Effective reducibility of quasiperiodic linear equations close to constant coefficients
Abstract
Let us consider the differential equation $$ \dot{x}=(A+\varepsilon Q(t,\varepsilon))x, \;\;\;\; |\varepsilon|\le\varepsilon_0, $$ where $A$ is an elliptic constant matrix and $Q$ depends on time in a quasiperiodic (and analytic) way. It is also assumed that the eigenvalues of $A$ and the basic frequencies of $Q$ satisfy a diophantine condition. Then it is proved that this system can be reduced to $$ \dot{y}=(A^{*}(\varepsilon)+\varepsilon R^{*}(t,\varepsilon))y, \;\;\;\; |\varepsilon|\le\varepsilon_0, $$ where $R^{*}$ is exponentially small in $\varepsilon$, and the linear change of variables that performs such reduction is also quasiperiodic with the same basic frequencies than $Q$. The results are illustrated and discussed in a practical example.
Full text
EFFECTIVE REDUCIBILITY OF QUASIPERIODIC LINEAR EQUATIONS CLOSE TO CONSTANT COEFFICIENTS ANGEL JORBA y ,RAFAEL RAMREZ-ROS y , AND JORDI VILLANUEVA y . Abstract. Let us consider the dierential equation _ x =( A + "Q ( t " )) x j " j " 0 where A is an elliptic constantmatr ix and Q depends on time in a quasiper io dic (andanalytic) way. It is also assumed thatthe e igenvalues of A andthe basic f requencie s of Q satisfy a diophantine condition. Then it is proved thatthis system can be re duce d to _ y =( A ( " )+ "R ( t " )) y j " j " 0 where R is exp onentially small in " ,andthelinear change of var iables that performs suchreduction is also quasiper io dic withthesame basic f requencie s than Q .The re sults are illustrated and discuss e d in a practical example. Key words. quasiper io dic Flo quet theorem, quasiper io dic perturbations, re ducibilityof linear equations. AMS(MOS) sub ject class ications. 34A30, 34C20, 34C27, 34C50, 58F30 1. Intro duction. Thewell-known Flo quet theorem states thatanylinear p er io dic system, _ x = A ( t ) x , can b e re duce d toconstant co ecients, _ y = By ,bymeans of a p er io dic change of var iable s. Moreover, thi s change of var iable s can b e taken, over C ,withthesameperiodthan A ( t ). Anatural extens ion i s toconsider thecaseinwhichthematr ix A ( t )dep ends on time in a quas ip er io dic way. Before startingthe di scuss ion of thi s i ssue, let us recall thedenition and bas ic prop ertie s of quas ip er io dic functions. Definition 1.1. A function f is a quasiperiodic function with vectorofbasic frequencies ! =( ! 1 :::! r ) if f ( t )= F ( 1 ::: r ) , where F is 2 periodic in al l its arguments and j = ! j t for j =1 :::r .Moreover, f is cal led analytic on a strip of width if F is analytical on an open set containing j Im j j for j =1 :::r . It i s also known thatananalytic quas ip er io dic function f ( t ) on a str ip of width has Four ier co ecientsdened by f k = 1 (2 ) r Z T r F ( 1 ::: r ) e ; ( k ) p ; 1 d suchthat f can b e expanded as f ( t )= X k 2 Z r f k e ( k! ) p ; 1 t for all t suchthat j Im t j = k ! k 1 .Wedenoteby k f k the norm k f k = X k 2 Z r j f k j e j k j anditisnotdiculttocheckthatitiswell dened for anyanalytical quas ip er io dic function dene d on a str ip of width .Finally,todeneananalytic quas ip er io dic This paper was wr itten on January 20th, 1995. y Dept. deMatematica Aplicada I, ETSEIB, UniversitatPolitecnica deCatalunya, Diagonal 647, 08028 Barcelona, Spain (E-mails: [email protected] , [email protected] , [email protected] ). 1
