EFFECTIVE REDUCIBILITY OF QUASIPERIODIC LINEAR
EQUATIONS CLOSE TO CONSTANT COEFFICIENTS


ANGEL JORBA
y
,RAFAEL RAMREZ-ROS
y
,
AND
JORDI VILLANUEVA
y
.
Abs ac .
Le us conside he die en ial equa ion
_
x
=(
A
+
"Q
(
 "
))
x
j
"
j
"
0

whe e
A
is an ellip ic cons an ma ix and
Q
depends on ime in a quasipe io dic (andanaly ic) way.
I is also assumed ha he e igen alues o
A
and he basic equencie s o
Q
sa is y a diophan ine
condi ion. Then i is p o ed ha his sys em can be e duce d o
_
y
=(
A

(
"
)+
"R

(
 "
))
y
j
"
j
"
0

whe e
R

is exp onen ially small in
"
,and helinea change o a iables ha pe o ms such educ ion
is also quasipe io dic wi h hesame basic equencie s han
Q
.The e sul s a e illus a ed and discuss e d
in a p ac ical example.
Key wo ds.
quasipe io dic Flo que heo em, quasipe io dic pe u ba ions, e ducibili yo linea
equa ions.
AMS(MOS) sub jec class ica ions.
34A30, 34C20, 34C27, 34C50, 58F30
1. In o duc ion.
Thewell-known Flo que heo em s a es ha anylinea p e i-
o dic sys em, _
x
=
A
(
)
x
, can b e e duce d ocons an co ecien s, _
y
=
By
,bymeans o
a p e io dic change o a iable s. Mo eo e , hi s change o a iable s can b e aken, o e
C
,wi h hesamepe iod han
A
(
).
Ana u al ex ens ion i s oconside hecaseinwhich hema ix
A
(
)dep ends on
ime in a quas ip e io dic way. Be o e s a ing he di scuss ion o hi s i ssue, le us ecall
hedeni ion and bas ic p op e ie s o quas ip e io dic unc ions.
De ini ion 1.1.
A unc ion
is a quasipe iodic unc ion wi h ec o o basic
equencies
!
=(
!
1
:::!
)
i
(
)=
F
(

1
:::
)
, whe e
F
is
2

pe iodic in al l i s
a gumen s and

j
=
!
j
o
j
=1
:::
.Mo eo e ,
is cal led analy ic on a s ip o
wid h

i
F
is analy ical on an open se con aining
j
Im

j
j

o
j
=1
:::
.
I i s also known ha ananaly ic quas ip e io dic unc ion
(
) on a s ip o wid h

has Fou ie co ecien sdened by
k
=
1
(2

)
Z
T
F
(

1
:::
)
e
;
(
k
)
p
;
1
d 
such ha
can b e expanded as
(
)=
X
k
2
Z
k
e
(
k!
)
p
;
1

o all
such ha
j
Im
j
=
k
!
k
1
.Wedeno eby
k
k

he no m
k
k

=
X
k
2
Z
j
k
j
e
j
k
j


andi isno dicul ocheck ha i iswell dened o anyanaly ical quas ip e io dic
unc ion dene d on a s ip o wid h

.Finally, odeneananaly ic quas ip e io dic

This pape was w i en on Janua y 20 h, 1995.
y
Dep . deMa ema ica Aplicada I, ETSEIB, Uni e si a Poli ecnica deCa alunya, Diagonal 647,
08028 Ba celona, Spain (E-mails:
jo [email protected]
,
a ael@ e e.upc.es
,
jo di@ e e.upc.es
).
1
ma ix, we no e ha all hese deni ions hold when
is a ma ix- alue d unc ion. In
hi s cas e, odene
k
k

weuse heinni y no m ( ha willbedeno ed by
jj
1
) o
hema ice s
k
.
A e hos e deni ions andp op e ie s, le us e u n o he p oblem o he edu-
cibili yo a linea quas ip e io dic equa ion, _
x
=
b
A
(
)
x
, o cons an co ecien s. The
app oacho hi s wo kis o assume ha he sys em i s clos e ocons an co ecien s,
ha is,
b
A
(
)=
A
+
"Q
(
 "
), whe e
"
is small. Thi s cas e has al eady b een cons ide e d
in many pap e s (s ee 2], 8]and9]amongo he s), and he esul scanbesumma ize d
as ollows: le

i
be he e igen alue s o
A
,and

ij
=

i
;

