# On the Effective Reducibility of a Class of Quasi-Periodic Linear Hamiltonian Systems Close to Constant Coefficients

Authors: Nina Xue; Wencai Zhao
Journal: Journal of Function Spaces
 (2018)

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## Abstract

In this paper, we consider the effective reducibility of the quasi-periodic linear Hamiltonian systemx˙=A+εQt,εx, ε∈0,ε0, where A is a constant matrix with possible multiple eigenvalues and Q(t,ε) is analytic quasi-periodic with respect to t. Under nonresonant conditions, it is proved that this system can be reduced to y˙=A⁎ε+εR⁎t,εy, ε∈0,ε⁎, where R⁎ is exponentially small in ε, and the change of variables that perform such a reduction is also quasi-periodic with the same basic frequencies as Q.


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## Body

## 1. Introduction

The question about the reducibility of quasi-periodic systems plays an important role in the theory of ordinary differential equations. In general, in order to understand the qualitative behavior of a system, we need to obtain the information about the existence and stability of solutions. During the last two decades, the study of the existence of solutions for differential equations has attracted the attention of many researchers; see [1–10] and the references therein. Some classical tools have been used to study the existence of solutions for differential equations in the literature, including the method of upper and lower solutions, degree theory, some fixed point theorems in cones for completely continuous operators, Schauder’s fixed point theorem, and a nonlinear Leray-Schauder alternative principle.Compared with the existence of solutions, the study on the dynamical stability behaviors of such equations is more difficult, and the results are fewer in the literature. Here we refer the reader to [11–16].Before stating our problem, we give some definitions and notations. A functionf is said to be a quasi-periodic function with a vector of basic frequencies ω=(ω1,ω2,…,ωr) if f(t)=F(θ1,θ2,…,θr), where F is 2π periodic in all its arguments and θj=ωjt for j=1,2,…,r. Moreover, if F(θ)(θ=(θ1,θ2,…,θr)) is analytic on Dρ={θ∈Cr:|Imθj|≤ρ,j=1,2,…,r}, we say that f(t) is analytic quasi-periodic on Dρ.It is well known that an analytic quasi-periodic functionf(t) can be expanded as Fourier series (1)ft=∑k∈Zrfkek,ω-1twith Fourier coefficients defined by (2)fk=12πr∫TrFθ1,θ2,…,θre-k,θ-1dθ.We denote by fρ the norm (3)fρ=∑k∈Zrfkekρ.An n×n matrix Q(t)=(qij)1≤i,j≤n is said to be analytic quasi-periodic on Dρ with frequencies ω=(ω1,ω2,…,ωr), if all qij(i,j=1,2,…,n) are analytic quasi-periodic on Dρ with frequencies ω=(ω1,ω2,…,ωr). Define the norm of Q by (4)Qρ=max1≤i≤n⁡∑j=1nqijρ.It is easy to see that Q1Q2ρ≤Q1ρQ2ρ. If Q is a constant matrix, write Q=Qρ for simplicity. Denote the average of Q(t) by Q¯=(q¯ij)1≤i,j≤n, where (5)q¯ij=limT⟶∞⁡12T∫-TTqijtdt,for the existence of the limit, see [17].LetA(t) be an n×n quasi-periodic matrix; the differential equations x˙=A(t)x,x∈Rn, are called reducible if there exists a nonsingular quasi-periodic change of variables x=ϕ(t)y, such that ϕ(t) and ϕ-1(t) are quasi-periodic and bounded, which changes x˙=A(t)x to y˙=By, where B is a constant matrix. The well-known Floquet theorem states that any periodic differential equations x˙=A(t)x can be reduced to constant coefficient differential equations y˙=By by means of a periodic change of variables with the same period as A(t). But this is not true for the quasi-periodic coefficient system; see [18]. Johnson and Sell [19] proved that x˙=A(t)x is reducible if the quasi-periodic coefficient matrix A(t) satisfies "full spectrum" condition.Recently, many authors [20–23] considered the reducibility of the following system which is close to constant coefficients matrix:(6)x˙=A+εQtx.This problem was first considered by Jorba and Simó in [20]. Suppose that A is a constant matrix with different