# On the Reducibility of Quasiperiodic Linear Hamiltonian Systems and Its Applications in Schrödinger Equation

Authors: Nina Xue; Wencai Zhao
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In this paper, we consider the reducibility of the quasiperiodic linear Hamiltonian systemẋ=A+εQt, where A is a constant matrix with possible multiple eigenvalues, Qt is analytic quasiperiodic with respect to t, and ε is a small parameter. Under some nonresonant conditions, it is proved that, for most sufficiently small ε, the Hamiltonian system can be reduced to a constant coefficient Hamiltonian system by means of a quasiperiodic symplectic change of variables with the same basic frequencies as Qt. Applications to the Schrödinger equation are also given.


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## Body

## 1. Introduction

In this paper, we are concerned with the reducibility of the quasiperiodic linear Hamiltonian system(1)ẋ=A+εQtx,x∈Rn,where A is a constant matrix with possible multiple eigenvalues, Qt is analytic quasiperiodic with respect to t, and ε is a small parameter.Firstly, let us recall the definition of the reducibility for quasiperiodic linear systems. LetAt be an n×n quasiperiodic matrix, the differential equation
(2)ẋ=Atx,x∈Rnis called reducible, if there exists a nonsingular quasiperiodic change of variables
(3)x=φty,where φt and φ−1t are quasiperiodic and bounded, which changes (2) into
(4)ẏ=By,y∈Rn,where B is a constant matrix.The well-known Floquet theorem states that every periodic differential equation (2) can be reduced to a constant coefficient differential equation (4) by means of a periodic change of variables with the same period as At. However, this is not true for the quasiperiodic linear system; one can see [1] for more details. In 1981, Johnson and Sell [2] proved that the quasiperiodic linear system (2) is reducible if the quasiperiodic coefficient matrix At satisfies the “full spectrum” condition.A typical example of quasiperiodic linear systems comes from the (continuous time) quasiperiodic Schrödinger operators, which are defined onL2R as
(5)Lyt=−y″t+qθ+ωtyt,where q:Tn⟶R is called the potential and θ∈Tn is called the phase. It is well known that the spectrum of L does not depend on the phase when ω is rationally independent, but it is closely related to the dynamics of the Schrödinger equations
(6)Lyt=−y″t+qθ+ωtytEytor equivalently the dynamics of the linear systems
(7)ẋ=VE,qθx,θ̇=ω,where
(8)VE,qθ=01qt−E0∈sl2,ℝ.Dinaburg and Sinai [3] proved that linear systems (7) are reducible for most E>E∗q,α,τ, which are sufficiently large, if ω is fixed and satisfies the Diophantine condition
(9)k,ω≥αkτ,k∈ℤr\0,where α,τ are positive constants. The result was generalized by Rüssmann [4] for ω satisfying the Bruno condition.Eliasson [5] proved a full measure reducibility result for quasiperiodic linear Schrödinger equations. More precisely, Eliasson proved that (7) is reducible for almost all E>E∗q,ω in Lebesgue measure sense, where ω is a fixed Diophantine vector.In the case thatn=2, a stronger reducibility result, called a nonperturbative reducibility, is available. The nonperturbative reducibility means that the smallness of the perturbation does not depend on the Diophantine constant α. Hou and You [6] proved, besides other results, the nonperturbative reducibility for (7).The reducibility of quasiperiodic linear systems with coefficients inglm,R was considered by Jorba and Simó [7]. 