# Reducibility for a Class of Almost-Periodic Differential Equations with Degenerate Equilibrium Point under Small Almost-Periodic Perturbations

Authors: Wenhua Qiu; Jianguo Si
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper focuses on almost-periodic time-dependent perturbations of an almost-periodic differential equation near the degenerate equilibrium point. Using the KAM method, the perturbed equation can be reduced to a suitable normal form with zero as equilibrium point by an affine almost-periodic transformation. Hence, for the equation we can obtain a small almost-periodic solution.


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## Body

## 1. Introduction and Main Result

Reducibility of nonautonomous finite-dimensional systems with quasiperiodic coefficients has basic importance in the analysis of dynamical systems; see [1, 2]. Unfortunately, we cannot guarantee in general such reducibility. In the last years, establishing the reducibility of finite-dimensional systems by means of the KAM tools is an active field of research, and many authors are devoted to the study of reducibility of such systems. In 1996, Jorba and Simó [3] considered reducibility of the following nonlinear quasiperiodic system near an elliptic equilibrium point:
(1)x˙=(A+ϵQ(t,ϵ))x+ϵg(t,ϵ)+h(x,t,ϵ),x∈ℝd,
where A is assumed to be elliptic. Q(t,ϵ), g(t,ϵ) and h(x,t,ϵ) depend on time in a quasiperiodic way with basic frequencies (ω1,…,ωr), h(x,ωt)=O(x2) as x→0. Under a nondegenerate condition and a nonresonant condition, using KAM iteration they proved that for most sufficiently small ϵ by an affine quasiperiodic transformation the system (1) is reducible to the following form:
(2)x˙=A*(ϵ)x+h*(x,t,ϵ),x∈ℝd,
where A* is a constant matrix close to A and h*(x,t,ϵ)=O(x2)(x→0) is a high-order term close to h. Therefore, the system (1) has a quasiperiodic solution near the zero equilibrium point. Some similar results were obtained in [4].Recently, Xu and Jiang [5] considered the following nonlinear quasiperiodic differential equation:
(3)x˙=x2n+1+h(x,ωt)+f(x,ωt),x∈ℝ,
where n≥0 is an integer, h=O(x2n+2)  (x→0) is a higher-order term, f is a small perturbation term, and h and f are all real analytic in x and t, quasiperiodic in t with frequency ω. Under the Diophantine condition, when f is sufficiently small the differential equation (3) can be reduced to a suitable normal form with zero as equilibrium point by an affine quasiperiodic transformation, so it has a quasiperiodic solution near zero.In 1996, Xu and You [6] considered the following linear almost-periodic differential equation:
(4)x˙=[A+ϵQ(t)]x,x∈ℝd,
where A is a constant d×d matrix with different eigenvalues λ1,…,λd and Q(t) is a d×d almost-periodic matrix with the frequency ω=(ω1,ω2,…). Under some small divisor condition, they proved that, for most sufficiently small ϵ the system (4) is reducible to the form y˙=By by an affine almost-periodic transformation.By the above inspired works [5, 6], we consider the following nonlinear almost-periodic differential equation:
(5)x˙=(A+ϵa(t))x2n+1+h(x,t,ϵ)+f(x,t,ϵ),x∈ℝ,
where n≥0 is an integer, A is a positive number, ϵ is a small parameter, h is a higher-order term, and f is a small perturbation term. Under some suitable conditions, we show that the differential equation (5) can be reduced to a suitable normal form with zero as equilibrium point by an affine almost-periodic transformation, so it has an almost-periodic solution near zero.For our purpose, we first introduce some definitions and notations.Definition 1.
The functionf(t) is called a quasiperiodic function of t with frequencies ω1,ω2,…,ωm, if there is a function F(θ)=F(θ1,θ2,…,θm), which is 2π-periodic in all its arguments θi  (i=1,…,m), such that f(t)=F(ωt)=F(ω1t,ω2t,…,ωmt).
IfF(θ) is analytic on a strip domain 𝕋s={θ∈ℂm/2πℤm∣|Imθj|≤s,j=1,2,…,m}, we say that f(t) is analytic quasiperiodic on 𝕋s. Expand f(t) as a Fourier series f=∑k∈ℤmfkei〈k,θ〉, where i=-1 and θ=ωt. Define ∥f∥𝕋s=∑k∈ℤm|fk|es|k|.
WriteD(0,r)={x∈ℂ|x|≤r} and Δr,s=D(0,r)×𝕋s.Definition 2.
LetP(x,t) be real analytic in x and t on Δr,s, and let P(x,t) be quasiperiodic with respect to t with the frequency ω. Then P can be expanded as a Fourier series as follows:
(6)P(x,t)=∑k∈ℤm‍Pk(x)ei〈k,θ〉.
Define a norm by
(7)∥P∥Δr,s=∑k∈ℤm‍|Pk|re|k|s,
where Pk(x)=∑l=0∞‍Plkxl and |Pk|r=supx∈D(0,r)∑l=0∞‍|Plk||x|l. It is easy to see that
(8)∥P1P2∥Δr,s≤∥P1∥Δr,s·∥P2∥Δr,s.Definition 3 (see [7]).