j
, o
i
6
=
j
.Then, i all he
alue s Re

ij
a e die en om ze o, he educ ion can b e p e o med o
j
"
j
<"
0
,
"
0
sucien ly small (s ee 2]). I someo heRe

ij
a e ze o ( hi s happ ens, o ins ance,
i
A
i s ellip ic, ha i s, i all he

i
a eon heimagina y axi s) mo e hyp o hesis a e
nee ded. The usual oneisadiophan inecondi ion in ol ing he

ij
and hebasic e-
quencie s o
Q
(
 "
), and o assumeanondegene acy condi ion wi h e sp ec o
"
on he
co e sp onding

ij
(
"
)o hema ix
A
+
"Q
(
"
)(
Q
(
"
)deno es hea e age o
Q
(
 "
)).
Thi s allows o p o e(see 9] o hede ails) ha he e exi s saCan o ian s e
E
such
ha he educ ion can b e p e o me d o all
"
2E
. Mo eo e , he ela i emeasu e o
he s e 0
"
0
]
nE
in 0
"
0
] i s exp onen ially small in
"
0
.
Ou pu p os e he e i s a li le bi die en : ins ead o lo oking o a o al e duc ion o
cons an co ecien s( hi s s eems oleadus o elimina ea dens e s e o alue s o
"
,see
8]o 9]), we y o minimize he quas ip e io dic pa , wi hou akingou any alue o
"
.The e sul ob ained is ha he quasipe iodic pa can be made exp onen ially small.
As all he p o o i s cons uc i e(and i can b e ca ie d ou wi h a ni enumbe o
s eps), i can b e applie d o p ac ical example s in o de o do an eec i e" e duc ion:
i
"
is small enough, he emainde will b e so small ha , o p ac ical pu p os e s, i
can b e aken equal o ze o. The e o p o duce d wi h hi s d opping can b e b ounded
eas ily,bymeans o heG onwall lemma. Finally,wewan o s e ss ha weha ealso
elimina ed henondegene acy hyp o he s i s o p e ious pap e s (8], 9]).
Be o e ni shing hi s in o duc ion, wewan omen ion some s imila e sul sob-
ained when hedynamics o he sys em i s slow: _
x
=
"
(
A
+
"Q
(
 "
))
x
. Thi s cas e
is con aine d in 14], whichisanex ens ion o 12]. The esul ob ained is also ha
he quas ip e io dic pa can b e made exp onen ially small in
"
.To al e ducibili yhas
b een also cons ide e d in hi s cas e: in 15]iss a ed ha he educ ion can b e p e o med
excep o a s e o alue s o
"
o measu e exp onen ially small.
The e a e manyo he e sul s o he educibili y p oblem. Fo ins ance, in he
cas e o heSch odinge equa ion wi h quasipe iodic po en ial wecanmen ion 3], 4],
5], 10], 11]and13]. Ano he class ical and ema kable pap e i s 7], whe e hegene al
cas e ( ha i s, wi hou asking o b e clos e o cons an co ecien s) i s cons ide e d. Finally,
he class ical e sul s o quas ip e io dic sys ems can b e oundin6].
In o de o s impli y he eading, hepape has b een di ided in sec ions as ollows:
Sec ion 2 con ains he exp os i ion (wi hou echnical de ails) o hemain ideas and
me ho dology, Sec ion 3 con ains hemain heo em, Sec ions 4 and5 a e de o ed o
he p o o s and, nally,Sec ion 6 con ains an example oshowhow hese esul s can
be applie d o a conc e e p oblem.
2. Theme hod.
Theme hod used is based on hesameinduc i escheme ha
8]. Le us w i e ou equa ion as
_
x
=(
A
+
"Q
(
 "
))
x
(1)
2
whe e
A
i s an ellip ic
d

d
ma ix and
Q
(
 "
) i s quas ip e io dic wi h
!
=(
!
1
:::!
)
as ec o o bas ic equencie s, andanaly ic on a s ip o wid h

. Fi s o all, le us
ew i e hi s equa ion as
_
x
=(
A
0
(
"
)+
"
e
Q
(
 "
))
x
whe e
A
0
(
"
)=
A
+
Q
(
"
)and
e
Q
(
 "
)=
Q
(
 "
)
;
Q
(
"
). Now le us assume ha wea e
able ond a quasipe iodic
d

d
ma ix
P
(wi h he same bas ic equencie s han
Q
)
e i ying
_
P
=
A
0
(
"
)
P
;
PA
0
(
"
)+
e
Q
(
 "
)

