≤l≤n2andω∈ᏻ1
m−1,
∂s
ω1bij≤∂
s
ωB(ω)ᏻ
1
m−1≤1/2. (3.21)
Thus, forω∈ᏻ1
m−1and 1≤s≤n2,
ds
dωs1
b1j1···bljl≤
s1+···+sl=s
ds1
dωs1 1
b1j1
···
dsl
dωsl 1
b1jl
≤(1/2)(s1+···+sl)
s1+···+sl=s
1
≤21/2s
,
(3.22)
and therefore,
dωdss
1
φl(ω)
≤ n2 l
21/2s. (3.23)
Without loss of generality, assume that|k| = |k1| + ··· + |kr| ≤r|k1|. Hence,
for everyω∈ᏻ1
m−1,
d
n2
dωn12
1≤l≤n2−1
φl(ω)(k·ω)n
2−l
≤
1≤l≤n2−1
dn2
dωn12
φl(ω)(k·ω)n
2−l
≤
1≤l≤n2−1
l≤s≤n2 n2 s
ds2
dωs12
φl(ω)
dn2−s
dωn12−s
(k·ω)n2−l
≤
1≤l≤n2−1
l≤s≤n2 n2 l n2 s
21/2sk 1
n2−s
|k·ω|s−ln2!
≤C41/2k1
n2−1
n2!,
(3.24)
whereC4is some constant which depends only onn,r, and on the maximum
of|ω|inΠ0. Obviously,
dn2
dωn12
(k·ω)n2=n2!k1
n2
Thus, inᏻ1m−1, we have
d
n2
dωn12
ᏹk(ω)
≥n2!k1
n2
1−C41/2k1− 1
>1 2n
2!k 1
n2
(3.26)
provided thatis small enough so thatC41/2<1/2. Using (3.26) andLemma
A.3, we get
Meask(m)≤n2
n2+1
γm
|k|τ1 1/n2
diamΠ0 r−1
≤Kγ1/n 2
|k|r+1 ·
1 m2.
(3.27)
This completes the proof.
ByLemma 2.1, the nested sequence of closed sets
Ω0=Π0⊃Π1⊃ ··· ⊃Πm⊃ ··· (3.28)
is defined inductively. The following lemma is a corollary ofLemma 3.2.
Lemma3.3. Let
Π∞= ∞
m=0
Πm. (3.29)
ThenMeasΠ∞=(MeasΠ0)(1−O(γ1/n 2
)).
Appendix
LemmaA.1. Forδ >0andν >0, the following inequality holds true:
k∈ZN
e−2|k|δ|k|ν≤ν
e
ν 1
δν+N(1+e)
N. (A.1)
Proof. This lemma can be found in [1]. We will find the value of z≥1
yielding a maximum value for the expressionνlnz−δz. Differentiating it in zand equating the result to zero, we get thatν/z=δandz=ν/δ >1. From this it follows that
νlnz−δz≤ν
lnν δ−1
. (A.2)
This expression yields
zν≤exp(δz)exp
ν
lnν δ−1
=ν
e
νexp(δz)
Thus,
k∈ZN
e−2|k||k|ν≤
ν
e
ν 1
δν
k
e−|k|δ
= ν
e
ν 1
δν
1+exp(−δ)
1−exp(−δ)
N
≤ ν
e
ν 1
δν
1+e
δ
N
.
(A.4)
LemmaA.2. LetA,B, andCber×r,s×s, andr×smatrices, respectively;
and letXbe anr×sunknown matrix. Then the matrix equation
AX+XB=C (A.5)
is solvable if and only if the vector equation
Es⊗A+BT⊗Er
X=C (A.6)
is solvable, whereX=(XT
1, . . . , XsT)T,C=(C1T, . . . , CsT)Tif we writeX=(X1, . . . ,
Xs)andC=(C1, . . . , Cs). Moreover,
X ≤Es⊗A+BT⊗Er−1C (A.7)
if the inverse exists.
Proof. This lemma can be found in many textbooks on matrix theory; for
example, [4, page 256].
The following lemma can be found in [7, page 23].
LemmaA.3. LetᏵbe an interval inR1and¯Ᏽits closure. Suppose thatg: ¯Ᏽ→
Cisktimes continuously differentiable. LetᏵh= {x∈Ᏽ¯:|g(x)| ≤h},h >0. If,
for some constantd >0,|dkg(x)/dxk| ≥dfor anyx∈Ᏽ, thenMeasI
h≤ch1/k
withc=2(2+3+···+k+d−1).
Acknowledgment. This work was carried out during the first author’s
stay at the Centro de Matemática e Aplicações Fundamentais (CMAF), Universi-dade de Lisboa with a Postdoctoral Fellowship Grant from the Fundação para a Ciência e Tecnologia.
References
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[3] T. Kato,Perturbation Theory for Linear Operators, corrected printing of the 2nd ed., Springer-Verlag, Berlin, 1980.
[4] P. Lancaster,Theory of Matrices, Academic Press, New York, 1969.
[5] V. V. Nemytskii and V. V. Stepanov,Qualitative Theory of Differential Equations, Princeton Mathematical Series, no. 22, Princeton University Press, New Jer-sey, 1960.
[6] J. Xu and Q. Zheng, On the reducibility of linear differential equations with quasiperiodic coefficients which are degenerate, Proc. Amer. Math. Soc.126
(1998), no. 5, 1445–1451.
[7] J. You,Perturbations of lower-dimensional tori for Hamiltonian systems, J. Differ-ential Equations152(1999), no. 1, 1–29.
Xiaoping Yuan: Department of Mathematics, Fudan University, Shanghai 200433, China
Current address: Centro de Matemática e Aplicações Fundamentais, Universidade de Lisboa, 1649-003 Lisboa, Portugal
Ana Nunes: Centro de Matemática e Aplicações Fundamentais, Universidade de Lis-boa, 1649-003 LisLis-boa, Portugal