A functionΔ is called an approximation function if it satisfies the following:(1)
Δ:[0,+∞)→[1,+∞), Δ(0)=1, and Δ is a nondecreasing function;(2)
logΔ(t)/t is a decreasing function in [0,+∞);(3)
∫0∞‍logΔ(t)/t2<+∞.
Obviously, any positive power of an approximation function is again an approximation function, so is the product of two such functions.Definition 4 (see [8]).
Suppose thatℕ is the natural number set and τ is a set composed of the subset of ℕ. We say that (τ,[·]) is the finite spatial structure on ℕ if τ satisfies the following:(1)
the empty set∅∈τ;(2)
ifΛ1,Λ2∈τ, then Λ1∪Λ2∈τ;(3)
⋃Λ∈τΛ=ℕ, [∅]=0 and [Λ1∪Λ2]≤[Λ1]+[Λ2] ([·] is called a weight function defined in τ).Definition 5.
Assumek=(k1,k2,…)∈ℤ∞, the set
(9)suppk={(i1,i2,…,in)∣kj≠0asj=i1,i2,…,in;kj=0asj=otherwise}
is called the support set of k. Consider
(10)[k]≕infsuppk⊂Λ,Λ∈τ[Λ]
is called the weight of k, and |k|≕∑i=1∞‍|ki|.Definition 6.
IfQ(t)=∑Λ∈τQΛ(t) with QΛ(t) is quasiperiodic function with the frequency ωΛ={ωi∣i∈Λ}, then Q(t) is said to be almost-periodic function with the finite spatial structure (τ,[·]). If ω=(ω1,ω2,…) is the biggest subset of ∪ωΛ in the sense of integer modulus, then ω is called to be the frequency of Q(t).
If almost-periodic functionQ(t) has a rapidly converging Fourier series expansion
(11)Q(t)=∑k∈Z∞‍Ukei〈k,ωt〉,
where ω=(ω1,ω2,…) is the frequency and k=(k1,k2,…) have only finitely many nonzero components, then Q(t) is analytic in t.Definition 7.
LetQ(t)=∑Λ∈τQΛ(t) with the frequency ω=(ω1,ω2,…). For m≥0, s≥0,
(12)|||Q(t)|||m,s=∑Λ∈τ‍em[Λ]∥QΛ(t)∥𝕋s
is called the weight norm of Q(t) in the finite spatial structure (τ,[·]).From Definition7, we know that |Q(t)|≤|||Q(t)|||0,s≤|||Q(t)|||m,s, for m≥0, s≥0.Definition 8.
LetP(x,t)=∑Λ∈τPΛ(x,t) with the frequency ω=(ω1,ω2⋯), for m≥0, s≥0,
(13)|||P(x,t)|||m,Δr,s=∑Λ∈τ‍em[Λ]∥PΛ(x,t)∥Δr,s
is called the weight norm of P(x,t) in the finite spatial structure (τ,[·]).LetQ(t)=∑Λ∈τQΛ(t) be almost-periodic function; then
(14)[Q]=limT→∞∫-TT‍Q(t)dt
is called the mean value of Q(t). The existence of the limit can be found in [9].Throughout this paper, we assume that the following hypotheses hold(H1)
the functionsa, h, and f are real analytic in all variables and almost-periodic in t with common frequency vector ω=(ω1,ω2,…); they also have the the finite spatial structure (τ,[·]);(H2)
|〈k,ω〉|≥α/Δ4(|k|)Δ4([k]), k∈ℤ∞∖{0}, where α>0 is a constant and Δ is an approximation function;(H3)
h=𝒪(x2n+2)  (x→0), where n≥0, and for fixed m0, s>0, we have
(15)|||a(t)|||m0,s<+∞.Now we are ready to state the main result of this paper.Theorem 9.
Suppose that conditions(H1)–(H3) hold. Then there exists sufficiently small ϵ>0, such that if
(16)|||f|||m0,Δr,s≤ϵ,
then there exists an affine real analytic almost-periodic transformation of the form x=y+u(t) such that the differential equation (5) is changed to
(17)y˙=A*y2n+1+f*(y,t),
where f*(y,t)=O(y) as y→0. Moreover, u is a real analytic almost-periodic in t with
(18)|||u(t)|||0,s/2=O(ϵ1/(2n+2)),
and x=u(t) is also an almost-periodic solution of (5).


## 2. Normal Form for an Almost-Periodic Equation with Parameters

The proof of Theorem9 is based on a norm form theorem for an almost-periodic equation with parameters. In this section, we first consider the following real almost-periodic differential equation with two parameters:
(19)x˙=(A+ϵ[a])ξ2n+1-λ+(A+ϵa(t))Ω(ξ)x+P(x,t;ξ,λ),
where [a] is the mean value of a(t). λ∈J=[-1,1], ξ∈I=[-δ,δ] are parameters and Ω(ξ)=(2n+1)ξ2n.LetT⊂ℝ2 and p=(ξ,λ). Denote by
(20)B(T,σ)={p∈ℂ2∣dist(p,T)≤σ}
the complex σ-neighborhood of T in the two-dimensional complex space ℂ2.We will invoke the KAM iteration technique to prove the following normal form theorem.Theorem 10.