(2)
such ha
k
"P
(
 "
)
k

<
1, o some
>
0. In hi s cas e, i i s no dicul ocheck ha
hechange o a iable s
x
=(
I
+
"P
(
 "
))
y
ans o ms equa ion (1) in o
_
y
=(
A
0
(
"
)+
"
2
(
I
+
"P
(
 "
))
;
1
e
Q
(
 "
)
P
(
 "
))
y:
(3)
As hi s equa ion i s like(1) bu wi h
"
2
ins ead o
"
, heinduc i escheme s eems clea :
oa e age he quas ip e io dic pa o (3) and o es a hi s p o ce ss. Themain dicul y
ha app ea in hi s p o ce ss come s om equa ion (2), b ecaus e hesolu ion con ains he
denomina o s

i
(
"
)
;

j
(
"
)+
p
;
1(
k !
), 1

i j

d
,whe e

i
(
"
)a e heeigen alue s
o
A
0
(
"
)( hi s i s shown ins ide he p o o o Lemma 4.2). Thi s di i so app ea s in he
k
h
Fou ie co ecien o
P
.No e ha i he alue s

i
(
"
)
;

j
(
"
) a e ou side heimagina y
axi s, he (mo dulus o he) di i so can b e b ounded om below, b e ingeasy op o e he
con e gence. On heo he hand, he alue

i
(
"
)
;

j
(
"
)+
p
;
1(
k !
) can b e a bi a ily
small gi ing ise o con e gence p oblems.
2.1. A oiding hesmall di i so s.
Le us s a assuming ha heeigen alue s

i
o he o iginal unp e u b e d ma ix
A
(s ee equa ion (1)) and he bas ic equencie s
o
Q
sa i s y hediophan ine condi ion
j

i
;

j
+
p
;
1(
k !
)
j
c
j
k
j


8
k
2
Z
n
0
g
:
(4)
whe e
j
k
j
=
j
k
1
j
+

+
j
k
j
.No e ha , in p inciple, we can no gua an ee ha in
equa ion (2) hi s condi ion holds, b ecaus e he e igen alue s o
A
0
(
"
)ha e b een change d
wi h e sp ec o heones o
A
(in an amoun o
O
(
"
)) andsomeo hedi iso scanbe
e y small o e en ze o.
Thekey p oin is o ealize ha , as heeigen alue s o
A
mo ein an amoun o
O
(
"
)a mos , hequan i ie s

i
(
"
)
;

j
(
"
) a e con aine d in a (complex) ball
B
ij
(
"
)
cen e e d in

i
;

j
andwi h adius
O
(
"
). As hecen e o heballsa i se s condi ion
(4), he alue s (
k !
) can no b e ins ide ha balli
j
k
j
i s le ss han some alue
M
(
"
).
Thi s implie s ha i i s p oss ible o cancel all heha monics such ha 0
<
j
k
j
<M
(
"
),
b ecaus e hey do no p o duce small di i so s (no e ha we can only ha e e sonance s
when (
k !
)isinside
B
ij
(
"
)). Theha monics wi h
j
k
j
M
(
"
) a e exp onen ially small
in
M
(
"
) (when
M
(
"
)
!1
), hi s i s, exp onen ially small in
"
(when
"
!
0), so wedo
no nee d o elimina e hem.
Theidea o cons ide ing only equencie s le ss han some h e shold
M
has al eady
b een applie d b e o e in o he con ex s (s ee, o ins ance, 1]).
2.2. Thei e a i escheme.
Toapply heconside a ions abo ewedene, as
b e o e,
A
0
(
"
)=
A
+
"Q
(
"
),
e
Q
(
 "
)=
Q
(
 "
)
;
Q
(
"
)andwe spli
e
Q
(
 "
)in hesum
3
o woma ice s
Q
0
(
 "
),
R
0
(
 "
):
Q
0
(
 "
)con ains heha monics
Q
k
e
(
k!
)
p
;
1
wi h
j
k
j
<M
(
"
)and
R
0
(
 "
) heone s wi h
j
k
j
M
(
"
). So, (1) can b e ew i ed as
_
x
=(
A
0
(
"
)+
"Q
0
(
 "
)+
"R
0
(
 "
))
x
(5)
Now heidea i s o cancel
Q
0
(
 "
)and o lea e
R
0
(
 "
) (i i s al eady exp onen ially
small wi h
"
). So, we compu e
P
0
such ha
_
P
0
=
A
0
(
"
)
P
0
;
P
0
A
0
(
"
)+
Q
0
(
 "
)
:
Then, hechange
x
=(
I
+
"P
0
(
 "
))
y
gi es
_
y
=