matr ix, we notethatallthese denitions hold when f is a matr ix-value d function. In thi s cas e, todene k f k weuse theinnity norm (that willbedenoted by jj 1 )for thematr ice s f k . After thos e denitions andprop ertie s, let us retur n tothe problem of thereducibilityofa linear quas ip er io dic equation, _ x = b A ( t ) x ,to constant co ecients. The approachofthi s workisto assumethatthe system i s clos e toconstant co ecients, thatis, b A ( t )= A + "Q ( t " ), where " is small. Thi s cas e has already b een cons idere d in many pap ers (s ee 2], 8]and9]amongothers), andtheresultscanbesummar ize d as follows: let i be the e igenvalue s of A ,and ij = i ; j ,for i 6 = j .Then, if all the value s Re ij are dierent f rom zero, thereduction can b e p erformed for j " j <" 0 , " 0 suciently small (s ee 2]). If someof theRe ij are zero (thi s happ ens, for instance, if A i s elliptic, that i s, if all the i areontheimaginary axi s) more hyp othesis are nee ded. The usual oneisadiophantinecondition involvingthe ij andthebasicfrequencie s of Q ( t " ), andto assumeanondegeneracy condition with re sp ect to " on the corre sp onding ij ( " )ofthematr ix A + "Q ( " )( Q ( " )denotes theaverage of Q ( t " )). Thi s allows to prove(see 9] for thedetails) thatthere exi stsaCantor ian s et E such thatthereduction can b e p erforme d for all " 2E . Moreover, therelativemeasure of the s et 0 " 0 ] nE in 0 " 0 ] i s exp onentially small in " 0 . Our purp os e here i s a little bit dierent: instead of lo okingfor a total re duction to constant co ecients(thi s s eems toleadusto eliminatea dens e s et of value s of " ,see 8]or 9]), wetry to minimize the quas ip er io dic part, withouttakingoutanyvalue of " .The re sultobtained is thatthe quasiperiodic part can be made exp onentially small. As all the pro of i s constructive(and it can b e carr ie d outwith a nitenumber of steps), it can b e applie d to practical example s in order to do an \eective" re duction: if " is small enough, theremainder will b e so small that, for practical purp os e s, it can b e taken equal to zero. The error pro duce d withthi s dropping can b e b ounded eas ily,bymeans of theGronwall lemma. Finally,wewantto stre ss thatwehavealso eliminated thenondegeneracy hyp othe s i s of previous pap ers (8], 9]). Before ni shingthi s intro duction, wewanttomention some s imilar re sultsobtained when thedynamics of the system i s slow: _ x = " ( A + "Q ( t " )) x . Thi s cas e is containe d in 14], whichisanextens ion of 12]. Theresultobtained is also that the quas ip er io dic part can b e made exp onentially small in " .Total re ducibilityhas b een also cons idere d in thi s cas e: in 15]isstated thatthereduction can b e p erformed except for a s et of value s of " of measure exp onentially small. There are manyother re sultsforthereducibility problem. For instance, in the cas e of theSchrodinger equation with quasiperiodic potential wecanmention 3], 4], 5], 10], 11]and13]. Another class ical and remarkable pap er i s 7], where thegeneral cas e (that i s, without askingto b e clos e to constant co ecients) i s cons idere d. Finally, the class ical re sults for quas ip er io dic systems can b e foundin6]. In order to s implify the reading, thepaperhas b een divided in sections as follows: Section 2 contains the exp os ition (withouttechnical details) of themain ideas and metho dology, Section 3 contains themain theorem, Sections 4 and5 are devoted to the pro ofs and, nally,Section 6 contains an example toshowhowthese results can be applie d to a concrete problem. 2. Themethod. Themethod used is based on thesameinductiveschemethat 8]. Let us wr ite our equation as _ x =( A + "Q ( t " )) x (1) 2