LetM0=B(I×J,σ0). Suppose that Ω and P are analytic on M0 and Δr0,s0×M0, respectively. Let ρ0=s0/8 and let E0>0. Let K0>0 such that E0=e-ρ0K0. There exists a sufficiently small E0>0 such that if
(21)ϵ0E0σ0≤14,ϵ0r0σ0≤116,E0r0≤1,(22)|||P|||m0,Δr0,s0×M0≤ϵ0=αr0E0Γ(m0/2)Γ(ρ0),
where
(23)Γ(ρ)≕supt≥0[Δ4(t)e-ρt],
then there exists a real C∞-smooth curve in M0,
(24)T:λ=λ(ξ)=(A+ϵ[a])ξ2n+1+N^(ξ),ξ∈I.
And for every p∈T, there exists an affine analytic almost-periodic transformation
(25)Φ(·,t;p):x=y+u*(t;p),
which changes (19) to
(26)y˙=Ω*(p)y+P*(y,t,p)
with P*(y,t,p)=O(y2)  (y→0). Moreover,
(27)|N^(ξ)|≤4ϵ0,|Ω*(p)-AΩ(p)|≤2ϵ0r0,∀p=(ξ,λ)∈T.
Furthermore, forp∈T, x=u*(t,p) is an analytic almost-periodic solution of the differential equation (19) with λ=λ(ξ).Lemma 11.
LetN(p)=(A+ϵ[a])ξ2n+1-λ+h(p), p=(ξ,λ)∈M. Suppose that h is real analytic on M⊂ℂ2 with
(28)∥𝒟ph∥M=∥hλ∥M+∥hξ∥M<12.
Suppose that, on the domain M, N(ξ,λ)=0 determines implicitly a real analytic curve T:λ=λ(ξ), ξ∈Iσ-/2, such that B(T,ϵ-/E-)⊂M, where ϵ-, E- and σ- are supposed to be well defined. Let N+(p)=(A+ϵ[a])ξ2n+1-λ+h+(p), where h+=h+h^ with ∥h^∥M≤ϵ. Suppose that ϵ/E≤ϵ-/4E- with E>0. Then there exists a domain of ℂ2, M+⊂M with dist(M+,∂M)≥ϵ-/4E-, such that N+ is real analytic on M+ and h^ satisfies ∥𝒟ph^∥M+≤ϵE-/ϵ-. If
(29)∥𝒟ph+∥M+≤12,
then, on the domain M+, N+(p)=0 determines implicitly a real analytic curve T+:λ=λ+(ξ), ξ∈Iσ/2  (σ<σ-), such that
(30)|λ+(ξ)-λ(ξ)|≤2ϵ,ξ∈Iσ/2,
and B(T+,ϵ/E)⊂M+. Moreover, one has
(31)|N+(p)|≤8ϵE,∀p∈M+.Proof.
LetM+=B(T,3ϵ/E)∩(Iσ/2+ϵ/E×ℂ). Since ϵ/E≤ϵ-/4E- and σ≤σ-, by B(T,ϵ-/E-)⊂M it follows easily that M+⊂M and dist(M+,∂M)≥ϵ-/4E-. Using Cauchy’s estimate we have
(32)∥𝒟ph^∥M+≤4ϵE-ϵ-.
By condition (29), the equation
(33)N+(p)=(A+ϵ[a])ξ2n+1-λ+h+(p)=0
determines implicitly an analytic curve on M+,
(34)T+:λ=λ+(ξ),ξ∈Iσ/2.
It follows that
(35)|λ+(ξ)-λ(ξ)|=|h(ξ,λ+)-h(ξ,λ)+h^(ξ,λ+)|,
so
(36)|λ+(ξ)-λ(ξ)|≤2ϵ,ξ∈Iσ/2.
Thus, B(T+,ϵ/E)⊂M+. Let E≤1/2. For each p=(ξ,λ)∈M+, we have
(37)|λ-λ+(ξ)|≤|λ-λ(ξ)|+|λ(ξ)-λ+(ξ)|≤3ϵE+2ϵ≤4ϵE.
Noting that |N+λ(p)|≤2, for all p∈M+, and N+(p)=0, for all p=(ξ,λ+(ξ)), we have
(38)|N+(p)|≤8ϵE,∀p∈M+.
Thus we prove Lemma 11.Lemma 12.
Assume thatR0(t)=∑Λ∈τR0Λ(t) is an analytic almost-periodic function with respect to t with the frequency ω=(ω1,ω2,…); it has the finite spatial structure (τ,[·]), and
(39)|||R0(t)|||m,s<+∞,m>0,s>0.