eigenvalues; they proved that if the eigenvalues of A and the frequencies of Q satisfy some nonresonant conditions, then for sufficiently small ε0>0, there exists a nonempty Cantor set E⊂(0,ε0), such that, for any ε∈E, system (6) is reducible. Moreover, the relative measure of the set (0,ε0)\E in (0,ε0) is exponentially small in ε0. In [23], Xu obtained the similar result for the multiple eigenvalues case.In [21], Jorba and Simó extended the conclusion of the linear system to the nonlinear system(7)x˙=A+εQt,εx+εgt+hx,t,x∈Rn.Suppose that A has n different nonzero eigenvalues; they proved that, under some nonresonant conditions and nondegeneracy conditions, there exists a nonempty Cantor set E⊂(0,ε0), such that, for all ε∈E, system (7) is reducible. Later, in [24], Wang and Xu considered the nonlinear quasi-periodic system(8)x˙=Ax+fx,t,x∈R2,and they proved without any nondegeneracy condition that one of two results holds: (1) system (8) is reducible to y˙=By+O(y) for all ε∈(0,ε0); (2) there exists a nonempty Cantor set E⊂(0,ε0), such that system (8) is reducible to y˙=By+O(y2) for all ε∈E.These papers above all deal with a total reduction to constant coefficients. In [25], instead of a total reduction to constant coefficients, Jorba, Ramirez-ros, and Villanueva considered the effective reducibility of the following quasi-periodic system:(9)x˙=A+εQt,εx,ε≤ε0,where A is a constant matrix with different eigenvalues. They proved that, under nonresonant conditions, by a quasi-periodic transformation, system (9) is reducible to a quasi-periodic system (10)y˙=A∗ε+εR∗t,εy,ε≤ε∗≤ε0,where R∗ is exponentially small in ε. In [26], Li and Xu obtained the similar result for Hamiltonian systems.In this paper, we consider the case thatA has multiple eigenvalues. Under some nonresonant conditions, we can obtain the effective reducibility for system (9) similar to [25, 26].Now we are in a position to state the main result.Theorem 1.
Consider the following linear Hamiltonian system:(11)x˙=A+εQt,εx,x∈Rn,where A is a constant matrix with eigenvalues λ1,…,λn, Q(t,ε) is an analytic quasi-periodic function on Dρ with the frequencies ω=(ω1,…,ωr), and ε∈(0,ε0) is a small parameter.
Ifλ=(λ1,λ2,…,λn) and ω=(ω1,ω2,…,ωr) satisfy the nonresonant conditions, (12)k,ω-1-λi+λj≥αkτfor all k∈Zr\{0}, 0≤i,j≤n, where α>0 is a small constant and τ>r-1. In addition, we assume that A+εQ¯ has n different eigenvalues μ1,…,μn, and δ:=min{ε-1|μi-μj|:0≤i,j≤n,i=j} is a positive constant independent of ε.
Then there exists someε∗>0 such that, for any ε∈(0,ε∗), there is an analytic quasi-periodic symplectic transformation x=ψ(t,ε)y on Dρ, where ψ(t,ε) has same frequencies as Q(t,ε), which changes system (11) into the following linear system:(13)y˙=A∗ε+εR∗t,εy,where A∗ is a constant matrix with (14)A∗-A≤eβ+1qe-1ε,R∗(t,ε) is an analytic quasi-periodic function on Dρ with the frequencies ω, and (15)R∗t,ερ-s≤e2β2qe-1exp⁡-dε1/21/τs,s∈0,ρ.Furthermore, a general explicit computation of ε∗ and d is possible: (16)ε∗=min⁡ε0,δe-13n-1eβq2,1β2,d=α12eβq,where β is the condition number of a matrix S such that S-1(A+εQ¯)S is diagonal, that is, β=C(S)=S-1S, and the constant q is the bound of Q(t,ε) on Dρ, that is, Q(t,ε)ρ≤q.Remark 2.
In general,Q depends on ε, so does the average Q¯. Below for simplicity, we do not indicate this dependence explicitly.Remark 3.
In Hamiltonian system (11), n is an even number. In fact, a Hamiltonian system is 2m-dimensional; moreover, the eigenvalues λ1,…,λ2m of a 2m×2m Hamiltonian matrix may be ordered so that λk+m=-λk(k=1,…,m).Now we give some remarks on this result. Firstly, here we deal with the Hamiltonian system and have to find thesymplectic transformation, which is different from that in [20, 23, 25]. Secondly, compared with [26], we can allow the matrix A to have multiple eigenvalues. Of course, if the eigenvalues of A are different, the nondegeneracy condition holds naturally, then our result is just the same as in [26].