where A i s an elliptic d d matr ix and Q ( t " ) i s quas ip er io dic with ! =( ! 1 :::! r ) as vector of bas ic f requencie s, andanalytic on a str ip of width . First of all, let us rewr itethi s equation as _ x =( A 0 ( " )+ " e Q ( t " )) x where A 0 ( " )= A + Q ( " )and e Q ( t " )= Q ( t " ) ; Q ( " ). Now let us assumethatweare able tond a quasiperiodic d d matr ix P (withthe same bas ic f requencie s than Q ) ver ifying _ P = A 0 ( " ) P ; PA 0 ( " )+ e Q ( t " ) (2) suchthat k "P ( t " ) k < 1, for some > 0. In thi s cas e, it i s not diculttocheckthat thechange of var iable s x =( I + "P ( t " )) y transforms equation (1) into _ y =( A 0 ( " )+ " 2 ( I + "P ( t " )) ; 1 e Q ( t " ) P ( t " )) y: (3) As thi s equation i s like(1) butwith " 2 instead of " ,theinductivescheme s eems clear: toaverage the quas ip er io dic part of (3) andtorestart thi s pro ce ss. Themain diculty thatapp ear in thi s pro ce ss come s f rom equation (2), b ecaus e thesolution contains the denominators i ( " ) ; j ( " )+ p ; 1( k ! ), 1 i j d ,where i ( " )aretheeigenvalue s of A 0 ( " )(thi s i s shown ins idethe pro of of Lemma 4.2). Thi s divi sor app ears in the k th Four ier co ecientof P .Notethatifthevalue s i ( " ) ; j ( " ) are outsidetheimaginary axi s, the (mo dulus of the) divi sor can b e b ounded from below, b e ingeasy toprovethe convergence. On theother hand, thevalue i ( " ) ; j ( " )+ p ; 1( k ! ) can b e arbitrar ily small givingrise to convergence problems. 2.1. Avoidingthesmall divi sors. Let us start assumingthattheeigenvalue s i of the or iginal unp erturb e d matr ix A (s ee equation (1)) andthe bas ic f requencie s of Q sati sfy thediophantine condition j i ; j + p ; 1( k ! ) j c j k j 8 k 2 Z r nf 0 g : (4) where j k j = j k 1 j + + j k r j .Notethat, in pr inciple, we can not guarantee thatin equation (2) thi s condition holds, b ecaus e the e igenvalue s of A 0 ( " )have b een change d with re sp ect totheones of A (in an amountof O ( " )) andsomeof thedivisorscanbe very small or even zero. Thekey p ointisto realize that, as theeigenvalue s of A movein an amountof O ( " )at most, thequantitie s i ( " ) ; j ( " ) are containe d in a (complex) ball B ij ( " ) centere d in i ; j andwith radius O ( " ). As thecentre of theballsati se s condition (4), thevalue s ( k ! ) can not b e ins idethatballif j k j i s le ss than somevalue M ( " ). Thi s implie s that it i s p oss ible to cancel all theharmonics suchthat0 < j k j <M ( " ), b ecaus e they do not pro duce small divi sors (notethatwe can only have re sonance s when ( k ! )isinside B ij ( " )). Theharmonics with j k j M ( " ) are exp onentially small in M ( " ) (when M ( " ) !1 ), thi s i s, exp onentially small in " (when " ! 0), so wedo not nee d to eliminatethem. Theidea of cons ider ing only f requencie s le ss than somethre shold M has already b een applie d b efore in other contexts (s ee, for instance, 1]). 2.2. Theiterativescheme. Toapply theconsiderations abovewedene, as b efore, A 0 ( " )= A + "Q ( " ), e Q ( t " )= Q ( t " ) ; Q ( " )andwe split e Q ( t " )inthesum 3