If
(40)|i〈k,ω〉|≥αΔ4(|k|)Δ4([k]),∀k∈ℤ∞∖{0},
then there exists an almost-periodic function u(t) with the same spatial structure as R0(t), which satisfies
(41)∂ωu=AΩu+R0K-[R0],
where ∂ωu=〈ω,∇θu〉 is the direction derivative of u along with ω  (θ=ωt).   R0K is the truncation of R0 with order K.
Moreover, for0<m-<m, 0<s-<s,
(42)|||u(t)|||m-m-,s-s-≤Γ(m-)Γ(s-)α|||R0(t)|||m,s.Proof.
We assume(43)R0ΛK(t)=∑suppk⊂Λ,|k|≤K‍R0Λkei〈k,ωt〉,uΛ(t)=∑suppk⊂Λ‍uΛkei〈k,ωt〉.
Insert the above formulas into the equation ∂ωu=AΩu+R0K-[R0], and compare the coefficients on both sides, thus we can find
(44)uΛk=R0Λki〈k,ω〉-AΩ,0<|k|≤K,suppk⊂Λ.
Then
(45)∥uΛ(t)∥𝕋s-s-≤∑suppk⊂Λ,0≤|k|≤K‍Δ4(|k|)e-s-|k|αΔ4([k])|R0Λk|es|k|≤Γ(s-)Δ4([Λ])α∥R0Λ∥𝕋s.
From Definition 7,
(46)|||u(t)|||m-m-,s-s-=∑Λ∈τ‍∥uΛ∥𝕋s-s-e(m-m-)[Λ]≤∑Λ∈τ‍Γ(s-)Δ4([Λ])α∥R0Λ∥𝕋s≤Γ(m-)Γ(s-)α∑Λ∈τ‍∥R0Λ∥𝕋sem[Λ]=Γ(m-)Γ(s-)α|||R0(t)|||m,s.
Thus, u(t)=∑Λ∈τuΛ(t) is convergent in the smaller domain Ds-s- with the norm |||·|||m-m-,s-s-.
Now we consider the following real analytic almost-periodic differential equation with parameters(47)x˙=N(p)+AΩ(p)x+ϵS(t,p)x+P(x,t,p)
with S(t,p)=a(t)Ω(p).Lemma 13.
Consider the above equation, whereN(p)=(A+ϵ[a])ξ2n+1-λ+h(p). Let h^=[P|x=0] and Ω^=(Px|x=0)K. Let h+=h+h^, Ω+=AΩ+Ω^ and N+(p)=(A+ϵ[a])ξ2n+1-λ+h+(p). Assume the following hold.(1)
P is real analytic on Δr,s×M and satisfies
(48)|||P|||m,Δr,s×M≤ϵ=αrEΓ(m/2)Γ(ρ)with0≤E<1/2, 0<ρ<s/2, and E=e-ρK.(2)
4ϵ/E≤σ.(3)
N, h^, and h+ are real analytic on M⊂C2 and all the assumptions of Lemma 11 hold. Let M+⊂M be the domain defined in Lemma 11.
Then, for anyp∈M+, there exists an affine analytic almost-periodic transformation
(49)Φ(·,t;p):x+∈D(0,r+)⟶x=(1+ϵQ)x++u(t;p)∈D(0,r),
where Φ is real analytic on D(r+,s+)×M+, such that the above differential equation is transformed to
(50)x˙+=N+(p)+Ω+(p)x++ϵ2S+(t,p)x++P+(x+,t,p),
where S+ and P+ will be get in the proof.
Moreover, one has(51)|||Φ-id|||m,Δr,s-2ρ×M+≤c(r+r2)E,|||DxΦ-1|||m,Δr,s-2ρ×M+≤crE.
The new perturbation termP+ satisfies
(52)|||P+|||m+,Δr+,s+×M+≤ϵ+=αr+E+Γ(m+/2)Γ(ρ+)
with(53)s+=s-2ρ,η=E1/2,ρ+=12ρ,r+=ηρ,E+=cE3/2,m+=12m.Proof.
The proof is the standard KAM step and we divide it into several parts.
(A) Truncation. Let R=R0+R1x with R0=P|x=0 and R1=Px|x=0. It follows easily that
(54)|||R0|||m,Δr,s×M≤ϵ,|||R1|||m,Δr,s×M≤ϵr.
Hence |||R|||m,Δr,s×M≤2ϵ. Let
(55)R=∑Λ∈τ,suppk⊂Λ,k∈Z∞‍Rk(x;p)ei〈k,ωt〉,RK=∑Λ∈τ,suppk⊂Λ,|k|≤K‍Rk(x;p)ei〈k,ωt〉.
By definition, we have
(56)|||R-RK|||m,Δr,s-ρ×M≤2ϵe-Kρ.
(B) Construnction of the Transformation. Define the transformation ϕ1:x=u(t)+y, where u satisfies
(57)∂ωu=AΩu+R0K-[R0].