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 there exists sufficiently small ε0>0 and a nonempty Cantor set E⊂0,ε0, such that for any ε∈E, system (1) is reducible. Moreover, the relative measure of the set 0,ε0E in 0,ε0 is exponentially small in ε0. Junxiang [8] obtained the similar result for the multiple eigenvalue case. Later, many authors [7–10] paid attention to the reducibility of the quasiperiodic linear system (1), which is close to a constant coefficient linear system.In 1996, Jorba and Simó [10] extended the conclusion of the linear system to the nonlinear system
(10)ẋ=A+εQt,εx+εgt+hx,t,x∈Rn.Suppose thatA 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 (10) is reducible. Later, Wang and Xu [11] further investigated the nonlinear quasiperiodic system
(11)ẋ=Ax+ft,x,ε,x∈R2,where A is a real 2×2 constant matrix, and ft,0,ε=Oε,∂xft,0,ε=Oε as ε⟶0. They proved without any nondegeneracy condition, one of two results holds: (1) system (11) is reducible to ẏ=By+Oy for all ε∈0,ε0; (2) there exists a nonempty Cantor set E⊂0,ε0, such that system (11) is reducible to ẏ=By+Oy2 for all ε∈E.In [12], Her and You considered one-parameter family of quasiperiodic linear system
(12)ẋ=Aλ+gω1t,⋯,ωlt,λx,where A∈CωΛ,glm,ℂCωΛ,glm,ℂ be the set of m×m matrices Aλ depending analytically on parameter λ in a closed interval (Λ⊂ℝ), and g is analytic and small. They proved that under some nonresonant conditions and nondegeneracy conditions, there exists an open and dense set A in CωΛ,glm,ℂ, such that for each A∈A, system (12) is reducible for almost all λ∈Λ.Instead of a total reduction to a constant coefficient linear system, Jorba et al. [13] investigated the effective reducibility of the following quasiperiodic system:
(13)ẋ=A+εQt,εx,ε≤ε0,where A is a constant matrix with different eigenvalues. They proved that under nonresonant conditions, by a quasiperiodic transformation, system (13) is reducible to a quasiperiodic system
(14)ẏ=A∗ε+εR∗t,εy,ε≤ε∗≤ε0,where R∗ is exponentially small in ε. Li and Xu [14] obtained the similar result for Hamiltonian systems. Later, Xue and Zhao [15] extended the result to the case of multiple eigenvalues.In this paper, we will study the reducibility of quasiperiodic linearHamiltonian system (1), where matrix A may have multiple eigenvalues. To this end, the following assumptions are made.Assumption 1.
(nonresonant condition). Let all eigenvalues of matrixA be λ1,⋯,λn,Qt be an analytic quasiperiodic function on Dρ=θ∈ℂr:Imθj≤ρ,j=1,2,⋯,r with the frequencies ω=ω1,⋯,ωr. Suppose that λ=λ1,⋯,λn and ω=ω1,⋯,ωr satisfy the nonresonant conditions
(15)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.Assumption 2.
(nondegeneracy condition). Assume thatA+εQ¯ has ndifferent eigenvalues μ1,⋯,μn with μi≥2δε, μi−μj≥2δε, i≠j, 0≤i, j≤n, where δ is a positive constant independently of ε. Here, we denote the average of Qt by Q¯, that is,
(16)Q¯=limT→∞12T∫−TTQtdt.
We are in a position to state the main result.Theorem 3.