of twomatr ice s Q 0 ( t " ), R 0 ( t " ): Q 0 ( t " )contains theharmonics Q k e ( k! ) p ; 1 t with j k j <M ( " )and R 0 ( t " )theone s with j k j M ( " ). So, (1) can b e rewr itted as _ x =( A 0 ( " )+ "Q 0 ( t " )+ "R 0 ( t " )) x (5) Nowtheidea i s to cancel Q 0 ( t " )andto leave R 0 ( t " ) (it i s already exp onentially small with " ). So, we compute P 0 suchthat _ P 0 = A 0 ( " ) P 0 ; P 0 A 0 ( " )+ Q 0 ( t " ) : Then, thechange x =( I + "P 0 ( t " )) y gives _ y = A 0 + " 2 ( I + "P 0 ) ; 1 Q 0 P 0 + " ( I + "P 0 ) ; 1 R 0 ( I + "P 0 ) y: Thi s equation can b e rewr itten tobelike (5) to rep eatthe pro ce ss. Notethatthe s ize of theharmonics with0 < j k j <M ( " )has b een square d. As we will s ee in the pro ofs, thi s i s enough to guarantee convergence of thos e terms to zero. Thus, thenal equation has a purely quas ip er io dic part exp onentially small with " . 2.3. Remarks. It i s intere stingtonotethat it i s enough toapply a nitenumber of steps of theinductive pro ce ss: wedonotnee d to cancel completely theharmonics with0 < j k j <M ( " )butwe can stopthe pro ce ss when they are of thesamesize of theones of R (f rom the pro of it can b e s een thatthenumberofsteps nee ded to achievethi s i s of order j ln j " jj ). Thi s allows toapply (withthehelp of a computer) thi s pro ce dure on a practical example. Another remarkable p ointisaboutthediophantinecondition: notethatweonly nee d the condition up to a nite order ( M ( " ), thatisoforder (1 = j " j ) 1 = ,asweshall see in the pro ofs). Thi s means that, in a practical example when theperturbing f requencie s are known withnite preci s ion, the diophantine condition can b e checked eas ily. 3. TheTheorem. In what follows, Q d ( ! )states for the s et of theanalytic quas ip er io dic d d matr ice s on a str ip of width andhaving ! as vector of bas ic f requencie s. Moreover, i will denote p ; 1. Theorem 3.1. Consider the equation _ x =( A + "Q ( t " )) x , j " j " 0 ,and x 2 R d , where 1. A isaconstant d d matrix with dierent eigenvalues 1 ::: d . 2. Q ( " ) 2Q d ( ! ) with k Q ( " ) k q , 8j " j " 0 ,forsome ! 2 R r ,and q > 0 . 3. The vector ! satises the diophantine conditions j j ; ` + i ( k ! ) j c j k j 8 k 2 Z r nf 0 g 8 j ` 2f 1 :::d g (6) for some constants c> 0 , >r ; 1 . As usual, j k j = j k 1 j + + j k r j . Then there exist positive constants " , a , r and m such that for al l " , j " j " ,the initial equation can betransformed into _ y =( A ( " )+ "R ( t " )) y (7) where: 1. A isaconstant matrix with j A ( " ) ; A j 1 a j " j . 2. R ( " ) 2Q d ( ! ) and k R ( " ) k ; r exp ; m j " j 1 = , 8 2 ]0 ] . 4
Furthermore the quasiperiodic change of variables that performs this transformation is also an element of Q d ( ! ) . Final ly, a general explicit computation of " , a , r and m is possible: " = min " 0 eq (3 d ; 1) a = eq 2 e ; 1 r = ea m = c 10 eq where e = exp (1) , =min j 6 = ` ( j j ; ` j ) and is the condition number of a regular matrix S such that S ; 1 AS is diagonal, that is, = C ( S )= j S ; 1 j 1 j S j 1 . Remark 3.1. For xed values of 1 ::: d and hypothesis 3 is not satised for any c> 0 only for a set of values of ! of zeromeasureif >r ; 1 . Remark 3.2. In case that the eigenvalues of the perturbed matrices move on bal ls of radius O ( " p ) (that is, if the nondegeneracy hypothesis needed in 8] or 9] is not satised), it is not dicult to show that the bound of the exponential can be improved: k R ( " ) k ; r exp ( ; ( m= j " j ) p= ) .Theproof is very similar, but using M ( " )=( m= j " j ) p= instead of ( m= j " j ) 1 = . Thi s last remark s eems toshowthatthi s nondegeneracy hyp othesis is not nece ssary,anditisonlyused fortechnical reasons. In f act, theresultsseemtobebetter when thi s hyp othesis is not sati se d. Remark 3.3. If the unperturbed matrix A has multiple eigenvalues (that is, if hypothesis 1 is not satised) the theorem is stil l true, but the exponent of " in the exponential of the remainder is slightly worse. This happens because the (smal l) divisors are now raisedtoa power that increases with the multiplicity of the eigenvalues. The proof is not included, sinceitdoes not introduce new ideas and the technical details arerather tedious. Remark 3.4. The values of " , a , r and m given in the theorem arerather pessimistic. In the proof, we have preferred to use simple (but rough) bounds instead of cumbersome but moreaccurate ones. If one is interestedinrealistic bounds for a given problem, the best thing to do is to rewrite the proof for that particular case. We have done this in Section 6 where, with the help of a computer program, we have applied some steps of the method to an example. This al lows not only to obtain better bounds, but also to obtain (numerical ly) the reduced matrix as wel l as the corresponding change of variables. 4. Lemmas. We will us e some lemmas to s implify the pro of of thetheorem. 4.1. Bas ic lemmas. Lemma 4.1. Let Q ( t )= X k 2 Z r Q k e i ( k! ) t be an element of Q d ( ! ) and M> 0 . Let us dene Q = Q 0 , e Q ( t )= Q ( t ) ; Q 0 , Q M ( t )= X k 2 Z r j k j M Q k e i ( k! ) t and e Q <M = e Q ; Q M . Then we have the 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 ] . Proof . Itisanimme diatecheck. Thenext lemmais used tocontrol thevar iation of the e igenvalue s of a p erturb e d diagonal matr ix. 5
Lemma 4.2. Let D bea d d diagonal matrix with dierent eigenvalues 1 ::: d and = min j 6 = ` ( j j ; ` j ) .Thenif A veries j A ; D j 1 b 3 d ; 1 , the fol lowing conditions hold: 1. A has dierent eigenvalues 1 ::: d and j j ; j j b if j =1 :::d . 2. There exists a regular matrix S such that S ; 1 AS = D =diag( 1 ::: d ) satisfying C ( S ) 2 . Proof .Itiscontaine d in 8]. Lemma 4.3. Let ( q n ) n , ( a n ) n and ( r n ) n besequences denedby q n +1 = q 2 n a n +1 = a n + q n +1 r n +1 = 2+ q n 2 ; q n r n + q n +1 : with initial values q 0 = a 0 = r 0 = e ; 1 . Then ( q n ) n is decreasing to zero and ( a n ) n , ( r n ) n are increasing and convergent to some values a 1 and r 1 respectively, with a 1 < 1 e ; 1 , r 1 < e e ; 1 . Proof . It i s immediatethat q n goes to zero quadratically andthi s implie s that a n i s convergenttothevalue a 1 : a 1 = 1 X j =0 q j < 1 X j =1 e ; j = 1 e ; 1 : Then r n p 0 @ r 0 + n X j =1 q j 1 A pa 1 where p = Q 1 j =0 2+ q j 2 ; q j . Thi s pro duct i s convergent, in f act: 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 ,where wehaveused that ln(1 + x ) x and ; ln(1 ; x ) 2 x ,for x 2 (0 1 = 2). 