ON THE REDUCIBILITY OF LINEAR DIFFERENTIAL EQUATIONS WITH QUASIPERIODIC COEFFICIENTS WHICH ARE DEGENERATE

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1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 5, May 1998, Pages S (98) ON THE REDUCIBILITY OF LINEAR DIFFERENTIAL EQUATIONS WITH QUASIPERIODIC COEFFICIENTS WHICH ARE DEGENERATE XU JUNXIANG AND ZHENG QIN (Communicated by Hal L Smith) Abstract This paper proves the reducibility of a class of linear differential equations with quasiperiodic coefficients which are degenerate with respect to a small perturbation parameter Our results generalize some that were obtained by Jorba and Simó 1 Introduction and main results Notation and definitions The function f(t) is called a quasiperiodic function of t with frequencies ω 1, ω 2,, ω r, if there is a function F (θ 1,θ 2,,θ r ), which is 2π-periodic in all its arguments θ i (i =1,2,,r), such that f(t) = F(ω 1 t, ω 2 t,,ω r t) If F (θ) =F(θ 1,θ 2,,θ r ) (θ =(θ 1,θ 2,,θ r )) is analytic on a strip D ρ = {θ C r Imθ j ρ, j =1,2,,r},wesaythatf(t) is analytic quasiperiodic in D ρ Denote the sup-norm of f on D ρ by f ρ =sup θ Dρ F(θ) An n n matrix Q(t) =(q ij (t)) 1 i,j n is called analytic quasiperiodic on D ρ with frequencies ω 1,ω 2,,ω r if q ij (i, j =1,2,,n) are all analytic quasiperiodic on D ρ with the frequencies ω 1,ω 2,,ω r Define a matrix norm of Q by Q ρ = n max 1 i,j n q ij ρ It is easy to see that Q 1 Q 2 ρ Q 1 ρ Q 2 ρ If Q is a constant matrix, set Q = Q ρ for simplicity Write the avarage of Q(t) as 1 T Q=( q ij ) 1 i,j n,where q ij = lim T 2T T q ij(t) dt; for the existence of the limit, see [1] Let A(t) beann nquasiperiodic matrix The differential equations ẋ = A(t)x, x R n, are called reducible if there exists a nonsingular quasiperiodic change of variables x =Φ(t)ysuch that Φ(t) andφ 1 (t) are quasiperiodic and bounded, and such that it changes the equations to ẏ = By, where B is a constant matrix Problems In this paper we consider the reducibility of the following linear differential equations: (11) ẋ =(A+ɛQ(t))x, x R n, Received by the editors October 22, Mathematics Subject Classification Primary 34D20; Secondary 34C05 Key words and phrases Linear differential equations, reducibility, quasiperiodic, KAM iteration 1445 c 1998 American Mathematical Society

2 1446 XU JUNXIANG AND ZHENG QIN where A is an n n constant matrix with different eigenvalues λ 1,λ 2,,λ n,q(t) is an n n quasiperiodic matrix of time t with frequencies ω 1,ω 2,,ω r,andɛis a small perturbation parameter This problem was considered by Jorba and Simó in [2] They proved that if (12) ω, k 1+λ i λ j α k τ, 0 k Zr, i, j =1,2,,n, and d (13) dɛ ( λ i (ɛ) λ j (ɛ)) ɛ=0 0, i j, where α>0andτ>r 1areconstants and λ i (ɛ) (i=1,2,,n) are eigenvalues of A+ɛ Q, then, for sufficiently small ɛ 0 > 0, there exists a nonempty Cantor subset E (0,ɛ 0 ) such that for ɛ (0,ɛ 0 ) the equations (11) are reducible The method used in [2] is a kind of KAM iteration In the KAM