A
0
+
"
2
(
I
+
"P
0
)
;
1
Q
0
P
0
+
"
(
I
+
"P
0
)
;
1
R
0
(
I
+
"P
0
)

y:
Thi s equa ion can b e ew i en obelike (5) o ep ea he p o ce ss. No e ha he
s ize o heha monics wi h0
<
j
k
j
<M
(
"
)has b een squa e d. As we will s ee in he
p o o s, hi s i s enough o gua an ee con e gence o hos e e ms o ze o. Thus, henal
equa ion has a pu ely quas ip e io dic pa exp onen ially small wi h
"
.
2.3. Rema ks.
I i s in e e s ing ono e ha i i s enough oapply a ni enumbe
o s eps o heinduc i e p o ce ss: wedono nee d o cancel comple ely heha monics
wi h0
<
j
k
j
<M
(
"
)bu we can s op he p o ce ss when hey a e o hesamesize
o heones o
R
( om he p o o i can b e s een ha henumbe o s eps nee ded o
achie e hi s i s o o de
j
ln
j
"
jj
). Thi s allows oapply (wi h hehelp o a compu e )
hi s p o ce du e on a p ac ical example.
Ano he ema kable p oin isabou hediophan inecondi ion: no e ha weonly
nee d he condi ion up o a ni e o de (
M
(
"
), ha iso o de (1
=
j
"
j
)
1
=
,asweshall
see in he p o o s). Thi s means ha , in a p ac ical example when hepe u bing
equencie s a e known wi hni e p eci s ion, he diophan ine condi ion can b e checked
eas ily.
3. TheTheo em.
In wha ollows,
Q
d
(
 !
)s a es o he s e o heanaly ic
quas ip e io dic
d

d
ma ice s on a s ip o wid h

andha ing
!
as ec o o bas ic
equencie s. Mo eo e ,
i
will deno e
p
;
1.
Theo em 3.1.
Conside he equa ion
_
x
=(
A
+
"Q
(
 "
))
x
,
j
"
j
"
0
,and
x
2
R
d
,
whe e
1.
A
isacons an
d

d
ma ix wi h die en eigen alues

1
:::
d
.
2.
Q
(

"
)
2Q
d
(
 !
)
wi h
k
Q
(

"
)
k


q
,
8j
"
j
"
0
, o some
!
2
R
,and
q  >
0
.
3. The ec o
!
sa ises he diophan ine condi ions
j

j
;

`
+
i
(
k !
)
j
c
j
k
j


8
k
2
Z
n
0
g

8
j `
2
1
:::d
g

(6)
o some cons an s
c>
0
,
>
;
1
. As usual,
j
k
j
=
j
k
1
j
+

+
j
k
j
.
Then he e exis posi i e cons an s
"

,
a

,

and
m
such ha o al l
"
,
j
"
j
"

, he
ini ial equa ion can be ans o med in o
_
y
=(
A

(
"
)+
"R

(
 "
))
y
(7)
whe e:
1.
A

isacons an ma ix wi h
j
A

(
"
)
;
A
j
1

a

j
"
j
.
2.
R

(

"
)
2Q
d
(
 !
)
and
k
R

(

"
)
k

;



exp

;

m
j
"
j

1
=


,
8

2
]0

]
.
4
Fu he mo e he quasipe iodic change o a iables ha pe o ms his ans o ma ion
is also an elemen o
Q
d
(
 !
)
. Final ly, a gene al explici compu a ion o
"

,
a

,

and
m
is possible:
"

= min

"
0


eq 
(3
d
;
1)

a

=
eq 
2
e
;
1


=
ea

m
=
c
10
eq 
whe e
e
= exp (1)
,

=min
j
6
=
`
(
j

j
;

`
j
)
and

is he condi ion numbe o a egula
ma ix
S
such ha
S
;
1
AS
is diagonal, ha is,

=
C
(
S
)=
j
S
;
1
j
1
j
S
j
1
.
Rema k 3.1.
Fo xed alues o

1
:::
d
and

hypo hesis 3 is no sa ised
o any
c>
0
only o a se o alues o
!
o ze omeasu ei
>
;
1
.
Rema k 3.2.
In case ha he eigen alues o he pe u bed ma ices mo e on
bal ls o adius
O
(
"
p
)
( ha is, i he nondegene acy hypo hesis needed in 8] o 9]
is no sa ised), i is no dicul o show ha he bound o he exponen ial can be
imp o ed:
k
R