From Lemma 12, we have
(58)|||u(t)|||m-(m/2),s-2ρ≤Γ(m/2)Γ(2ρ)α|||R0(t)|||m,s≤Γ(m/2)Γ(2ρ)αϵ<crE.
By the transformation ϕ1, the equation becomes
(59)u˙+y˙=N(p)+AΩ(p)(u+y)+ϵa(t)Ω(p)(u+y)+P(u+y,t,p),(60)y˙=N(p)+[R0]+(AΩ(p)+R1K)y+ϵa(t)Ω(p)y+R(u+y,t)-RK(u+y,t)+R1Ku+ϵa(t)Ω(p)u+P(u+y,t)-R(u+y,t);
then
(61)y˙=N+(p)+(AΩ(p)+R1K)y+ϵa(t)Ω(p)y+R(u+y,t)-RK(u+y,t)+R1Ku+ϵa(t)Ω(p)u+P(u+y,t)-R(u+y,t).
Define the transformationϕ2:y=(1+ϵQ)x+; Q satisfies
(62)Q˙=a(t)Ω(p).
We assume
(63)Q(t)=∑Λ∈τ‍QΛ(t),QΛ(t)=∑suppk⊂Λ,‍QΛkei〈k,ωt〉,a(t)=∑Λ∈τ‍aΛ(t),aΛ(t)=∑suppk⊂Λ,‍aΛkei〈k,ωt〉.
Then(64)QΛk={0,|k|=0,suppk⊂Λ,aΛkΩi〈k,ω〉,|k|>0,suppk⊂Λ.
Similar to Lemma 12, we have
(65)|||Q(t)|||m-(m/2),s-2ρ≤Γ(m/2)Γ(2ρ)α|||a(t)|||m,s|Ω|≤cΓ(m/2)Γ(2ρ)α,|||ϵQ(t)|||m-(m/2),s-2ρ≤crE.
By the transformationϕ2, the equation becomes
(66)(1+ϵQ)x˙++ϵQ˙x+=N+(p)+(AΩ(p)+R1K)(1+ϵQ)x++ϵa(t)Ω(p)(1+ϵQ)x++R(u+(1+ϵQ)x+,t)-RK(u+(1+ϵQ)x+,t)+R1Ku+ϵa(t)Ω(p)u+P(u+(1+ϵQ)x+,t)-R(u+(1+ϵQ)x+,t);
then
(67)x˙+=N+(p)+(AΩ(p)+R1K)x++ϵ2a(t)Ω(p)Q1+ϵQx++(11+ϵQ-1)N++11+ϵQR(u+(1+ϵQ)x+,t)-11+ϵQRK(u+(1+ϵQ)x+,t)+11+ϵQR1Ku+ϵa(t)Ω(p)u1+ϵQ+11+ϵQP(u+(1+ϵQ)x+,t)-11+ϵQR(u+(1+ϵQ)x+,t).
Thus, by the transformationΦ=ϕ1∘ϕ2:x=u+(1+ϵQ)x+, the equation is transformed to
(68)x˙+=N++Ω+x++ϵ2S+x++P+(x+,t;p),
where
(69)P+=(11+ϵQ-1)N++11+ϵQR(u+(1+ϵQ)x+,t)-11+ϵQRK(u+(1+ϵQ)x+,t)+11+ϵQR1Ku+ϵa(t)Ω(p)u1+ϵQ+11+ϵQP(u+(1+ϵQ)x+,t)-11+ϵQR(u+(1+ϵQ)x+,t),S+=a(t)Ω(p)Q1+ϵQ.
With the estimates of u and Q, we have
(70)|||Φ-id|||m,Δr,s-2ρ×M+≤r|||ϵQ|||m,Δr,s-2ρ×M++|||u|||m,Δr,s-2ρ×M+≤c(r+r2)E,|||DxΦ-1|||m,Δr,s-2ρ×M+≤|||ϵQ|||m,Δr,s-2ρ×M+≤crE.
Letη=E1/2≤1/4, s+=s-2ρ, and r+=ηr. Then the transformation Φ:x+∈D(0,r+)→D(0,2r+) is analytic almost-periodic on 𝕋s+ with respect to t and affine in x+.
(C) Estimates of Error Terms. Because |||ϵQ|||m+,Δr+,s+×M+<1, then 1/(1+ϵQ)=1-ϵQ+(ϵQ)2-(ϵQ)3+⋯+(-ϵQ)n+⋯. Thus,
(71)|||11+ϵQ|||m+,Δr+,s+×M+≤1+|||ϵQ|||m+,Δr+,s+×M++|||ϵQ|||m+,Δr+,s+×M+2+⋯+|||ϵQ|||m+,Δr+,s+×M+n+⋯≤2.