iteration the difficulty is caused by a small divisor ω, k + λ i λ j The small divisor conditions (12) are necessary to overcome the difficulty of the small divisor Since the frequencies ω are invariant in the iteration, which is different from the usual KAM iteration [4], one must adjust the small parameter ɛ (0,ɛ 0 ) to guarantee the small divisor conditions So Jorba and Simó needed the nondegeneracy conditions (13) to guarantee existence of the nonempty Cantor subset E, on which all the small divisor conditions in the KAM iterations hold If the nondegeneracy conditions (13) do not hold, we say λ i (ɛ) λ j (ɛ) are degenerate This degenerate case is mentioned in [2], but there is no any result given Like the motivation of [2], in this paper we want to consider this degenerate case We will prove a similar result under weaker nondegeneracy conditions Moreover, in the situation of the usual nondegeneracy conditions (13), our result is just the same as that of [2], but the proof is simpler Main result Theorem Suppose A = diag(λ 1,λ 2,,λ n ) with λ i λ j for i j, 1 i, j n, and Q(t) =(q ij (t)) 1 i,j n = k Z Q ke i k,ω t is an analytic quasiperiodic matrix r on D ρ with frequencies ω 1,ω 2,,ω r, where the Fourier coefficients Q k depend analytically on the small parameter ɛ Let Q d 0= diag( q 11, q 22,, q nn ), where q ii is the average of q ii (t) Suppose that for i j, ɛ( q ii q jj ) has one of the following forms: µ 1 ɛ l1 + o(ɛ l1 ), µ 2 ɛ l2 +o(ɛ l2 ),,µ p ɛ lp +o(ɛ lp ), where µ i 0,i =1,2,,p, 1 l 1 <l 2 < <l p,ando(ɛ l )is of order smaller than ɛ l as ɛ 0 Suppose (1) (nonresonance conditions) λ =(λ 1,λ 2,,λ n ) and ω =(ω 1,ω 2,,ω r ) satisfy k, ω + λ i λ j α k τ, 0 k Zr, 1 i, j n, where α>0,τ>r 1 (2) Q is l p order continuously differentiable with respect to sufficiently small ɛ and dl Q dɛ l D ρ M l, l =0,1,2,,l p,

3 REDUCIBILITY OF LINEAR DIFFERENTIAL EQUATIONS 1447 where Q = Q Q d 0 Then, there exist N 1,N 2,,N p,n i depending on M 1,M 2,, M lj,α,τ,n,l p and µ 1 ɛ l1 + o(ɛ l1 ),,µ j ɛ lj +o(ɛ lj ), where l j l i 2, such that if µ i >N i,i=1,2,,p, then for sufficiently small ɛ 0 > 0, there exists a nonempty Cantor subset E (0,ɛ 0 ) with positive Lebesgue measure such that for ɛ E the equations (11) are reducible, ie, there exists a nonsingular quasiperiodic transformation x =Φ(t)ythat changes (11) to ẏ = By, where B is a constant matrix If ɛ 0 is small enough, the relative measure of E in (0,ɛ 0 ) is close to 1 Moreover, the quasiperiodic matrix Φ(t) has the same frequencies as Q(t) Remark 1 If l 1 =1,wecanchooseN 1 =0 Ifɛ( q ii q jj ) take only the form µɛ + o(ɛ), this corresponds to the nondegenerate case of [2] Remark 2 There are many ω and λ satisfying the nonresonance conditions in the theorem We refer to [2], [3] for detailed discussions about nonresonance conditions or small divisor conditions 2 Proof of the main result In this section we prove the theorem by the same idea of [2] with a little modification, which can simplify