(

"
)
k

;



exp (
;
(
m=
j
"
j
)
p=

)
.Thep oo is e y simila , bu using
M
(
"
)=(
m=
j
"
j
)
p=
ins ead o
(
m=
j
"
j
)
1
=
.
Thi s las ema k s eems oshow ha hi s nondegene acy hyp o hesis is no nece s-
sa y,andi isonlyused o echnical easons. In ac , he esul sseem obebe e
when hi s hyp o hesis is no sa i se d.
Rema k 3.3.
I he unpe u bed ma ix
A
has mul iple eigen alues ( ha is, i
hypo hesis 1 is no sa ised) he heo em is s il l ue, bu he exponen o
"
in he ex-
ponen ial o he emainde is sligh ly wo se. This happens because he (smal l) di iso s
a e now aised oa powe ha inc eases wi h he mul iplici y o he eigen alues. The
p oo is no included, sincei does no in oduce new ideas and he echnical de ails
a e a he edious.
Rema k 3.4.
The alues o
"

,
a

,

and
m
gi en in he heo em a e a he
pessimis ic. In he p oo , we ha e p e e ed o use simple (bu ough) bounds ins ead o
cumbe some bu mo eaccu a e ones. I one is in e es edin ealis ic bounds o a gi en
p oblem, he bes hing o do is o ew i e he p oo o ha pa icula case. We ha e
done his in Sec ion 6 whe e, wi h he help o a compu e p og am, we ha e applied
some s eps o he me hod o an example. This al lows no only o ob ain be e bounds,
bu also o ob ain (nume ical ly) he educed ma ix as wel l as he co esponding change
o a iables.
4. Lemmas.
We will us e some lemmas o s impli y he p o o o he heo em.
4.1. Bas ic lemmas.
Lemma 4.1.
Le
Q
(
)=
X
k
2
Z
Q
k
e
i
(
k!
)
be an elemen o
Q
d
(
 !
)
and
M>
0
.
Le us dene
Q
=
Q
0
,
e
Q
(
)=
Q
(
)
;
Q
0
,
Q

M
(
)=
X
k
2
Z
j
k
j
M
Q
k
e
i
(
k!
)

and
e
Q
<M
=
e
Q
;
Q

M
. Then we ha e he bounds
1.
j
Q
j
1
,
k
e
Q
k

,
k
e
Q
<M
k

k
Q
k

.
2.
k
Q

M
k

;

k
Q
k

e
;
M
,
8

2
]0

]
.
P oo
. I isanimme dia echeck.
Thenex lemmais used ocon ol he a ia ion o he e igen alue s o a p e u b e d
diagonal ma ix.
5

Lemma 4.2.
Le
D
bea
d

d
diagonal ma ix wi h die en eigen alues

1
:::
d
and

= min
j
6
=
`
(
j

j
;

`
j
)
.Theni
A
e ies
j
A
;
D
j
1

b


3
d
;
1
, he ol lowing
condi ions hold:
1.
A
has die en eigen alues

1
:::
d
and
j

j
;

j
j
b
i
j
=1
:::d
.
2. The e exis s a egula ma ix
S
such ha
S
;
1
AS
=
D

=diag(

1
:::
d
)
sa is ying
C
(
S
)

2
.
P oo
.I iscon aine d in 8].
Lemma 4.3.
Le
(
q
n
)
n
,
(
a
n
)
n
and
(
n
)
n
besequences denedby
q
n
+1
=
q
2
n
 a
n
+1
=
a
n
+
q
n
+1

n
+1
=
2+
q
n
2
;
q
n
n
+
q
n
+1
:
wi h ini ial alues
q
0
=
a
0
=
0
=
e
;
1
. Then
(
q
n
)
n
is dec easing o ze o and
(
a
n
)
n
,
(
n
)
n
a e inc easing and con e gen o some alues
a
1
and
1
espec i ely, wi h
a
1
<
1
e
;
1
,
1
<
e
e
;
1
.
P oo
. I i s immedia e ha
q
n
goes o ze o quad a ically and hi s implie s ha
a
n
i s con e gen o he alue
a
1
:
a
1
=
1
X
j
=0
q
j
<
1
X
j
=1
e
;
j
=
1
e
;
1
:
Then
n

p
0
@
0
+
n
X
j
=1
q
j
1
A

pa
1

whe e
p
=
Q
1
j
=0
2+
q
j
2
;
q
j
. Thi s p o duc i s con e gen , in ac :
ln
p
=
1
X
j
=0
ln(1 +
q
j
=
2)
;
ln(1
;
q
j
=
2)]