Suppose that Hamiltonian system (1) satisfies Assumptions 1 and 2. Then there exist some sufficiently small ε0>0 and a nonempty Cantor subset Eε0⊂0,ε0 with positive Lebesgue measure, such that for ε∈Eε0, Hamiltonian system (1) is reducible, i.e., there is an analytic quasiperiodic symplectic transformation x=ψty, where ψt has same frequencies as Qt, which changes (1) into the Hamiltonian system ẏ=By, where B is a constant matrix. Moreover, if ε0 is small enough, the relative measure of Eε0 in 0,ε0 is close to 1.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 [7, 8]. Secondly, we consider the reducibility, other than the effective reducibility in [13, 14]. The last but not the least, we can allow matrix A to have multiple eigenvalues. Of course, if the eigenvalues of A are different, the nondegeneracy condition holds naturally.As an example, we apply Theorem3 to the following Schrödinger equation:
(17)x¨+εatx=0,where at is analytic quasiperiodic with the frequencies ω=ω1,⋯,ωr. Denote the average of at by a¯. If a¯>0 and the frequencies ω of at satisfy the Diophantine condition
(18)k,ω≥αkτ,k∈ℤr\0,where α>0 is a small constant and τ>r−1, then there exists some sufficiently small ε0>0, equation (17) is reducible and the equilibrium of (17) is stable in the sense of Lyapunov for most sufficiently small ε∈0,ε0. Moreover, all solutions of equation (17) are quasiperiodic with the frequencies Ω=ω1,⋯,ωr,b for most sufficiently small ε∈0,ε0, where b=a¯ε+Oε2 as ε⟶0. Here, we remark that if we rewrite equation (17) into Hamiltonian system (1), we find that
(19)A=0100,which has multiple eigenvalues λ1=λ2=0. One can see Section 4 for more details about this example.There are plenty of works about the stability of all kinds of equations, one can refer to [16–23] for a detailed description. In particular, for quasiperiodic equations, in order to determine the type of stability of the equilibria of quasiperiodic Hamiltonian systems, the authors need to assume that the corresponding linearized system is reducible, and some conditions were added to the system after the reducibility. However, as far as we know, the case that the conditions are added to the original system has not been considered in the literature up to now, which we will study in the future.The paper is organized as follows. In Section2, we list some basic definitions and results that will be useful in the proof of the main result. In Section 3, we will prove Theorem 3. Equation (17) will be analyzed in Section 4.


## 2. Some Preliminaries

We first give the definition of quasiperiodic functions.Definition 4.
A functionf is said to be a quasiperiodic function with a vector of basic frequencies ω=ω1,ω2,⋯,ωr, if ft=Fθ1,θ2,⋯,θr, where F is 2π periodic in all its arguments and θj=ωjt for j=1,2,⋯,r. Moreover, if Fθθ=θ1,θ2,⋯,θr is analytic on Dρ=θ∈ℂr:Imθj≤ρ,j=1,2,⋯,r, we say that ft is analytic quasiperiodic on Dρ.It is well known that an analytic quasiperiodic functionft can be expanded as Fourier series
(20)ft=∑k∈ℤrfkek,ω−1t,with Fourier coefficients defined by
(21)fk=12πr∫TrFθe−k,θ−1dθ.Denote byfρ the norm
(22)fρ=∑k∈ℤrfkekρ.Definition 5.
Ann×n matrix Qt=qijt1≤i,j≤n is said to be analytic quasiperiodic on Dρ with frequencies ω=ω1,ω2,⋯,ωr, if all qijti,j=1,2,⋯,n are analytic quasiperiodic on Dρ with frequencies ω=ω1,ω2,⋯,ωr.Define the norm ofQ by
(23)Qρ=max1≤i≤n∑j=1nqijρ..It is easy to see that(24)Q1Q2ρ≤Q1ρQ2ρ.IfQ is a constant matrix, write Q=Qρ for simplicity. Denote the average of Qt by Q¯=q¯ij1≤i,j≤n, where
(25)q¯ij=limT→∞12T∫−TTqijtdt.See [24] for the existence of the limit.Also, we need two lemmas which are provided in this section for the proof of Theorem3 that were proved in [10].Lemma 6.
Leth:Bσ0⊂ℝn⟶ℝn be a C2 function that satisfies h0=0, Dxh0=0, Dxxhx≤K, x∈Bσ0. Then, hx≤K/2x2,Dxhx≤Kx.Lemma 7.
Suppose thatB0 is an n×n matrix with different nonzero eigenvalues satisfying μi0>γ, μi0−μj0>γ, i≠j,1≤i,j≤n, and S0 is a regular matrix such that S0−1B0S0=diagμ10,⋯,μn0. Set β0=maxS0,S0−1, and choose b such that
(26)0<b<γ3n−1β02.
