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

Definition 1. The function f(t) is called a quasiperiodic function of t with frequencies ω₁, ω₂, …, ω_m, if there is a function F(θ) = F(θ₁, θ₂, …,θ_m), which is 2π-periodic in all its arguments θ_i  (i = 1, …, m), such that f(t) = F(ωt) = F(ω₁t, ω₂t, …, ω_mt).

If F(θ) is analytic on a strip domain 𝕋_s = {θ ∈ ℂ^m/2πℤ^m∣|Imθ_j| ≤ s,  j = 1,2, …, m}, we say that f(t) is analytic quasiperiodic on 𝕋_s. Expand f(t) as a Fourier series $$, where $$ and θ = ωt. Define $$.

Write D(0, r) = {x ∈ ℂ | x | ≤ r} and Δ_r,s = D(0, r) × 𝕋_s.

Definition 2. Let P(x, t) be real analytic in x and t on Δ_r,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:

Define a norm by where $$ and $$. It is easy to see that

Definition 3 (see [7].)A function Δ is called an approximation function if it satisfies the following:

• (1)

  Δ : [0, +∞)→[1, +∞), Δ(0) = 1, and Δ is a nondecreasing function;

• (2)

  log Δ(t)/t is a decreasing function in [0, +∞);

• (3)

  $$.

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 that ℕ 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)

  if Λ₁, 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. Assume k = (k₁, k₂, …) ∈ ℤ^∞, the set

is called the support set of k. Consider is called the weight of k, and $$.

Definition 6. If Q(t) = ∑_Λ∈τQ_Λ(t) with Q_Λ(t) is quasiperiodic function with the frequency ω_Λ = {ω_i∣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 ∪ω_Λ in the sense of integer modulus, then ω is called to be the frequency of Q(t).

If almost-periodic function Q(t) has a rapidly converging Fourier series expansion

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. Let Q(t) = ∑_Λ∈τQ_Λ(t) with the frequency ω = (ω₁, ω₂, …). For m ≥ 0, s ≥ 0,

is called the weight norm of Q(t) in the finite spatial structure (τ, [·]).

Definition 8. Let P(x, t) = ∑_Λ∈τP_Λ(x, t) with the frequency ω = (ω₁, ω₂⋯), for m ≥ 0, s ≥ 0,

is called the weight norm of P(x, t) in the finite spatial structure (τ, [·]).

Theorem 9. Suppose that conditions (H1)–(H3) hold. Then there exists sufficiently small ϵ > 0, such that if

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 where f_*(y, t) = O(y) as y → 0. Moreover, u is a real analytic almost-periodic in t with and x = u(t) is also an almost-periodic solution of (5).

Theorem 10. Let M₀ = B(I × J, σ₀). Suppose that Ω and P are analytic on M₀ and $$, respectively. Let ρ₀ = s₀/8 and let E₀ > 0. Let K₀ > 0 such that $$. There exists a sufficiently small E₀ > 0 such that if

where then there exists a real C^∞-smooth curve in M₀, And for every p ∈ T, there exists an affine analytic almost-periodic transformation which changes (19) to with P_*(y, t, p) = O(y²)  (y → 0). Moreover,

Furthermore, for p ∈ T, x = u_*(t, p) is an analytic almost-periodic solution of the differential equation (19) with λ = λ(ξ).