the proof in the nondegenerate case A Outline of the proof Write the equations (11) as (21) ẋ =(A + +ɛ Q(t))x, where A + = diag(λ + 1,λ+ 2,,λ+ n )=A+ɛQd 0, Q = Q(t) ɛq d 0 From [2] we know that under the change of variables x =(I+ɛP (t))y, where P satisfies (22) P = A + P PA + + Q, the equations (21) are changed to ẏ =(A + +ɛ 2 Q + (t)p)y, where Q + =(I+ɛP ) 1 QP If the above process can go on, then the perturbation term ɛ 2 Q + (t) becomes smaller and smaller and the equations converge to constant coefficient equations The key to the iteration is to solve the equation (22) for P Denote by P = (p ij (t)) 1 i,j n, Q =( qij (t)) 1 i,j n Expanding them in Fourier series and substituting them into the equations (22), by comparing the coefficients of both sides of the equations, we see formally that q k ij p k ij = k, ω 1+λ + i λ +, k Z r, i j + k 0 j If ω, k 1+λ + i λ + j α +, 0 k Z r, i, j =1,2, n, k τ where τ =3τ, then p k ij /α ( k τ + ) q ij k Since Q is analytic on D ρ,wehave Q k Q Dρ e k ρ So (23) P Dρ s P k e k (ρ s) ( 1 δ + k Z r 0 k Z r e k τ s k ) α Q Dρ c + α + s v Q D ρ,

4 1448 XU JUNXIANG AND ZHENG QIN where v = τ + r 1, 0 <s 1 2 ρ, δ =min i j λ + i λ + j,and c depends on τ,r,n Thus, If ɛ is sufficiently small that (I + ɛp ) 1 Dρ s 2, then (24) Q + Dρ s 2c α + s v Q 2 D ρ B Iteration step Consider the following equations: (25) ẋ m =(A m +ɛ 2m Q m (t))x m, m 0 where A m = diag(λ m 1,λ m 2,,λ m n ), and Q m (t) is an analytic quasiperiodic matrix on D ρm with the frequencies ω 1,ω 2,,ω r Let A m+1 = diag(λ m+1 1,λ m+1 2,,λ m+1 n ), where λ m+1 i = λ m i +ɛ 2m q ii m,i=1,2,,n,with qm ii being the average of qm ii (t) Let Q m (t) =Q m (t) ɛ 2m diag( q 11 m, qm 22,, qm nn ) If (26) ω, k 1+λ m+1 j α m k 3τ, k + i j 0, then by the above discussions we have an analytic quasiperiodic matrix P m on D ρm s m with the frequencies ω 1,ω 2,,ω r, such that under the change of variables x m =(I+ɛ 2m P m )x m+1, the equations (25) are changed to ẋ m+1 =(A m+1 + ɛ 2m+1 Q m+1 (t))x m+1 Moreover, we have (27) P m Dρm sm c α m s v Q m 2 D ρm m If ɛ is sufficiently small that (I + ɛ 2m P m ) 1 Dρm sm 2, we have (28) Q m+1 Dρm sm 2c α m s v Q m 2 D ρm m Now we prove this iteration is convergent At the first step, let A 0 = A, Q 0 (t) = Q(t),α 0 = α/2,ρ 0 = ρ, s 0 = ρ/4,d 0 = D ρ0,f 0 =(ɛ Q 0 D0 )/(α 0 s v 0 )At the mth step we choose α m = α 0 /(1 + m) 2, s m = s m 1 /2, ρ m+1 = ρ m s m,d m = D ρm,f m =(ɛ 2m Q m Dm )/(α m s v m ) If (29) (I + ɛ 2m P m ) 1 Dm+1 2, m 0, then by (28) we have F m+1 cfm, 2 where c =2 v+1 c So cf m+1 ( cf m ) 2 If cf 0 < 1 2, then cf m ( 1 2 By (27) it follows that ɛ )2m 2m P m Dm+1 cf m ( 1 2 So, we have )2m (I + ɛ 2m P m ) 1 Dm+1 1+ ɛ 2m P m l D m+1 2 Let ɛ 1 > 0besosmallthat cf for ɛ (0,ɛ 1) Thus, for ɛ (0,ɛ 1 ) satisfying all the small divisor conditions (26), all the above estimates hold Let D = m=0 D m,p m =(I+ɛ 2m P m )(I +ɛ 2m 1 P m 1 ) (I+ɛP 0 ) Obviously, D 1 2 ρ D and P m is convergent as m under the norm D Let Φ = lim m P m From the above discussion, it follows that A m A m 1 ɛ 2m Q m Dm So, A m is also convergent Let lim m A m = B Obviously, lim m ɛ 2m Q m D =0 12 ρ l=1