IfB1 verifies B1−B0≤b, then the following conclusions hold:
(1)
B1 has n different nonzero eigenvalues μ11,⋯,μn1
(2)
There exists a regular matrixS1 such that S1−1B1S1=diagμ11,⋯,μn1, which satisfies S,S1−1≤β1, where β1=2β0.The next lemma is used to perform a step of the inductive procedure in the proof of Theorem3.Lemma 8.
Consider the differential equation(27)Ṗt=ΛPt−PtΛ+Rt,where Λ is a constant Hamiltonian matrix with n different eigenvalues ν1,⋯,νn, R is an analytic quasiperiodic Hamiltonian matrix on Dρ with frequencies ω satisfying R¯=0.
If(28)k,ω−1−vi+vj≥αk3τfor all 0≠k∈ℤr, and νi≥δε, νi−νj≥δε, for i≠j, 0≤i, j≤n, where δ is a positive constant independent of ε, then equation (27) has a unique analytic quasiperiodic Hamiltonian solution Pt with P¯=0, where Pt has frequencies ω and satisfies
(29)Pρ−s≤cαsvRρwith ν=3τ+r and 0<s<ρ, where the constant c depends only on τ and r.Proof.
ChoosingS such that S−1ΛS=D=diagν1,⋯,νn, making the change of variable Pt=SXtS−1 and defining Yt=S−1RtS, equation (27) becomes
(30)Ẋt=DXt−XtD+Yt,Y¯=0.
ExpandingX and Y into Fourier series yields that
(31)Xt=∑k∈ℤrXkek,ω−1t,Yt=∑k∈ℤrYkek,ω−1t,where Xk=xijk1≤i,j≤n and Yk=yijk1≤i,j≤n.By comparing the coefficients of (30), we obtain that
(32)xijk=yijkk,ω−1−vi+vj,1≤i,j≤n,k≠0,(33)x0ij=0,1≤i,j≤n.SinceR is analytic on Dρ, Y is also analytic on Dρ. Therefore, we have
(34)Yk≤Yρe−kρ.Hence,(35)Xρ−s=∑k∈ℤrXkekρ−s≤∑0≠k∈ℤrk3τe−skαYρ≤cαsvYρ,where ν=3τ+r and 0<s<ρ. Here and hereafter, we always use the same symbol c to denote different constants in estimates. Hence,
(36)Pρ−s≤cXρ−s≤cαsvYρ≤cαsvRρ.Now, we prove thatP is Hamiltonian. Since Λ and R are Hamiltonian, then Λ=JΛJ and R=JRJ, where ΛJ and RJ are symmetric. Let PJ=J−1P, if PJ is symmetric, then P is Hamiltonian. Below, we prove that PJ is symmetric. Substituting P=JPI into equation (27) yields
(37)ṖJ=ΛJJPJ−PJJΛJ+RJ,and transposing equation (37), we get
(38)ṖJT=ΛJJPJT−PJTJΛJ+RJ.It is easy to see thatJPJ and JPJT are solutions of (27); moreover, JPJ¯=JPJT¯=0. Since the solution of (27) with P¯=0 is unique, we have JPJ=JPJT, which implies that P is Hamiltonian. Up to now, we have finished the proof of this lemma.