Lemma 11. Let N(p) = (A + ϵ[a])ξ²ⁿ⁺¹ − λ + h(p), p = (ξ, λ) ∈ M. Suppose that h is real analytic on M ⊂ ℂ² with

Suppose that, on the domain M, N(ξ, λ) = 0 determines implicitly a real analytic curve T : λ = λ(ξ), $$, such that B(T, ϵ₋/E₋) ⊂ M, where ϵ₋, E₋ and σ₋ are supposed to be well defined. Let N₊(p) = (A + ϵ[a])ξ²ⁿ⁺¹ − λ + h₊(p), where $$ with $$. Suppose that ϵ/E ≤ ϵ₋/4E₋ with E > 0. Then there exists a domain of ℂ², M₊ ⊂ M with dist (M₊, ∂M) ≥ ϵ₋/4E₋, such that N₊ is real analytic on M₊ and $$ satisfies $$. If then, on the domain M₊, N₊(p) = 0 determines implicitly a real analytic curve T₊ : λ = λ₊(ξ), ξ ∈ I_σ/2  (σ < σ₋), such that and B(T₊, ϵ/E) ⊂ M₊. Moreover, one has

Proof. Let M₊ = B(T, 3ϵ/E)∩(I_σ/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

By condition (29), the equation determines implicitly an analytic curve on M₊, It follows that so Thus, B(T₊, ϵ/E) ⊂ M₊. Let E ≤ 1/2. For each p = (ξ, λ) ∈ M₊, we have Noting that |N_+λ(p)| ≤ 2, for all p ∈ M₊, and N₊(p) = 0, for all p = (ξ, λ₊(ξ)), we have Thus we prove Lemma 11.

Lemma 12. Assume that R₀(t) = ∑_Λ∈τR_0Λ(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

If then there exists an almost-periodic function u(t) with the same spatial structure as R₀(t), which satisfies where ∂_ωu = 〈ω, ∇_θu〉 is the direction derivative of u along with ω  (θ = ωt).   $$ is the truncation of R₀ with order K.

Moreover, for $$, $$,

Proof. We assume

Insert the above formulas into the equation $$, and compare the coefficients on both sides, thus we can find Then From Definition 7, Thus, u(t) = ∑_Λ∈τu_Λ(t) is convergent in the smaller domain $$ with the norm $$.

Now we consider the following real analytic almost-periodic differential equation with parameters

with S(t, p) = a(t)Ω(p).

Lemma 13. Consider the above equation, where N(p) = (A + ϵ[a])ξ²ⁿ⁺¹ − λ + h(p). Let $$ and $$. Let $$, $$ and N₊(p) = (A + ϵ[a])ξ²ⁿ⁺¹ − λ + h₊(p). Assume the following hold.

• (1)

  P is real analytic on Δ_r,s × M and satisfies

  $$ ()

•  

  with 0 ≤ E < 1/2, 0 < ρ < s/2, and E = e^−ρK.

• (2)

  4ϵ/E ≤ σ.

• (3)

  N, $$, 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 any p ∈ M₊, there exists an affine analytic almost-periodic transformation

where Φ is real analytic on D(r₊, s₊) × M₊, such that the above differential equation is transformed to where S₊ and P₊ will be get in the proof.

Moreover, one has

The new perturbation term P₊ satisfies

with

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 = P_x|_x=0. It follows easily that

Hence $$. Let By definition, we have

(B) Construnction of the Transformation. Define the transformation ϕ₁ : x = u(t) + y, where u satisfies

From Lemma 12, we have By the transformation ϕ₁, the equation becomes then

Define the transformation ϕ₂ : y = (1 + ϵQ)x₊; Q satisfies

We assume

Then

Similar to Lemma 12, we have

By the transformation ϕ₂, the equation becomes

then

Thus, by the transformation Φ = ϕ₁∘ϕ₂ : x = u + (1 + ϵQ)x₊, the equation is transformed to

where With the estimates of u and Q, we have

Let η = E^1/2 ≤ 1/4, s₊ = s − 2ρ, and r₊ = ηr. Then the transformation Φ : x₊ ∈ D(0, r₊) → D(0,2r₊) is analytic almost-periodic on $$ with respect to t and affine in x₊.

