Boundedness of Solutions for a Class of Sublinear Reversible Oscillators with Periodic Forcing

Remark 1. Using the method of this paper we also can consider the more general equation

provided of adding suitable conditions for g(x). For convenience, we only consider the case g(x) ≡ x.

Remark 2. Adding the perturbation term ɛe(t) | x|^α−1x will lead to a new difficulty for estimating |S(θT₀)|^α−1C(θT₀) appeared in (86). Fortunately, we can easily verify that $$ is bounded by a constant (see in the proof of Lemma 12).

Theorem 3. Assume that B ≠ 0 and (A1)-(A2) hold. Then there exists an 0 < ɛ^** < ɛ^* such that for any 0 < ɛ < ɛ^**, every solution of (8) is bounded if and only if B > 0.

Theorem 4. Under the conditions of Theorem 3, there is an ɛ₀ > 0 such that, for any ω ∈ (n, n + ɛ₀), (8) has a solution $$ of Mather type with rotation number ω. More precisely:

• (i)

  if ω = p/q is rational, the solutions $$, 1 ≤ i ≤ q − 1, are periodic solutions of period q; moreover, in this case

  $$ ()

• (ii)

  if ω is irrational, the solution $$ is either a usual quasi-periodic solution or a generalized one.

We recall that a solution is called generalized quasi-periodic if the closed set

is a Denjoys minimal set.

Lemma 5. There exists a G-invariant diffeomorphism (x, y)→(x, z) such that (13) is transformed into the following system:

where $$.

Proof. Introduce a transformation Ψ:

where U(x, t) will be determined later. Under this transformation, the system (13) is transformed into a new system as follows: Now, we define the function U(x, t) by Since $$, so we can obtain U(x, t) = ɛE(t)|x|^α−1x. Then the new system can be expressed as in (14) by direct computation.

It is easy to know that U(−x, −t) = U(x, t) by (A1), then we can obtain that the transformation Ψ is a G-invariant diffeomorphism.

Lemma 6. For 0 ≤ k + m ≤ 4, the following inequalities hold:

• (1)

  |(∂^k/∂ρ^k)l₁(ρ, φ)| ≤ Cρ^{−k+2−γ−(5/2)b},

• (2)

  |(∂^k+m/∂ρ^k∂t^m)l₂(ρ, φ, t)| ≤ Cρ^−k+a,

• (3)

  |(∂^k+m/∂ρ^k∂t^m)l₃(ρ, φ, t)| ≤ Cρ^−k+3−4b,

• (4)

  |(∂^k+m/∂ρ^k∂t^m)l₄(ρ, φ, t)| ≤ Cρ^−k+3−4b,

• (5)

  |(∂^k/∂ρ^k)h₁(ρ, φ)| ≤ Cρ^{−k+1−γ−(5/2)b},

• (6)

  |(∂^k+m/∂ρ^k∂t^m)h₂(ρ, φ, t)| ≤ Cρ^−k+τ,

• (7)

  |(∂^k+m/∂ρ^k∂t^m)h₃(ρ, φ, t)| ≤ Cρ^−k+2−4b,

where γ = βb, a = max (3 − (9/2)b − γ, 1 − b), and τ = max (3 − 6b, −b).

Proof. (1) It is easy to know that (∂^k/∂ρ^k)l₁(ρ, φ) is a sum of terms of the form

where 0 ≤ i₁, i₂, i₃ ≤ k. Meanwhile, $$ is a sum terms of the form Hence, we obtain by the assumptions on f(x) and the definitions of x(ρ, φ) and y(ρ, φ).

(2) From the expression of l₂(ρ, φ, t), we have

We can find that

where a = max (3 − 9b/2 − γ, 1 − b).

(3) From the expression of l₃(ρ, φ, t), we have

(4) From the expression of l₄(ρ, φ, t), we can obtain that

(5) From the definition of h₁(ρ, φ), we have

(6) From the definition of h₂(ρ, φ, t), we can obtain

Hence, we can know that where τ = max (3 − 6b, −b).

(7) From the expression of h₃(ρ, φ, t), we have

Lemma 7. There exists a G-invariant diffeomorphism Ψ₁:

such that $$ for some I₋ < I₀ < I₊. Under this transformation, (24) is transformed into the system where with

Proof. Define a transformation Φ₁ by

By we get Let $$, φ = θ. The system (24) is transformed into (41).