## 3. Proof of Theorem3

From the assumptions of Theorem3, it follows that A+εQ¯ is a Hamiltonian matrix with n different eigenvalues μ1,⋯,μn, and μi≥2δε, μi−μj≥2δε, i≠j, 0≤i, j≤n, where δ is a positive constant independent of ε. We rewrite Hamiltonian system (1) into
(39)ẋ=A+εQ¯+εQt−Q¯x≔A1+εQ~tx,where A1=A+εQ¯, Q~t=Qt−Q, and Q~t=0.Introduce the change of variablesx=eεPtx1, where Pt will be determined later, under this symplectic transformation, Hamiltonian system (39) is changed into the new Hamiltonian system
(40)ẋ1=e−εPtA1+εQ~−εṖeεPtx1.ExpandeεP and e−εP into
(41)eεP=I+εP+B,e−εP=I−εP+B~,where
(42)B=εP22!+εP33!+⋯,B~=εP22!−εP33!+⋯.Then, system (40) can be rewritten
(43)ẋ1=I−εP+B~A1+εQ~−εṖI+εP+Bx1=A1+εQ~−εṖ+εA1P−εPA1+ε2Q1x1,where
(44)Q1=−PQ~−Ṗ+Q~−ṖP−PA1+εQ~−εṖP+I−εPA1+εQ~−εṖBε2+eεPA1+εQ~−εṖB~ε2.We would like to have(45)Q~−Ṗ+A1P−PA1=0,which is equivalent to
(46)Ṗ=A1P−PA1+Q~.By Assumption2 of Theorem 3, it is easy to see that the inequalities
(47)μi≥δε,μi−μj≥δε,i≠j,0≤i,j≤nhold. Moreover, if the equalities
(48)k,ω−1−μi+μj≥α0k3τ,0≠k∈ℤr,also hold, where α0=α/2, thus, by Lemma 8, (46) is solvable for P on a smaller domain, that is, there is a unique quasiperiodic Hamiltonian matrix Pt with frequencies ω on Dρ−s, which satisfies P¯=0 and
(49)Pρ−s≤cα0svQ~ρ≤cα0svQρ,where s=1/2ρ.Therefore, by (46), Hamiltonian system (43) becomes
(50)ẋ1=A1+ε2Q1x1,where
(51)Q1=PA1P−PA1+PA1−A1PP−PA1−εPA1−A1PP+I−εPA1−εPA1−A1PBε2+B~ε2A1−εPA1−A1PeεP.From Lemma6, it follows that
(52)Bρ−s≤cεPρ−s2,B~ρ−s≤cεPρ−s2.Therefore, ifε is sufficiently small, we have
(53)Q1ρ−s≤cPρ−s2≤cα02s2vQρ2.Now we consider the iteration step. In themth step, we consider the Hamiltonian system
(54)ẋm=Am+ε2mQmtxm,m≥1,where Am has n different eigenvalues λ1m,⋯,λnm with
(55)λim≥δε,λim−λjm≥δε,i≠j,1≤i,j≤n.Here, we define λi1=μi, i=1,⋯,n.LetAm+1=Am+ε2mQ¯m, then system (54) becomes
(56)ẋm=Am+1+ε2mQ~mxm,m≥1,where Q~m=Qmt−Q¯m. We need to solve
(57)Ṗm=Am+1Pm−PmAm+1+Q~m.If(58)k,ω−1−λim+1+λjm+1≥αmk3τ,0≠k∈ℤr,and Am+1 has n different eigenvalues λ1m+1,⋯,λnm+1 with
(59)λim+1≥δε,λim+1−λjm+1≥δε,i≠j,1≤i,j≤n,by Lemma 8, there is a unique quasiperiodic Hamiltonian matrix Pmt with frequencies ω on Dρm−sm, which satisfies
(60)Pmρm−sm≤cαmsmvQmρm.Thus, under the symplectic change of variablesxm=eε2mPmtxm+1, system (56) is changed into
(61)ẋm+1=Am+1+ε2m+1Qm+1xm+1,where
(62)Qm+1t=PmAm+1Pm−PmAm+1+PmAm+1−Am+1PmPm−PmAm+1+ε2mPmAm+1−Am+1PmPm+I−ε2mPmAm+1+ε2mPmAm+1−Am+1PmBmε2m+1+B~mε2m+1Am+1+ε2mPmAm+1−Am+1Pmε2mPm,e−ε2mPm=I+ε2mPm+Bm,e−ε2mPm=I−ε2mPm+B~m,Bm=ε2mPm22!+ε2mPm33!+⋯,B~m=ε2mPm22!+ε2mPm33!+⋯.From Lemma6, it follows that