(C) Estimates of Error Terms. Because $$, then 1/(1 + ϵQ) = 1 − ϵQ + (ϵQ) ² − (ϵQ) ³ + ⋯+(−ϵQ) ⁿ + ⋯. Thus,

Let η = E^1/2. Then, it follows that So where ρ₊ = (1/2)ρ, r₊ = ηr, E₊ = cE^3/2, 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

Let K₀ satisfy $$ and σ₀ = δ. Inductively, we define And K_j satisfies $$.

By $$, we have $$. Thus, if E₀ is sufficiently small, we have cE_j ≤ 1 and $$. Moreover, by definition it follows that

Thus ϵ_j+1/E_j+1 ≤ ϵ_j/E_j, j ≤ 0.

Now we prove that, for E₀ sufficiently small, 4ϵ_j/E_j ≤ σ_j hold for all j ≤ 0.

Let G_j = 4ϵ_j/E_jσ_j; from (21) we have G₀ ≤ 1. Moreover, we have

for all j ≤ 0. Thus, G_j ≤ G₀ ≤ 1. So the inequalities in the assumption 2 of Lemma 13 hold for all j ≤ 0.

Let M₀ = 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 M_j with M_j+1 ⊂ M_j and a sequence of affine transformations

We also have Let Φ^j = Φ₀∘Φ₁∘⋯Φ_j−1 with Φ₀ = id. Then, after the transformation Φ^j, (19) is changed to By the inductive assumptions of KAM iteration, we have $$.

The correction terms $$ and $$ satisfy

By Lemma 11, we have dist (M_j+1, ∂M_j) ≥ ϵ_j−1/4E_j−1.  For Cauchy’s estimate we have

Noting that M₀ = B(I × J, σ₀) and T₀ : λ = (A + ϵ[a])ξ²ⁿ⁺¹, |ξ | ≤ δ ≤ 1. Since σ₀ ≥ 4ϵ₀/E₀ and $$, it follows that M₁ ⊂ M₀ and dist (M₁, ∂M₀) ≥ σ₀/4. For Cauchy’s estimate we have

Let F_j = ϵ_jE_j−1/ϵ_j−1r_j; then

So $$. Obviously, we can choose E₀ sufficiently small so that R_j+1/R_j ≤ 1/2, R_j+1 ≤ (1/2)R_j. Noting that h₀ = 0 and ϵ₀/r₀σ₀ ≤ 1/16, we have So condition (29) holds for all j ≤ 0.

From Lemma 11, N_j(p) = (A + ϵ[a])ξ²ⁿ⁺¹ − λ + h_j(p) = 0 defines implicitly a real analytic curve T_j ⊂ M_j : λ = λ_j(ξ), $$, satisfying

Furthermore, $$.

Convergence of KAM Iteration. Now we prove the convergence of KAM iteration. By the definition of E_j, if E₀ is sufficiently small, it follows that

Therefore, we have

Let

We have

Thus

So we have the convergence of Φ^j to Φ on $$.

From (86) it is easy to show that λ_j is convergent on I. In fact, ϵ_j+1/ϵ_j ≤ cE_j ≤ 1/2. For i > j, it follows that

Let $$, ξ ∈ I. For λ₀(ξ) = (A + ϵ[a])ξ²ⁿ⁺¹, we have

Moreover, by Cauchy’s estimate we have

Let L_j = 4ϵ_j/σ_j; then L_j+1/L_j ≤ cE_j. Thus, it is easy to prove that $$ 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.

Since T_i ⊂ M_i ⊂ M_j, for all i ≥ j, letting i → ∞ we have T = {(ξ, λ)∣λ = λ(ξ),   ξ ∈ I ⊂ M_j} and T = M_* = ∩_j≤0M_j. Obviously, N_j(p) → 0, for p ∈ T. Let Ω_j → Ω_* and let P_j → P_*. By Cauchy’s estimate we have

Thus P_*|_x=0 = 0 and D_xP_*|_x=0 = 0. Hence P_*(x, t; p) = O(x²) for p ∈ T.

Noting that $$ and $$, we have

The proof of Theorem 10 is complete.