3
2
a
1

3
2(
e
;
1)
<
1

andso
p<e
,whe e weha eused ha ln(1 +
x
)

x
and
;
ln(1
;
x
)

2
x
, o
x
2
(0

1
=
2).
4.2. Theinduc i e lemma.
Thenex lemmais used odoas ep o heinduc i e
p o ce du e.
Be o e s a ing he e sul , le us in o duce some no a ion. Le
D
and

be likein
Lemma4.2 andle
"

,
q

,
L
and
M
(
"
) b e p os i i e cons an s. Weconside he equa ion
a hes ep
n
o hei e a i e p o ce ss:
_
x
n
=(
A
n
(
"
)+
"Q
n
(
 "
)+
"R
n
(
 "
))
x
n

j
"
j
"


(8)
whe e
Q
n
(

"
),
R
n
(

"
)
2Q
d
(
 !
)and
Q
n
(
"
)=
Q
n
(

"
)

M
(
"
)
=0. Weassume ha
o some
a
n
,
q
n
,
n

0and
j
"
j
<"

he ollowingbounds hold:
j
A
n
(
"
)
;
D
j
q

a
n
j
"
j

k
Q
n
(

"
)
k


q

q
n

k
R
n
(

"
)
k

;


q

n
e
;
M
(
"
)


whe e

is such ha 0
<


( hecons an
q

has b een in o duce d o s impli y,
la e , he p o o o he heo em). Wewan o s ee i i i s p oss ible oapply a s ep o he
i e a i e p o ce ss o equa ion (8) oob ain
_
x
n
+1
=(
A
n
+1
(
"
)+
"Q
n
+1
(
 "
)+
"R
n
+1
(
 "
))
x
n
+1

j
"
j
"


(9)
6
such ha
Q
n
+1
(

"
),
R
n
+1
(

"
)
2Q
d
(
 !
),
Q
n
+1
(
"
)=
Q
n
+1
(

"
)

M
(
"
)
=0. Wealso
wan o ela e hebounds
a
n
+1
,
q
n
+1
and
n
+1
o he e ms o hi s equa ion wi h he
co e sp ondingbounds o equa ion (8).
Lemma 4.4.
Le

(
n
)
1
(
"
)
:::
(
n
)
d
(
"
)
be he eigen alues o
A
n
(
"
)
. Unde he
p e ious no a ions, i
1.
L

8
q

,
"



q

(3
d
;
1)
,
2.
a
n

1
,
q
n

e
;
1
,
3. he condi ion
j

(
n
)
j
(
"
)
;

(
n
)
`
(
"
)+
i
(
k !
)
j
L
j
"
j

j
"
j
"


is sa ised o al l
j
,
`
and o al l
k
2
Z
such ha
0
<
j
k
j
<M
(
"
)
,
hen, equa ion (8) can be ans o med in o (9) and:
q
n
+1
=
q
2
n
 a
n
+1
=
a
n
+
q
n
+1

n
+1
=
2+
q
n
2
;
q
n
n
+
q
n
+1
:
The quasipe iodic change o a iables ha pe o ms his ans o ma ion is
x
n
=(
I
+
"P
n
(
 "
))
x
n
+1

(10)
whe e
P
n
(

"
)
is he (only) solu ion o
_
P
n
=
A
n
(
"
)
P
n
;
P
n
A
n
(
"
)+
Q
n
(
 "
)
 P
n
=0

(11)
ha belongs o
Q
d
(
 !
)
.Mo eo e ,
k
"P
n
(

"
)
k


q
n
=
2
<
1
=
2
.
Rema k 4.1.
A
n
,
Q
n
,
R
n
,
P
n
,
M
and

(
n
)
j
depend on
"
bu , o simplici y, we
wil l no w i e his explici ely.
P oo
. Le us s a s udying he solu ions o (11). Le
S
n
be hema ix oundin
Lemma 4.2 wi h
S
;
1
n
A
n
S
n
=
D
n
= diag (

(
n
)
1
:::
(
n
)
d
),
C
(
S
n
)

2. Thi s lemma can
be applie d b ecaus e
j
A
n
;
D
j
1

q

a
n
j
"
j
q

"



3
d
;
1
o all
j
"
j
"