(63)Bmρm−sm≤cε2mPmρm−sm2,B~mρm−sm≤cε2mPmρm−sm2.Therefore, ifε is sufficiently small, by (60) we have
(64)Qm+1ρm−sm≤cαm2sm2vQmρm2.Now we prove that the iteration is convergent asm⟶∞. When m=1, we choose
(65)α1=14α,ρ1=12ρ,s1=18ρ,F1=ε2Q1ρ1α12s12v.At themth step, we define
(66)αm=αm+12,sm=ρ2m+2,ρm=ρ1−s1+s2+⋯+sm−1Fm=ε2mQmρmαm2sm2v.By (64), we have
(67)Fm+1≤cε2m+1Qmρm2αm2sm2v2=cFm2,where the constant c depends only on α,ρ. Hence, it follows that
(68)cFm+1≤cFm2≤cF12m.IfcF1<1, then cFm⟶0 as m⟶∞. From (60), it follows that
(69)ε2mPmρm−sm<cFm.Thus, ifcF1<1/2, then
(70)e±ε2mPmρm≤2.Since(71)Am+1−Am=ε2mQ¯m≤ε2mQmρm<cFm,if cF1≤δε/3n−1βm2, it follows from (71) that
(72)Am+1−Am≤δε3n−1βm2,foranym≥1,where βm=maxSm,Sm−1 and Sm is the regular matrix in Lemma 7 such that
(73)Sm−1AmSm=diagλ1m,⋯,λnm.Thus, it follows from Lemma7 that Am+1 has n different eigenvalues λ1m+1,⋯,λnm+1.Moreover,(74)λim+1−λjm+1≥δε,i≠j,1≤i,j≤n,λim+1≥δε,i=1,⋯,n.In fact,(75)λim+1−λjm+1≥λi1−λj1−∑l=1mλil+1−λil+λjl+1−λjl≥λi1−λj1−2∑l=1mAl+1−Al≥λi1−λj1−2∑l=1mcFl≥2δε−2∑l=1mcFl.Moreover, we have(76)∑l=1mcFl≤∑l=1∞cFl≤∑m=0∞cF12m≤∑m=1∞cF1m=cF11−cF1<2cF1.Thus, ifcF1≤min1/2,1/4δε,δε/3n−1βm2, that is, 0<ε≤min1,c/Qρ,c/Qρ2, then by (68), we have
(77)λim+1−λjm+1≥2δε−4εcF1≥δε,i≠j,1≤n.In the same way as above, we have(78)λim+1≥δε,i=1,⋯,n.LetD∗=∩m+1∞Dρm=Dρ/4. By (69), the composition of all the changes eε2mPm converges to ψ as m⟶∞. Obviously,
(79)ε2mQmD∗≤cFm⟶0,m⟶∞.Furthermore, it follows from (71) that Am is convergent as m⟶∞. Define B=limm→∞Am. Then, under the symplectic change of variables x=ψty, m⟶∞ Hamiltonian system (1) is changed into ẏ=By.Now we prove that, for most sufficiently smallε, such symplectic transformation exists. From the above iteration, we need to prove that the nonresonant conditions
(80)k,ω−1−λim+1+λjm+1≥αmk3τ,for all 0≠k∈ℤr,1≤i,j≤n,m=0,1,2,⋯, hold for most sufficiently small ε.Letfε=k,ω−1−λim+1+λjm+1,i≠j,
(81)Oijmk=ε∈0,ε0:fε<αmk3τ,where we choose
(82)ε0=min1,cQρ,cQρ2,such that, for ε∈0,ε0, the above iteration is convergent, and
(83)dfdε=ddελim+1−λjm+1≥δ.Forε∈0,ε0, by (71), we have
(84)Am+1−A1≤Am+1−Am+⋯+A2−A1≤cFm+⋯+cF2≤2cF1≤12δε.Hence,(85)fε≥k,ω−1−λi+λj−λi1−λi−λj1−λj−λi1−λim+1−λj1−λjm+1≥αkτ−2qε−2Am+1−A1≥αkτ−2qε−δε≥αkτ−3Mε0,where M=maxq,δ.If1/kτ>6Mε0/α, then
(86)fε≥α2kτ>αmk3τ,and Oijmk=∅.Suppose that1/kτ<6Mε0/α. By (83), it follows that
(87)measOijmk<αmk3τδ.Thus,(88)meas∪i≠j∪0≠k∈ℤrOijmK≤n2αmδ∑k−τ<6Mε0/α1k3τ≤n2αmδ.36M2ε02α2∑k∈ℤr1kτ≤cε02m2.Let(89)Em=ε∈0,ε0:k,ω−1−λim+1+λjm+1>αmk3τ,0≠k∈ℤr,i≠j.Then(90)0,ε0−Em=∪i≠j∪0≠k∈ℤrOijmk.Thus(91)meas0,ε0−Em≤cε02m2.LetEε0=∩m=1∞Em, then