Let η=E1/2. Then, it follows that
(72)|||(11+ϵQ-1)N+|||m+,Δr+,s+×M+≤∥|ϵQN+|∥m+,Δr+,s+×M+≤cϵE,|||11+ϵQ(R∘Φ-RK∘Φ)|||m+,Δr+,s+×M+≤cϵe-Kρ,|||11+ϵQR1Ku|||m+,Δr+,s+×M+≤2ϵrcrE≤cϵE,|||ϵa(t)Ωu1+ϵQ|||m+,Δr+,s+×M+≤cϵE,|||11+ϵQ(P∘Φ-R∘Φ)|||m+,Δr+,s+×M+≤cϵr2r2η2=cη2ϵ.
So
(73)|||P+|||m+,Δr+,s+×M+≤cϵE+cϵe-Kρ+cη2ϵ=cϵE=αr+E+Γ(m+/2)Γ(ρ+)=ϵ+,
where ρ+=(1/2)ρ, r+=ηr, E+=cE3/2, and m+=(1/2)m. Thus we have proved Lemma 13.
Iteration. Now we choose some suitable parameters so that the above KAM step can be iterated infinitely. At the initial step, let
(74)s0=s,r0=r,E0>0,η0=E01/2,ρ0=s08,ϵ0=αr0E0Γ(m0/2)Γ(ρ0),m0=s08.
Let K0 satisfy E0=e-K0ρ0 and σ0=δ. Inductively, we define
(75)sj=sj-1-2ρj-1,ηj=Ej1/2,ρj=12ρj-1,rj=ηj-1rj-1,Ej=cEj-13/2,mj=12mj-1,ϵj=αrjEjΓ(mj/2)Γ(ρj),σj=σj-14.
And Kj satisfies Ej=e-Kjρj.
ByEj=cEj-13/2, we have Ej=c-2(c2E0)(3/2)j. Thus, if E0 is sufficiently small, we have cEj≤1 and ηj=Ej1/2≤1/4. Moreover, by definition it follows that
(76)(ϵj+1/Ej+1)(ϵj/Ej)=Γ(mj/2)Γ(ρj)Γ(mj+1/2)Γ(ρj+1)Ej1/2≤Ej1/2≤14≤1.
Thus ϵj+1/Ej+1≤ϵj/Ej, j≤0.
Now we prove that, forE0 sufficiently small, 4ϵj/Ej≤σj hold for all j≤0.
LetGj=4ϵj/Ejσj; from (21) we have G0≤1. Moreover, we have
(77)Gj+1Gj=σjϵj+1Ejσj+1ϵjEj+1=4αrj+1Ej+1Γ(mj+1/2)Γ(ρj+1)·Γ(mj/2)Γ(ρj)αrjEj·EjEj+1≤4ηjEj+1Ej·EjEj+1=4Ej1/2≤1,
for all j≤0. Thus, Gj≤G0≤1. So the inequalities in the assumption 2 of Lemma 13 hold for all j≤0.
LetM0=M, N0=(A+ϵ[a])ξ2n+1-λ, h0=0, Ω0=AΩ, S0=aΩ, and P0=P. By Lemmas 11, 12, and 13, we have a sequence of closed domains Mj with Mj+1⊂Mj and a sequence of affine transformations
(78)Φj:D(0,rj+1)⟶D(0,2rj+1)⊂D(0,rj),Φj:xj=(1+ϵ2jQj)xj+1+uj.
We also have
(79)|||Φj-id|||mj+1,Δrj+1,sj+1×Mj+1≤c(rj+rj2)Ej,|||DxΦj-1|||mj+1,Δrj+1,sj+1×Mj+1≤crjEj.
Let Φj=Φ0∘Φ1∘⋯Φj-1 with Φ0=id. Then, after the transformation Φj, (19) is changed to
(80)x˙=Nj(p)+Ωj(p)x+ϵ2jSjx+Pj(x,t;p).
By the inductive assumptions of KAM iteration, we have |||Pj|||mj,Δrj,sj×Mj≤ϵj.
The correction termsh^j and Ω^j satisfy
(81)∥h^j∥Mj≤ϵj,∥Ω^j∥Mj≤ϵjrj.
By Lemma 11, we have dist(Mj+1,∂Mj)≥ϵj-1/4Ej-1.  For Cauchy’s estimate we have
(82)∥Dph^j∥Mj+1≤4ϵjEj-1ϵj-1.
Noting thatM0=B(I×J,σ0) and T0:λ=(A+ϵ[a])ξ2n+1, |ξ|≤δ≤1. Since σ0≥4ϵ0/E0 and M1=B(T0,3ϵ0/E0)∩(Iσ0/2+ϵ0/E0×ℂ), it follows that M1⊂M0 and dist(M1,∂M0)≥σ0/4. For Cauchy’s estimate we have
(83)∥Dph^0∥M1≤4ϵ0σ0.
LetFj=ϵjEj-1/ϵj-1rj; then
(84)Fj=Ej-1rj·αrjEjΓ(mj-1/2)Γ(ρj-1)αrj-1Ej-1Γ(mj/2)Γ(ρj)≤Ej-1rj·αrjEjαrj-1Ej-1=cEj-12rj=Rj.