Lemma 8. For I large enough, the following conclusions hold:

• (i)

  |∂^kU₁(I, θ)/∂I^k| ≤ CI^{−k+1−γ−b/2},

• (ii)

  U₁(I, −θ) = U₁(I, θ).

Proof. In view of

we obtain By |(∂^k/∂ρ^k)V₁(ρ, φ)| ≤ Cρ^{−k+1−γ−b/2}, we have |(∂/∂ρ)V₁(ρ, φ)| ≤ Cρ^−γ−b/2 ≤ 1/2 for ρ large enough. Hence, U₁ is uniquely determined by the contraction mapping principle. Moreover, $$, for some I₀ > 0, as a consequence of the implicit function theorem and

Above all, if k = 1, from (47) and (49), we get

We note that

and the right side hand is sum of the term where 1 ≤ s ≤ k, k₁ + ⋯+k_s = k, k_i ≥ 1  (for  1 ≤ i ≤ s). The highest order term in U₁ is the one with s = 1, namely, (∂V₁/∂ρ)·(∂^k(I + U₁)/∂I^k). We move the part (∂V₁/∂ρ)·(∂^kU₁/∂I^k) to the left hand side of (52). Since |(∂/∂ρ)V₁(ρ, φ)| ≤ 1/2 for ρ large enough, this also provides immediately a bound on ∂^kU₁(I, θ)/∂I^k. The rest part |(∂V₁/∂ρ)·(∂^kI/∂I^k)| ≤ CI^{−k+1−γ−b/2}.

Now, we proceed inductively by assuming that for j ≤ k − 1 the estimates

hold and we wish to conclude that the same estimate holds for j = k.

Indeed, if s ≥ 2, we have

by This proves (i) of Lemma 8.

Now we check (ii). In fact, since

we have From (47), we have |(∂/∂ρ)V₁(ρ, φ)| ≤ 1/2 for I ≥ I₀ sufficiently large and therefore we obtain U₁(I, θ) = U₁(I, −θ).

Lemma 9. For 0 ≤ k + m ≤ 4, the following inequalities hold:

• (1)

  $$,

• (2)

  $$,

• (3)

  $$,

• (4)

  $$,

• (5)

  $$,

• (6)

  $$,

• (7)

  $$.

Proof. (1) From the estimates (1) and (5) of Lemmas 6 and 8, it follows that

(2) Since we can prove that Their proofs are similar to the proofs in (1).

Next, we check the last part of   $$. We get

by the estimate in Lemma 6 and the definition of a.

(3) It is clearly by (3) in Lemma 6.

(4) It is clearly by (4) in Lemmas 6 and 8.

(5) We have that

From the last inequalities and (5) in Lemma 6, we obtain (6) Since we just have to prove that In fact, so we have proved (6).

(7) We have

by (7) in Lemma 6.

Lemma 10. The following inequalities hold:

• (1)

  cI^2b−1≤|η₀(I)| ≤ CI^2b−1,

• (2)

  |(∂^k/∂I^k)η₁(I, θ)| ≤ CI^{−k−1−γ+3b/2},

• (3)

  |(∂^k+m/∂I^k∂t^m)η₂(I, θ, t)| ≤ CI^{−k+τ+4b−2},

• (4)

  |(∂^k+m/∂I^k∂t^m)η₃(I, θ, t)| ≤ CI^−k,

• (5)

  |(∂^k+m/∂I^k∂t^m)ξ₁(I, θ, t)| ≤ CI^{−k+a+2b−1},

• (6)

  |(∂^k+m/∂I^k∂t^m)ξ₂(I, θ, t)| ≤ CI^−k+2−2b,

• (7)

  |(∂^k+m/∂I^k∂t^m)ξ₃(I, θ, t)| ≤ CI^−k+2−2b, for 0 ≤ k + m ≤ 4.

Proof. (1) It is clear.

(2) Note that 1 − 2b > 1 − γ − 2b/5, and

it follows that as I ≫ 1.