:
Making hechange o a iable s
P
n
=
S
n
X
n
S
;
1
n
anddening
Y
n
=
S
;
1
n
Q
n
S
n
, equa ion
(11) b ecomes
_
X
n
=
D
n
X
n
;
X
n
D
n
+
Y
n
 Y
n
=0
:
As
D
n
i s a diagonal ma ix wecanhandle hi s equa ion as
d
2
unidimens ional equa ions,
ha can b e sol e d eas ily by expandinginFou ie se ies. I
X
n
=(
x
`jn
),
Y
n
=(
y
`jn
),
wi h
x
`jn
(
)=
X
k
2
Z
0
<
j
k
j
<M
x
k
`jn
e
i
(
k!
)
 y
`jn
(
)=
X
k
2
Z
0
<
j
k
j
<M
y
k
`jn
e
i
(
k!
)

he co ecien smus b e
x
k
`jn
=
y
k
`jn

(
n
)
j
;

(
n
)
`
+
i
(
k !
)

7
and, byhyp o hesis 3 hey can b e b ounded by
j
x
k
`jn
j
(
L
j
"
j
)
;
1
j
y
k
`jn
j
,and hi s implie s
k
P
n
k


C
(
S
n
)
k
X
n
k


C
(
S
n
)(
L
j
"
j
)
;
1
k
Y
n
k


C
(
S
n
)
2
(
L
j
"
j
)
;
1
k
Q
n
k



4(
L
j
"
j
)
;
1
q

q
n
j
"
j
;
1
q
n
2
:
Hence,
k
"P
n
k


q
n
=
2
<
1
=
2. Thus
I
+
"P
n
is in e ible and
k
(
I
+
"P
n
)
;
1
k


1
1
;k
"P
n
k

<
2
:
Now, applying hechange (10) o (8) anddening
Q

n
=
"
(
I
+
"P
n
)
;
1
Q
n
P
n
,
A
n
+1
=
A
n
+
"Q

n
,
Q
n
+1
=(
Q

n
)
<M
and
R
n
+1
=(
I
+
"P
n
)
;
1
R
n
(
I
+
"P
n
)+(
Q

n
)

M
,i is easy
ode i e equa ion (9). Finally we us e Lemma4.1 obound he e ms o hi s equa ion:
k
Q

n
k

 k
(
I
+
"P
n
)
;
1
k

k
Q
n
k

k
"P
n
k

k
Q
n
k

q
n

q

q
2
n
=
q

q
n
+1
k
Q
n
+1
k

 k
Q

n
k


q

q
n
+1
j
A
n
+1
;
D
j
1
 j
A
n
;
D
j
1
+
j
"Q

n
j
1

q

(
a
n
+
q
n
+1
)
j
"
j
=
q

a
n
+1
j
"
j
k
R
n
+1
k

;


1+
k
"P
n
k

1
;k
"P
n
k

k
R
n
k

;

+
k
(
Q

n
)

M
k

;




1+
q
n
=
2
1
;
q
n
=
2
n
+
q
n
+1

q

e
;
M
=
q

n
+1
e
;
M

8

2
]0

]
:
5. P o o o Theo em.
Le
S
b e a egula ma ix such ha
S
;
1
AS
=
D
=
diag (

1
:::
d
). Wedene
"

,

,

and
m
as in hes a emen o Theo em 3.1. We
also dene
q

=
e q
,
M
=
M
(
"
)=

m
j
"
j

1
=
and
L
=8
q

.
The (cons an ) change
x
=
Sx
0
ans o ms heini ial equa ion in o
_
x
0
=(
D
+
"Q

(
 "
))
x
0
(12)
whe e
Q

=
S
;
1
QS
andso
k
Q

k


e
;
1
q

o
j
"
j
"

.We spli equa ion (12) as
_
x
=(
A
0
+
"Q
0
(
)+
"R
0
(
))
x
0
whe e
A
0
=
D
+
"Q

,
Q
0
=
Q

<M
and
R
0
=
Q


M
.Using Lemma 4.1 i i s easy o
see ha
j
A
0
;
D
j
1

q

a
0
j
"
j

k
Q
0
k


q

q
0

k
R
0
k

;


q

0
e
;
M

8

2
]0

]

j
"
j
"

,i
a
0
=
q
0
=
0
=
e
;
1
.
We a e going oshow ha in all hes eps hehyp o hesis o Lemma 4.4 a e sa ised.
As hyp o hesis1and 2 a e easy ocheck, we o cus on hyp o hesis 3.
Now s ince
a
n

1and
j
"
j
"