So Rj+1/Rj=cEj1/6. Obviously, we can choose E0 sufficiently small so that Rj+1/Rj≤1/2, Rj+1≤(1/2)Rj. Noting that h0=0 and ϵ0/r0σ0≤1/16, we have
(85)∥Dphj∥Mj+1≤∥Dph^0∥Mj+1+∑l=1j-1‍∥Dph^l∥Mj+1≤4ϵ0σ0+∑l=1j-1‍4rjRj≤14+cE02≤12.
So condition (29) holds for all j≤0.
From Lemma11, Nj(p)=(A+ϵ[a])ξ2n+1-λ+hj(p)=0 defines implicitly a real analytic curve Tj⊂Mj:λ=λj(ξ), ξ∈Iσj/2, satisfying
(86)|λj+1(ξ)-λj(ξ)|≤2ϵj,∀ξ∈Iσj/2.
Furthermore, ∥Nj+1∥Mj+1≤8ϵj/Ej.
Convergence of KAM Iteration. Now we prove the convergence of KAM iteration. By the definition of Ej, if E0 is sufficiently small, it follows that
(87)|||DxΦj|||mj,Δrj,sj×Mj≤∏i=0j-1‍(1+criEi)≤2.
Therefore, we have
(88)|||Φj-Φj-1|||mj,Δrj,sj×Mj≤c(rj-1+rj-12)Ej-1,|||Dx(Φj-Φj-1)|||mj,Δrj,sj×Mj≤crj-1Ej-1.
Let(89)Δ*=⋂j=0∞Δrj,sj=Δ0,s0/2,M*=⋂j≥0Mj,Φ=limj→∞Φj.
We have
(90)|||Φj-Φ0|||mj,Δrj,sj×Mj≤|||Φj-Φj-1|||mj,Δrj,sj×Mj+|||Φj-1-Φj-2|||mj,Δrj,sj×Mj+⋯+|||Φ1-Φ0|||mj,Δrj,sj×Mj≤c(rj-1+rj-12)Ej-1+c(rj-2+rj-22)Ej-2+⋯+c(r0+r02)E0≤c(r0+r02)E0.
Thus(91)|||Φ-id|||m*,Δr*,s*×M*≤c(r0+r02)E0,|||DxΦ-id|||m*,Δr*,s*×M*≤cr0E0.
So we have the convergence of Φj to Φ on Δr0/2,s0/2.
From (86) it is easy to show that λj is convergent on I. In fact, ϵj+1/ϵj≤cEj≤1/2. For i>j, it follows that
(92)|λi(ξ)-λj(ξ)|≤∑l=ji-1‍2ϵl≤4ϵj,∀ξ∈I(1/2)σj.
Let λi(ξ)→λ(ξ)=(A+ϵ[a])ξ2n+1+N^(ξ), ξ∈I. For λ0(ξ)=(A+ϵ[a])ξ2n+1, we have
(93)|λi(ξ)-(A+ϵ[a])ξ2n+1|≤4ϵ0,∀ξ∈I(1/2)σj,|N^(ξ)|=|λ(ξ)-(A+ϵ[a])ξ2n+1|≤4ϵ0,∀ξ∈I.
Moreover, by Cauchy’s estimate we have(94)|λj+1′(ξ)-λj′(ξ)|≤4ϵjσj,∀ξ∈I.
LetLj=4ϵj/σj; then Lj+1/Lj≤cEj. Thus, it is easy to prove that {λj′(ξ)} is convergent uniformly on I, and so λ(ξ) is differentiable on I. In fact, in the same way as in [7], we can prove that λ(ξ) is C∞-smooth on I.
SinceTi⊂Mi⊂Mj, for all i≥j, letting i→∞ we have T={(ξ,λ)∣λ=λ(ξ),ξ∈I⊂Mj} and T=M*=∩j≤0Mj. Obviously, Nj(p)→0, for p∈T. Let Ωj→Ω* and let Pj→P*. By Cauchy’s estimate we have
(95)|DxPj|x=0|≤ϵjrj=αEjΓ(mj/2)Γ(ρj)⟶0.
Thus P*|x=0=0 and DxP*|x=0=0. Hence P*(x,t;p)=O(x2) for p∈T.
Noting that∥Ω^j∥Mj≤ϵj/rj and Ωj=AΩ+∑i=1j-1‍Ω^j, we have
(96)|Ω*(p)-AΩ(p)|≤2ϵ0r0,∀p∈Γ.
The proof of Theorem 10 is complete.


## 3. Proof of Theorem9

Letx=ξ+y with |ξ|≤δ and |y|≤r0≤δ. Then (5) becomes
(97)y˙=(A+ϵa(t))(y+ξ)2n+1+h(y+ξ,t)+f(y+ξ,t).
Let
(98)(y+ξ)2n+1=ξ2n+1+(2n+1)ξ2ny+g(y,ξ)
with
(99)g(y,ξ)=(2n+1)2n2ξ2n-1y2+⋯+(2n+1)ξy2n+y2n+1.