5 REDUCIBILITY OF LINEAR DIFFERENTIAL EQUATIONS 1449 Thus, if ɛ (0,ɛ 1 ) satisfy the small divisor conditions (26) for all m 0, then, under the change of variables x =Φy, the equations (11) become ẏ = By Moreover, Φ is an analytic quasiperiodic matrix on D 1 2 ρ with the frequencies ω 1,ω 2,,ω r To finish the proof of the theorem, it remains to prove that there exist N 1,N 2,, N p such that if µ i N i (i=1,2,,p), then there exist 0 <ɛ 0 <ɛ 1 and a nonempty Cantor subset E (0,ɛ 0 ) such that for ɛ E, the small divisor conditions (26) hold for all m 0 Now we first prove that there exist N 1,N 2,,N p such that if µ i N i,i =1,2,,p, then for all m 0andi j, there exist l {l 1,l 2,,l p } such that the lth derivative of λ m+1 i λ m+1 j with respect to ɛ at ɛ = 0 does not vanish Suppose that λ 1 i λ1 j = µ 1ɛ l1 + o(ɛ l1 ) Let N 1 0 be an integer such that 2 N 1 l 1 2 N 1+1 Let M i 1 d l1 2i Q i = ɛ=0,i=1,2,, dɛ l1 2i N 1 Dρi Then M i 1 (i =1,2,, N 1 ) only depends on τ,n,α,ρ, andm 0,M 1,,M l1 2 iby 1 the construction of the transformation, M i should depend on all derivatives up to the l 1 2 i -th of λ i j 1 (ɛ) λ i j 2 (ɛ) with respect to ɛ at ɛ =0But these derivatives only depend on M 0,M 1,,M l1 2 i and are independent of µ 1,µ 2,,µ p Let M 1 1 N 1 = + M M 1 N1 l 1! If µ 1 >N 1,we have at ɛ =0 d l1 (λ m+1 j ) l 1! µ 1 ( M 1 + M M N1 )>0 dɛ l1 Similarly, let N 2 N 1 be an integer such that 2 N 2 l 2 < 2 N 2+1 Let M 2 i = d l2 2i Q i ɛ=0,i=1,2,, dɛ l2 2i N 2 Dρi Then M i 2 (i =1,2,, N 2 ) only depends on τ,n,α,ρ,m 0,M 1,,M l2 2i and the l-th order derivatives of µ 1 ɛ l1 + o(ɛ l1 ) with respect to ɛ, wherel l 2 2 i Let N 2 = M M M 2 N2 l 2! If µ 2 >N 2,then, when λ 1 i λ1 j = µ 2ɛ l2 + o(ɛ l2 ), for all m 0, the l 2 -th derivative of λ m+1 j with respect to ɛ at ɛ = 0 does not vanish From the above we see that N 2 may depend on N 1, but N 1 is independent of N2 In the same way we can obtain N p, which depends on τ,n,α,ρ,m 0,M 1,,M lp 2 and the l-th derivatives of µ i ɛ li + o(ɛ li ) with respect to ɛ, i =1,2,,p 1, where l l p 2 If µ p >N p,then, when λ 1 i λ1 j consists of µ pɛ lp, for all m 0, the l p -th derivative of λ m+1 j with respect to ɛ at ɛ = 0 is not zero It is easy to see that N p

6 1450 XU JUNXIANG AND ZHENG QIN may depend on N 1,N 2,,N p 1 Thus,ifλ 1 i λ1 j =µ iɛ li +o(ɛ li ), then d li (λ m+1 j ) ɛ=0 0, m 0 dɛ li From the above iteration we see that the first step can only be done for all ɛ E 0 (0,ɛ 0 ), where E 0 is the set on which the small divisor conditions of the first step hold Let E m 1 be the set on which the small divisor conditions of the m-thstepholdthenthem-thstepcanonlybedoneforɛ E 0 E 1 E m 1 Thus, the above iteration can only be convegent on the set E = m=0 E m Since the small divisor conditions hold on a Cantor subset, in all the iteration