## 2. Some Lemmas

We need some lemmas which are provided in this section for the proof of Theorem1.Lemma 4.
LetQ(t)=∑k∈ZrQke〈k,ω〉-1t be analytic quasi-periodic on Dρ with frequencies ω. Let Q~(t)=Q(t)-Q¯, (17)Q≥Mt=∑k∈Zr,k≥MQkek,ω-1t, and Q~M=Q~-Q≥M, where M>0. Then we have the following results:
(1)R¯, Q~ρ, Q~Mρ≤Qρ.
(2)Q≥Mρ-s≤Qρe-Ms, ∀s∈(0,ρ].This lemma can be seen in [25].The next lemma will be used to show the convergence.Lemma 5.
Let(qm)m, (am)m, and (rm)m be sequences defined by (18)qm+1=qm2,am+1=am+qm+1,rm+1=2+qm2-qmrm+qm+1,with initial values q1=a1=r1=e-1. Then (qm)m is decreasing to zero and (am)m, (rm)m are increasing and convergent to some values a∞ and r∞, respectively, with a∞<1/e-1, r∞<e/e-1.The proof of this lemma can be found in [25].Lemma 6.
LetD be an n×n diagonal matrix with different eigenvalues λ1,…,λn, and δ=mini≠j(|λi-λj|). Then if A verifies A-D≤b≤δ/3n-1, the following conditions hold:
(1)A has n different eigenvalues μ1,…,μn and |λj-μj|≤b,j=1,…,n.
(2) There exists a regular matrixS such that S-1AS=D∗=diag(μ1,…,μn) satisfying C(S)≤2.This lemma can be seen in [20].