(92)meas0,ε0−Eε0≤cε02,limε0→0meas0,ε0−Eε0ε0=0.Therefore,Eε0 is a nonempty subset of 0,ε0. Thus, for ε∈Eε0, Hamiltonian system (1) is reducible, i.e., there exists a symplectic transformation x=ψty, which changes Hamiltonian system (1) into the Hamiltonian system ẏ=By. Thus, Theorem 3 is proved completely.


## 4. The Applications

As an example, we apply Theorem3 to the following Schrödinger equation
(93)x¨+εatx=0,where at is an analytic quasiperiodic function on Dρ with frequencies ω=ω1,⋯,ωr. Denote the average of at by a¯, and suppose a¯>0.Letẋ=y, then equation (93) can be rewritten in the equivalent form
(94)ẋ=y,ẏ=−εatx.To apply Theorem3, we express (94) in the form
(95)ż=A+εQtz,where
(96)z=xy,A=0100,Q=00‐at0.It is easy to see thatA has multiple eigenvalues λ1=λ2=0; moreover, A+εQ¯ has two different eigenvalues μ1=ia¯ε, μ2=−ia¯ε, where Q¯ stands for the average of the matrix Qt and i=−1. It is clear that
(97)μi=a¯ε≥2δε,i=1,2,(98)μ1−μ2=2a¯ε≥2δε,where we choose δ=1/2a¯>0, which is a constant independent of ε. Therefore, Theorem 3 can be applied. It follows from Theorem 3 that the following result holds.Theorem 9.
Assume thatat is an analytic quasiperiodic function on Dρ with frequencies ω=ω1,⋯,ωr, and a¯>0. If the frequencies ω of at satisfy the Diophantine condition
(99)k,ω≥αkτ,k∈ℤr\0,where α>0 is a small constant and τ>r−1.Then, there exist some sufficiently smallε0>0 and a nonempty Cantor subset Eε0⊂0,ε0 with positive Lebesgue measure, such that for ε∈Eε0, system (95) is reducible. Moreover, if ε0 is small enough, the relative measure of Eε0 in 0,ε0 is close to 1.Remark 10.
From Theorem9, it follows that equation (93) can be changed into a constant coefficient differential equation for most sufficiently small ε>0.Now we want to study the Lyapunov stability of the equilibrium of equation (93), using the results obtained in Section 3. If at is periodic in time (T is the period), one famous stability criterion was given by Magnus and Winkler [25] for Hill’s equation
(100)x¨+atx=0.That is, (100) is stable if
(101)at>0,∫0Tatdt≤4T,which can be shown using a Poincaré inequality. Such a stability criterion had been generalized and improved by Zhang and Li in [26], which now is the so-called Lp criterion. Recently, Zhang in [27] had extended such a criterion to the linear planar Hamiltonian system
(102)ẋ=mty,ẏ=−ntx,where mt, nt are continuous and T-periodic functions.However, for quasiperiodic equation (93), the results above cannot be applied directly. Now, we obtain a result about the stability of the equilibrium of equation (93).Theorem 11.