steps the differentiations with respect to ɛ are understood in Whitney s sense [5] By the Whitney extension theorem in [5], all differentiable functions on a closed set in Whitney s sense can be extended to the usual differentiable function on (, + ) and the differentiation in Whitney s sense can be treated as the usual differentiation Thus all the functions in the iteration step can be regarded as regular differentiable functions on (, + ), and so all the estimates can be obtained without any difficulty However, the iteration makes sense only for ɛ E Now we prove E is a nonempty set For this we prove that for most sufficiently small ɛ, the small divisor conditions (26) hold for all m 0 Let λ m+1 j = µ l ɛ l + o(ɛ l ), where l l p It is easy to see that the term µ l ɛ l is unvariant when m N p So there exists a sufficiently small ɛ 0 > 0 such that if ɛ ɛ 0,then λ m+1 i λ m+1 j 2 µ l ɛ l and d(λm+1 i λ m+1 j ) dɛ 1 2 µ l ɛ l 1 for all m 0 Let f(ɛ) = ω, k + λ m+1 j,i j,and Oijm k = {ɛ (0,ɛ 0 ) f(ɛ) < α m k 3τ} Since λ 0 i λ0 j 0, i j, we choose ɛ 0 so small that if ɛ ɛ 0, λ m+1 j ᾱ>0holdsforalli jand m 1 So we only consider k 0 Since ω, k + λ m+1 j ω, k 1+λ 0 i λ0 j 2µ lɛ l, by the nonresonance conditions of the theorem, if 1/ k τ > 4µ l ɛ l /α 0, then f(ɛ) α 0 /2 k τ >α m / k 3τ Suppose α 0 /4µ l k τ <ɛ l ɛ 0 Since df (ɛ) dɛ 1 2 µ l ɛ l 1, by the differentiation mean value theorem, we have meas(oijm) k 2α m 2 k 3τ µ l ɛ l 1 ɛ l µ l k τ (m +1) 2 So meas( Oijm) k 8n2 max l µ l ɛ2 0 α i j 0 k Z r m=0 α 0 8 µ l ɛ l+1 0 α(m +1) 2 k τ 1 1 (m +1) 2 k τ =cɛ2 0, 0 k Z r where c depends on n, τ, α and µ l Let E be the subset of (0,ɛ 0 )wherethesmall divisor conditions (26) hold for all m 0 Then E =(0,ɛ 0 ) m,i,j,k Ok ijm Thus meas(e) ɛ 0 cɛ 2 0 = ɛ 0(1 cɛ 0 ) If ɛ 0 is so small that 1 cɛ 0 > 0 then E is a nonempty set with positive Lebesgue measure Noticing that { k k 0 k Zr } is dense on the unit ball of R r, we conclude that E is a Cantor set

7 REDUCIBILITY OF LINEAR DIFFERENTIAL EQUATIONS 1451 Acknowledgement We would like to thank the referee for much helpful advice which we used in our revision References [1] N N Bogoljubov, J A Mitropoliski and A M Samoilenko, Methods of Accelerated Convergence in Nonlinear Mechanics, Springer-Veriage, New York (1976) MR 53:1156 [2] Angel Jorba and Carles Simó, On the Reducibility of Linear Differential Equations with Quasiperiodic Coefficients, J Diff Equa 98 (1992), MR 94f:34024 [3] V I Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, Russian Math Surveys 18 (1963), no 6, MR 30:943 [4] Pöschel, J, On elliptic lower dimensional tori in Hamiltonian systems, Math Z 202 (1989), MR 91a:58065 [5] Whitney, H, Analytical extensions of differentiable functions defined in closed sets, Trans AMS 36 (1934), Department of Mathematics and Mechanics, Southeast University, Nanjing , People s Republic of China address: xujun@seueducn Department of Mathematics, Nanjing University, Nanjing , People s Republic of China