We write h(y+ξ,t)=h(y,t,ξ) and f(y+ξ,t)=f(y,t,ξ) and decompose a(t) as a(t)=[a]+a(t)~, where [a] is the average of a(t) and a(t)~ has zero mean value. Then the differential equation (5) becomes
(100)y˙=(A+ϵ[a])ξ2n+1+(2n+1)(A+ϵa(t))ξ2ny+P(y,t,ξ),
where P=ϵa(t)~ξ2n+1+(A+ϵa(t))g+h+f.LetI=[-δ,δ] and Iσ={ξ∈ℂ∣dist(ξ,I)≤σ}. By assumption and the choice of r and δ, it is easy to see that g, h, and f are all real analytic on Δr0,s0×Iσ with r0≤δ, σ0≤δ and s0=s. Moreover, we have that
(101)|||(A+ϵa(t))g|||m0,Δr0,s0×Iσ0≤cr02δ,|||ϵa(t)~ξ2n+1|||m0,Δr0,s0×Iσ0≤cϵ|δ|2n+1≤cϵ,
where c is a constant depending on n. Note that we always use c to denote different constants in estimates. Similarly, we have |||h|||m0,Δr0,s0×Iσ0≤cδ2n+2 and |||f|||m0,Δr0,s0×Iσ0≤ϵ. Let δ=ϵ1/(2n+2) and let r0=ϵ2n+1/(4n+4). Then δ2n+2=ϵ and r0=δ2n+1/2. Let σ0=δ. Then it follows that
(102)|||P|||m0,Δr0,s0×Iσ0≤cr02δ+cδ2n+2+ϵ=cϵ=ϵ0.
Now (97) is equivalent to the following parameterized differential equation:
(103)y˙=(A+ϵ[a])ξ2n+1+(A+ϵa(t))Ω(ξ)y+P(y,t,ξ),
where Ω(ξ)=(2n+1)ξ2n.Now we want to prove that ifϵ is sufficiently small, then there exists ξ*∈I such that at ξ=ξ* the differential equation (103) is reducible to a normal form with zero as equilibrium point. We introduce an external parameter and consider the following almost-periodic differential equation:
(104)x˙=(A+ϵ[a])ξ2n+1-λ+(A+ϵa(t))Ω(ξ)y+P(y,t,ξ),
where λ∈J=[-1,1] is an external parameter. Obviously, (103) corresponds to (104) with λ=0.By Theorem10, we will prove that there exists a smooth curve T:λ=λ(ξ), ξ∈I, such that for (ξ,λ)∈T the differential equation (104) can be reduced to a normal form with zero equilibrium. Moreover, we can find ξ*∈I such that λ(ξ*)=0 and then come back to the original equation (103) with ξ=ξ*.To apply Theorem10 to (104), we verify all the assumptions. Note that
(105)δ=ϵ1/2n+2,r0=ϵ2n+1/4n+4,s0=s,σ0=δ,ϵ0=cϵ.
Let
(106)E0=ϵ0Γ(m0/2)Γ(ρ0)αr0=ϵ2n+3/4n+4Γ(m0/2)Γ(ρ0)α.
Thus when ϵ is small enough, E0 is also small. Moreover, we have
(107)ϵ0E0σ0=αϵ(2n-1)/(4n+4)Γ(m0/2)Γ(ρ0)≤14,ϵ0r0σ0=ϵ(2n+1)/(4n+4)≤116,E0r0=ϵ1/2n+2Γ(m0/2)Γ(ρ0)α≤1.
So all the inequalities of (21) hold. Moreover, we have
(108)|||P|||m0,Δr0,s0×M0≤ϵ0=αr0E0Γ(m0/2)Γ(ρ0).
Thus Theorem 10 holds for (104).Since|N^(±δ)|≤4ϵ0≤cδ2n+2, it follows that λ(±δ)=±(A+ϵ[a])δ2n+1+N^(±δ) must have a different sign for δ>0 sufficiently small. Thus there exists a ξ*∈I, such that λ(ξ*)=0. Moreover, we have |ξ*|≤δ=ϵ1/(2n+2) and p*=(ξ*,0)∈T. Hence, by the transformation x=Φ(y,t;p*) the differential equation (103) at ξ=ξ* is changed to
(109)y˙=A*Ω*y+P*(y,t,p*)
with P*(y,t,p*)=O(y2)  (y→0). Therefore by the transformation x=Φ(y+ξ*,t;p*) the system (5) is changed to the form of (17). Moreover, x(t)=Φ(ξ*,t;p*) is an almost-periodic solution with the frequency ω.SinceΦ(y+ξ*,t;p*)=y+u*(t;p)+ξ*, we have x=y+u(t) with u(t)=u*(t;p)+ξ*. Noting that |ξ*|≤ϵ1/(2n+2), it follows that, for p∈T, |||u|||0,s0/2=O(ϵ1/(2n+2)). Thus, Theorem 9 is proved.



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*Source: 386812-2013-11-24.xml*