Under the conditions of Theorem9, the equilibrium of equation (93) is stable in the sense of Lyapunov for most sufficiently small ε>0.Proof.
Theorem9 tells us that, for most sufficiently small ε∈0,ε0, there exists an analytic quasiperiodic symplectic transformation z=ψtz∞, where ψt has the same frequencies as Qt, which changes (95) into the Hamiltonian system
(103)ż∞=Bz∞,where B is a constant matrix. Moreover, from the proof of Theorem 3 in Section 3, it follows that B has two different eigenvaluesλ1∞,λ2∞, satisfying
(104)λi∞≥δεi=1,2,λ1∞−λ2∞≥δε.
Furthermore, by the proof of Theorem3, we have
(105)B−A+εQ¯≤cF1=Oε2.
Therefore, the two different eigenvalues ofB are pure imaginary and can be written in the form
(106)λi∞=±ib,i=1,2,where b can be written in the following form
(107)b=a¯ε+Oε2,which depends on a¯ and ε only.Thus, there exists a singular symplectic matrixS such that
(108)S−1BS=ib00−ib.Letz∞=Sz~∞, under this symplectic transformation, system (103) is changed into
(109)z~̇∞=S−1BSz~∞=ib00−ibz~∞.Hence, by an analytic quasiperiodic symplectic transformation, equation (93) is changed into
(110)x¨∞+bx∞=0.It is easy to see that equation (110) is elliptic. Therefore, the equilibrium of equation (93) is stable in the sense of Lyapunov for most sufficiently small ε>0.For the existence of quasiperiodic solution of equation (93), we have the following result.Theorem 12.
Under the conditions of Theorem9, all solutions of equation (93) are quasiperiodic with frequencies Ω=ω1,⋯,ωr,b for most sufficiently small ε>0, where b is given by (107).Proof.
By Theorem9, we know that, for most sufficiently small ε∈0,ε0, there exists an analytic quasiperiodic symplectic transformation which has the same frequencies as at; by this transformation, equation (93) is changed into (110). On the other hand, it is easy to see that all solutions of equation (110) are periodic, and the frequency of these solutions is b.Thus, we only need to prove that, for most sufficiently smallε∈0,ε0, the following nonresonant condition
(111)k1ω1+⋯+krωr+1b≥α0k5τ+4holds for all k=k1,⋯,kr+1∈ℤr+1\0, where α0 is defined in Section 3, that is, α0=1/2α, and ω=ω1,⋯,ωr are the frequencies of at.Ifkr+1=0, then from the Diophantine condition (99), it follows that (111) holds.Suppose thatkr+1≠0. Let gε=k1ω1+⋯+krωr+kr+1b, and
(112)Ok=ε∈0,ε0:gε<α0k5τ+4.It follows from the nondegeneracy condition that(113)dgdε=ddεkr+1b≥kr+1δ.By (107), we have
(114)b≤4δε.From the Diophantine condition (99), it follows that
(115)gε≥αk1+⋯+kr−kr+1b≥αkrτ−kr+1b≥αkτ−kr+14δε≥αkτ−4δkε0.If1/kτ+1≥8δε0/α, then
(116)gε≥α2kτ≥α0k5τ+4,and Ok=∅.Suppose that1/kτ+1<8δε0/α, it follows from (113) that
(117)measOk<α0k5τ+4kr+1δ.Thus,(118)meas∪0≠k∈ℤr+1Ok≤α0δ∑1/kτ+1<8δε0/α1k5τ+4kr+1≤α0δ8δ4ε02α4∑k∈ℤr+11kτkr+1≤cε02∑0≠kr+1∈ℤ1kr+1τ+1≤cε02.Then(119)limε0→0meas∪0≠k∈ℤr+1Okε0=0.Therefore, (111) holds for most sufficiently small ε∈0,ε0.Thus, all solutions of equation (93) are quasiperiodic with frequencies Ω=ω1,⋯,ωr,b for most sufficiently small ε>0.



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*Source: 6260253-2020-05-05.xml*