4.2. Theinductive lemma. Thenext lemmais used todoastep of theinductive pro ce dure. Before statingthe re sult, let us intro duce some notation. Let D and be likein Lemma4.2 andlet " , q , L and M ( " ) b e p os itive constants. Weconsider the equation atthestep n of theiterative pro ce ss: _ x n =( A n ( " )+ "Q n ( t " )+ "R n ( t " )) x n j " j " (8) where Q n ( " ), R n ( " ) 2Q d ( ! )and Q n ( " )= Q n ( " ) M ( " ) =0. Weassumethat for some a n , q n , r n 0and j " j <" the followingbounds hold: j A n ( " ) ; D j q a n j " j k Q n ( " ) k q q n k R n ( " ) k ; q r n e ; M ( " ) where is suchthat0 < (theconstant q has b een intro duce d to s implify, later, the pro of of thetheorem). Wewantto s ee if it i s p oss ible toapply a step of the iterative pro ce ss to equation (8) toobtain _ x n +1 =( A n +1 ( " )+ "Q n +1 ( t " )+ "R n +1 ( t " )) x n +1 j " j " (9) 6
suchthat Q n +1 ( " ), R n +1 ( " ) 2Q d ( ! ), Q n +1 ( " )= Q n +1 ( " ) M ( " ) =0. Wealso wantto relatethebounds a n +1 , q n +1 and r n +1 of theterms of thi s equation withthe corre sp ondingbounds of equation (8). Lemma 4.4. Let ( n ) 1 ( " ) ::: ( n ) d ( " ) be the eigenvalues of A n ( " ) . Under the previous notations, if 1. L 8 q , " q (3 d ; 1) , 2. a n 1 , q n e ; 1 , 3. the condition j ( n ) j ( " ) ; ( n ) ` ( " )+ i ( k ! ) j L j " j j " j " is satised for al l j , ` and for al l k 2 Z r such that 0 < j k j <M ( " ) , then, equation (8) can betransformed into (9) and: q n +1 = q 2 n a n +1 = a n + q n +1 r n +1 = 2+ q n 2 ; q n r n + q n +1 : The quasiperiodic change of variables that performs this transformation is x n =( I + "P n ( t " )) x n +1 (10) where P n ( " ) is the (only) solution of _ P n = A n ( " ) P n ; P n A n ( " )+ Q n ( t " ) P n =0 (11) that belongs to Q d ( ! ) .Moreover, k "P n ( " ) k q n = 2 < 1 = 2 . Remark 4.1. A n , Q n , R n , P n , M and ( n ) j depend on " but, for simplicity, we wil l not write this explicitely. Proof . Let us start studyingthe solutions of (11). Let S n be thematr ix foundin Lemma 4.2 with 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 for all j " j " : Makingthechange of var iable s P n = S n X n S ; 1 n anddening Y n = S ; 1 n Q n S n , equation (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 matr ix wecanhandle thi s equation as d 2 unidimens ional equations, that can b e solve d eas ily by expandinginFourier series. If X n =( x `jn ), Y n =( y `jn ), with x `jn ( t )= X k 2 Z r 0 < j k j <M x k `jn e i ( k! ) t y `jn ( t )= X k 2 Z r 0 < j k j <M y k `jn e i ( k! ) t the co ecientsmust b e x k `jn = y k `jn ( n ) j ; ( n ) ` + i ( k ! ) 7
and, byhyp othesis 3 they can b e b ounded by j x k `jn j ( L j " j ) ; 1 j y k `jn j ,andthi 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 invertible and k ( I + "P n ) ; 1 k 1 1 ;k "P n k < 2 : Now, applyingthechange (10) to (8) anddening Q n = " ( I + "P n ) ; 1 Q n P n , A n +1 = A n + "Q n , Q n +1 =( f Q n ) <M and R n +1 =( I + "P n ) ; 1 R n ( I + "P n )+( Q n ) M ,itis easy toder ive equation (9). Finally we us e Lemma4.1 toboundtheterms of thi s equation: 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 r n + q n +1 q e ; M = q r n +1 e ; M 8 2 ]0 ] : 5. Pro of of Theorem. Let S b e a regular matr ix suchthat S ; 1 AS = D = diag ( 1 ::: d ). Wedene " , , and m as in thestatementofTheorem 3.1. We also dene q = e q , M = M ( " )= m j " j 1 = and L =8 q . The (constant) change x = Sx 0 transforms theinitial equation into _ x 0 =( D + "Q ( t " )) x 0 (12) where Q = S ; 1 QS andso k Q k e ; 1 q for j " j " .We split equation (12) as _ x =( A 0 + "Q 0 ( t )+ "R 0 ( t )) x 0 where A 0 = D + "Q , Q 0 = f Q <M and R 0 = Q M .Using Lemma 4.1 it i s easy to see that j A 0 ; D j 1 q a 0 j " j k Q 0 k q q 0 k R 0 k ; q r 0 e ; M 8 2 ]0 ] j " j " ,if a 0 = q 0 = r 0 = e ; 1 . We are goingtoshowthat in all thesteps thehyp othesis of Lemma 4.4 are satised. As hyp othesis1and 2 are easy tocheck, we fo cus on hyp othesis 3. Now s ince a n 1and j " j " , j A n ; D j 1 q j " j 3 d ; 1 , Lemma 4.2 gives that j ( n ) j` ; j` j < 2 q j " j for all j ` j " j " where j` = j ; ` , ( n ) j` = ( n ) j ; ( n ) ` being ( n ) 1 ::: ( n ) d the e igenvalue s of A n ( " ). 8
Us inghyp othesis 3 of theTheorem we obtain that, if k 2 Z r 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 othesis 3 of Lemma 4.4 i s ver ie d. In cons equence theiterative pro ce ss can b e carr ie d outandLemma 4.3 ensure s the convergence of the pro ce ss. Thecomposition of all thechange s I + "P n i s convergent b ecaus e k I + "P n k 1+ q n = 2. Then thenal equation i s _ x 1 =( A 1 ( " )+ "R 1 ( t " )) x 1 j " j " (13) where j A 1 ( " ) ; D j 1 q a 1 j " j e e ; 1 q j " j ,and k R 1 ( " ) k ; q r 1 e ; M ( " ) e 2 e ; 1 q exp ( ; m j " j 1 = ) 8 2 ]0 ] : Toendupthe pro of, thechange x 1 = S ; 1 y transforms equation (13) into equation (7) withthebounds thatwewere lo oking for. 6. An example. Theresultsofthi s pap er can b e applie d in manyways, accordingtothe kind of problem weareintere ste d in. Let us illustratethi s withthehelp of an example. Let us cons ider the equation x +(1+ "q ( t )) x =0 (14) where q ( t )=cos( ! 1 t )+cos( ! 2 t ), b e ing ! 1 = p 2and ! 2 = p 3. Dening y as _ x we can rewr ite (14) as _ x _ y = 0 1 ; 1 0 + " 0 0 ; q ( t ) 0 x y : (15) As 1 2 = i ,thediophantinecondition (6) i s sati se d for = 1 (b ecaus e the f requencie s are quadratic irrationals). Thevalue of c will b e di scuss e d later. For thesakeof s implicity, let us take =2and =1. This implies that q = k Q k =2 e 2 . Itisnot diculttoder ive =2 and, nally, " =4 : 9787 ::: 10 ; 3 and r =2 : 5419 ::: 10 2 . Thevalue of c might b e calculated for all k =( k 1 k 2 ), butbetter (bigger) value s can be used since weonlynee d toconsider j k j up to a nite order. For instance, an easy computation shows thatfor j k j 125 c is 0 : 149. If j k j =126,then c must b e 0 : 013 at most, due tothe quas ire sonance pro duce d by k =(70 ; 56). In therange 126 j k j 10 5 there are no more relevantresonance s, so thevalue c =0 : 013 suce s. Tostart the di scuss ion, let us suppose thatthevalue of " in (15) i s " =2 10 ; 6 . If wetake c =0 : 149 weobtain that m =1 : 8545 ::: 10 ; 4 and M = 93 (recall that the pro ce ss cancels f requencie s suchthat j k j <M ( " )). If thevalue of M had b een bigger than 125, weshould haveused thevalue c =0 : 013 instead. So, we can re duce the system to constant co ecientswitha remainder R suchthat k R k ; 1 < 10 ; 37 . If the given value of " is smaller, for instance " =10 ; 7 ,the computed value of M if c =0 : 149 i s 1855, so c =0 : 013 must b e us e d. Thi s pro duce s M =162and k R k ; 1 < 10 ; 67 .Avalue of " =5 10 ; 8 implie s M = 324 and k R k ; 1 < 10 ; 138 . 9