## 3. Proof of Theorem1

By the assumptions of Theorem1, A+εQ¯ has n different eigenvalues μ1,⋯,μn, then there exists a symplectic matrix S such that (19)S-1A+εQ¯S=D=diagμ1,…,μn.Under the change of variables x=Sx1, system (11) is changed into(20)x˙1=D+εS-1Qt-Q¯Sx1=D+εQ~tx1,where Q~(t)=S-1(Q(t)-Q¯)S; it is easy to see that Q~¯=0.Now we can consider the iteration step.In them-th step, we consider the system(21)x˙m=Amε+εQmt+εRmtxm,m≥1,where A1=D, Q1=Q~M, R1=Q~≥M are Hamiltonian. Suppose Am, Qm, and Rm are Hamiltonian. Assume(22)Am-D≤q∗amε3/2,Qmρ≤q∗qm,Rmρ-s≤q∗rme-Mεs,where q∗=βeq, s∈(0,ρ], am,qm, and rm are defined in Lemma 5.Let the change of variables bexm=eεPmxm+1; under this symplectic transformation, system (21) is changed to (23)x˙m+1=e-εPmAm+εQm-εP˙meεPm+e-εPmεP˙meεPm-ddteεPm+εe-εPmRmeεPmxm+1=I-εPm+B~mAm+εQm-εP˙mI+εPm+Bm+e-εPmεP˙meεPm-ddteεPm+εe-εPmRmeεPmxm+1=Am+εQm-εP˙m+εAmPm-εPmAm+εQm∗+εe-εPmRmeεPmxm+1,where (24)εQm∗=-ε2PmQm-P˙m+ε2Qm-P˙mPm-ε2PmAm+εQm-εP˙mPm-εPmAm+εQm-εP˙mBm+Am+εQm-εP˙mBm+B~mAm+εQm-εP˙meεPm+e-εPmεP˙meεPm-ddteεPm,eεPm=I+εPm+Bm,e-εPm=I-εPm+B~m,and (25)Bm=εPm22!+εPm33!+⋯,B~m=εPm22!-εPm33!+⋯.We would like to have(26)Qm+AmPm-P˙m-PmAm=0,and this is equivalent to(27)P˙m=AmPm-PmAm+Qm.Now we want to solve (27) to obtain an analytic quasi-periodic Hamiltonian solution Pm(t) on Dρ with the frequencies ω.From (22), it follows that (28)Am-D≤q∗amε3/2≤δε3n-1,ε∈0,ε∗.Thus by Lemma 6, Am has n different eigenvalues λ1m,…,λnm and(29)λim-μi≤q∗amε3/2.Since Am is Hamiltonian, from the discussion in Section 15 of [17], it follows that there exists a symplectic matrix Sm such that (30)Sm-1AmSm=Dm=diagλ1m,…,λnm;moreover, C(Sm)≤2, where we let D1=D, λi1=μi(i=1,2,…,n).If(31)k,ω-1-λim+λjm≥Lε1/2for 0<|k|<M, 1≤i,j≤n, where L>0, M>0 are constants.Making the change of variablePm=SmXmSm-1 and defining Ym=Sm-1QmSm, (27) becomes(32)X˙m=DmXm-XmDm+Ym,Y¯m=0.Expand Xm and Ym into Fourier series (33)Xmt=∑k∈Zr,0<k<Mxmkek,ω-1t,Ymt=∑k∈Zr,0<k<Mymkek,ω-1t,where xmk=(xij,mk)1≤i,j≤n and ymk=(yij,mk)1≤i,j≤n.Thus the coefficients must be(34)xij,mk=yij,mkk,ω-1-λim+λjm.By (31), we have (35)Xmρ≤Lε1/2-1Ymρ≤Lε1/2-1CSmQmρ≤2Lε1/2-1Qmρ,which implies(36)Pmρ≤CSmXmρ≤4Lε1/2-1Qmρ.Now we prove thatPm is Hamiltonian. To this end, we only need to prove that Xm is Hamiltonian. Since Dm and Ym are Hamiltonian, then Dm=JDmJ and Ym=JYmJ, where DmJ and YmJ are symmetric. Let XmJ=J-1Xm, if XmJ is symmetric, then Xm is Hamiltonian. Below we prove that XmJ is symmetric. Substituting Xm=JXmJ into (32) yields that(37)X˙mJ=DmJJXmJ-XmJJDmJ+YmJ,and transposing (37), we get (38)X˙mJT=DmJJXmJT-XmJTJDmJ+YmJ.It is easy to see that JXmJ and JXmJT are solutions of (32); moreover, JXmJ¯=JXmJT¯=0. Since the solution of (32) with X¯m=0 is unique, we have that JXmJ=JXmJT, which implies that Xm is Hamiltonian. Since Sm is symplectic, it is easy