,
j
A
n
;
D
j
1

q

j
"
j

3
d
;
1
, Lemma 4.2 gi es ha
j

(
n
)
j`
;

j`
j
<
2
q

j
"
j
o all
j `
j
"
j
"


whe e

j`
=

j
;

`
,

(
n
)
j`
=

(
n
)
j
;

(
n
)
`
being

(
n
)
1
:::
(
n
)
d
he e igen alue s o
A
n
(
"
).
8
Us inghyp o hesis 3 o heTheo em we ob ain ha , i
k
2
Z
and0
<
j
k
j
<M
(
"
),
j

(
n
)
j`
+
i
(
k !
)
j  j

j`
+
i
(
k !
)
j;j

(
n
)
j`
;

j`
j
>
c
j
k
j

;
2
q

j
"
j
>
>

c
m
;
2
q


j
"
j
=
L
j
"
j

andhyp o hesis 3 o Lemma 4.4 i s e ie d.
In cons equence hei e a i e p o ce ss can b e ca ie d ou andLemma 4.3 ensu e s he
con e gence o he p o ce ss. Thecomposi ion o all hechange s
I
+
"P
n
i s con e gen
b ecaus e
k
I
+
"P
n
k


1+
q
n
=
2. Then henal equa ion i s
_
x
1
=(
A
1
(
"
)+
"R
1
(
 "
))
x
1

j
"
j
"


(13)
whe e
j
A
1
(
"
)
;
D
j
1

q

a
1
j
"
j
e
e
;
1
q
j
"
j
,and
k
R
1
(

"
)
k

;


q

1
e
;
M
(
"
)


e
2

e
;
1
q
exp
(
;

m
j
"
j

1
=

)

8

2
]0

]
:
Toendup he p o o , hechange
x
1
=
S
;
1
y
ans o ms equa ion (13) in o equa ion
(7) wi h hebounds ha wewe e lo oking o .
6. An example.
The esul so hi s pap e can b e applie d in manyways, acco d-
ing o he kind o p oblem wea ein e e s e d in. Le us illus a e hi s wi h hehelp o
an example.
Le us cons ide he equa ion

x
+(1+
"q
(
))
x
=0

(14)
whe e
q
(
)=cos(
!
1
)+cos(
!
2
), b e ing
!
1
=
p
2and
!
2
=
p
3. Dening
y
as _
x
we
can ew i e (14) as

_
x
_
y

=

0 1
;
1 0

+
"

0 0
;
q
(
) 0

x
y

:
(15)
As

1

2
=

i
, hediophan inecondi ion (6) i s sa i se d o

= 1 (b ecaus e he equen-
cie s a e quad a ic i a ionals). The alue o
c
will b e di scuss e d la e . Fo hesakeo
s implici y, le us ake

=2and

=1. This implies ha
q
=
k
Q
k

=2
e
2
. I isno
dicul ode i e

=2 and, nally,
"

=4
:
9787
:::

10
;
3
and

=2
:
5419
:::

10
2
.
The alue o
c
migh b e calcula ed o all
k
=(
k
1
k
2
), bu be e (bigge ) alue s
can be used since weonlynee d oconside
j
k
j
up o a ni e o de . Fo ins ance, an
easy compu a ion shows ha o
j
k
j
125
c
is 0
:
149. I
j
k
j
=126, hen
c
mus b e
0
:
013 a mos , due o he quas i e sonance p o duce d by
k
=(70

;
56). In he ange
126
j
k
j
10
5
he e a e no mo e ele an esonance s, so he alue
c
=0
:
013 suce s.
Tos a he di scuss ion, le us suppose ha he alue o
"
in (15) i s
"
=2

10
;
6
.
I we ake
c
=0
:
149 weob ain ha
m
=1
:
8545
:::

10
;
4
and
M
= 93 ( ecall ha
he p o ce ss cancels equencie s such ha
j
k
j
<M
(
"
)). I he alue o
M
had b een
bigge han 125, weshould ha eused he alue
c
=0
:
013 ins ead. So, we can e duce
he sys em o cons an co ecien swi ha emainde
R

such ha
k
R

k

;
1
<
10
;
37
.
I he gi en alue o
"
is smalle , o ins ance
"
=10
;
7
, he compu ed alue o
M
i
c
=0
:
149 i s 1855, so
c
=0
:
013 mus b e us e d. Thi s p o duce s
M
=162and
k
R

k

;
1
<
10
;
67
.A alue o
"
=5

10
;
8
implie s
M
= 324 and
k
R

k

;
1
<
10
;
138
.
9