Moreover, we also have

So From (72) and (74), it is easy to see that (3) We have By (72), 1 − 2b > τ (τ = max (3 − 6b, −b)) and 1 − 2b > 2 − 4b, we have for I ≫ 1.

Let $$. We find that

so

When m = 0, the proof of (3) is similar to the proof of (2).

When m > 0, then

(4) The proof of (4) is similar to the proof of (3).

(5) Let η₀(I) + η₁(I, θ) + η₂(I, θ, t) + ɛη₃(I, θ, t) = η(I, θ, t). By using the estimates on the functions $$ and η_j  (j = 0,1, 2,3), it follows that

when m ≠ 0.

When m = 0, then

by a > 2 − γ − 5b/2.

(6) By using the estimates on the functions $$ and η_i  (i = 0,1, 2,3), it follows that

(7) By using the estimates on the functions $$ and η_i  (i = 0,1, 2,3), it follows that

Lemma 11. For 0 ≤ k + m ≤ 4 and r ≫ 1, the following inequalities hold:

• (1)

  |(∂^k/∂r^k)F₀(r, θ)| ≤ Cr^{−k−(1 + γ − 3b/2)/(2b−1)},

• (2)

  |(∂^k+m/∂r^k∂t^m)F₁(r, θ, t)| ≤ C(r^{−k+(τ + 4b − 2)/(2b−1)} + ɛr^−k),

• (3)

  |(∂^k+m/∂r^k∂t^m)F₂(r, θ, t)| ≤ C(r^{−k+(a + 4b − 3)/(2b−1)} + ɛr^−k),

• (4)

  |(∂^k+m/∂r^k∂t^m)F₃(r, θ, t)| ≤ Cɛr^−k.

Proof. Above all, we know that r = η₀(I) = T₀I^2b−1, so we can get I = ((1/T₀)r) ^1/(2b−1). Then we have

(1) We have that

(2) We have that

(3) We have that

(4) We have that

Lemma 12. The time 1 map Φ¹ of the flow Φ^θ of the system (86) is of the form

where $$. And there exists a μ₀ > 0 such that, for 0 ≤ k + m ≤ 4, sufficiently large r and sufficiently small ɛ, hold. Moreover, the map Φ¹ is reversible with respect to the involution G : (t, r)↦(−t, r).

Proof. Since

then we get $$ is bounded.

Let αT₀ | S(υT₀)|^α−1C(υT₀) = S₁(υ). Set (r(θ), t(θ)) = Φ^θ(r, t) with Φ⁰ = id for the flow:

Since where X denotes the vector field of the system (86), we have which is equivalent to the following equations for D₁ and D₂:

Let D(r, t, θ) = (D₁(r, t, θ), D₂(r, t, θ)), $$. Define ∥D∥ = :|D₁ | /3 + 2 | D₁ | /3, and T(D) = :(T₁(D), T₂(D)), where

Next, we will prove that T is a contraction map. From the definition of T(D), we have by Lemma 11 and the boundedness of $$. Then we have by the definition of the norm ∥·∥.

Using the contraction principle, one verifies easily that for r ≥ r₀, (98) has a unique solution in the space {|D₁ | ≤ 1, |D₂ | ≤ 1}. Moreover, D₁ and D₂ are smooth.

Next, we will estimate Q₁(r, t) and Q₂(r, t) as follows:

In order to prove (93), we just need to prove that hold for k + m ≤ 4.

(1) When k + m = 0,

where μ₀ = min ((1 + γ − 3b/2)/(2b − 1), (2 − 4b − τ)/(2b − 1), (3 − 4b − a)/(2b − 1)).

(2) When m = 0 and k ≠ 0, we check the case when k = 1 firstly

Hence, Now, we proceed inductively by assuming that for j < k − 1 the estimates hold and we wish to conclude that the same estimate holds for j = k where s + ν ≤ 2. Hence,

(3) We can prove that

similarly to (2) when m ≠ 0.

(4) we have that

Hence,

(5) We can prove (103) similarly to (4) for the left k + m ≤ 4.

Proof of Boundedness. From Theorem 1.1 in [21] we can see Φ¹ possesses a sequence of invariant circles tending to infinity. So, in the original system (13), there exists a corresponding sequence of invariant tori in phase space $$. Then any solution of system (13) is bounded because it must stay within one of those tori.