to see that Pm=SmXmSm-1 is Hamiltonian.Thus, under the symplectic transformationxm=eεPmxm+1, system (21) is changed into the system(39)x˙m+1=Am+εQm∗+εe-εPmRmeεPmxm+1,where (40)εQm∗=-ε2Pm-AmPm+PmAm+ε2-AmPm+PmAmPm-ε2PmAm+εPmAm-εAmPmPm-εPmAm+εPmAm-εAmPmBm+Am+εPmAm-εAmPmBm+B~mAm+εPmAm-εAmPmeεPm+e-εPmεP˙meεPm-ddteεPm.System (39) can be written in the following system:(41)x˙m+1=Am+1+εQm+1+εe-εPmRmeεPmxm+1,where Am+1=Am+εQ¯m∗, Qm+1=(Qm∗-Q¯m∗)M, and Rm+1=(Qm∗-Q¯m∗)≥M+e-εPmRmeεPm are Hamiltonian and analytic quasi-periodic on Dρ with the frequencies ω.Now we prove the convergence of the iteration asm→∞.We first prove (22) holds by mathematical induction. By Lemma 4, it is easy to verify that (42)A1-D=0≤q∗a1ε3/2,Q1ρ≤q∗q1,R1ρ≤q∗r1e-Mεs,where a1=q1=r1=e-1, β=C(S), q∗=βeq, s∈(0,ρ], ε∈(0,ε∗).Assume that (22) holds at the m-th step. By (22) and (36), we have (43)Pmρ≤4Lε1/2-1Qmρ≤4Lε1/2-1q∗qm.Hence(44)ε1/2Pmρ≤qm2<12,ε∈0,ε∗,where L=8q∗ is a constant. It is easy to see that (45)εPmρ≤ε1/2Pmρ≤qm2<12,ε∈0,ε∗.Thus(46)e±εPmρ<2.From (22), (44), (46), Lemmas 4 and 5, it follows that (47)Qm∗ρ≤cε1/2q∗qm2ε1/2≤q∗qm2ε1/2=q∗qm+1ε1/2,where c is a positive constant, (48)Am+1-D≤Am-D+εQm∗ρ≤q∗am+qm+1ε3/2=q∗am+1ε3/2,Qm+1ρ≤Qm∗ρ≤q∗qm+1ε1/2≤q∗qm+1,Rm+1ρ-s≤Qm∗≥Mρ-s+1+εPm1-εPmRmρ-s≤qm+1+2+qm2-qmrmq∗e-Mεs=q∗rm+1e-Mεs,s∈0,ρ.By the mathematical induction, then (22) holds.Below we prove (31) holds. If k∈Zr and 0<|k|<M(ε)=d/ε1/21/τ, from the nonresonant conditions of Theorem 1 and (29), it follows that (49)k,ω-1-λim+λjm≥k,ω-1-λi+λj-λim-μi-λjm-μj-μi-λi-μj-λj≥αkτ-2q∗amε3/2-2qβε≥αkτ-2q∗a∞ε3/2-2qβε≥αkτ-2q∗a∞ε1/2-2qε1/2>αd-4q∗ε1/2=Lε1/2,where d=α/12q∗ and L=8q∗. So for any m≥1, (31) holds.Consequently, the iterative process can be carried out. The composition of all of the changeseεPm is convergent because eεPmρ≤1+qm. That is, there exists an analytic quasi-periodic function φ(t,ε) on Dρ with the frequencies ω, such that the composition of all of the changes eεPm converges to φ(t,ε) as m→∞.From (22) and Lemma 5, it follows that (50)limm⟶∞⁡Qm=0.By (22) and (41), we have (51)Am+1-Am≤εQ¯m∗≤εQm∗ρ≤q∗qm+1ε3/2.Hence, according to Lemma 5, Am and Rm are convergent as m→∞. Let (52)limm⟶∞⁡Am=A∞,limm⟶∞⁡Rm=R∞.Then the final equation is(53)x˙∞=A∞ε+εR∞t,εx∞,ε∈0,ε∗.By (22) and Lemma 5, we have (54)A∞ε-D≤q∗a∞ε3/2≤eβqe-1ε3/2and (55)R∞t,ερ-s≤q∗r∞e-Mεs≤e2βqe-1exp⁡-dε1/21/τs,s∈0,ρ,where d=α/12βeq.Under the change of variablesx∞=S-1y, system (53) is changed into (13) with (56)A∗=SA∞S-1,R∗=SR∞S-1.Moreover, (57)A∗-A=A∗-A+εQ¯+εQ¯≤SA∞S-1-SDS-1+εQρ≤βeβqe-1ε3/2+εq≤eβ+1qe-1ε,ε∈0,ε∗.R∗ρ-s≤e2β2qe-1exp⁡-dε1/21/τs,s∈0,ρ,ε∈0,ε∗,where d=α/12eβq.Thus, under the symplectic transformationx=φ(t,ε)S-1y=ψ(t,ε)y, Hamiltonian system (11) is changed into Hamiltonian system (13). Therefore, Theorem 1 is proved completely.



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