Boundedness of solutions for a class of impact oscillators with time-denpendent polynomial potentials

In this paper, we consider the boundedness of solutions for a class of impact oscillators with time dependent polynomial potentials,    ẍ+ x + 2n ∑ i=0 pi(t)x i = 0, for x(t) > 0, x(t) ≥ 0, ẋ(t+0 ) = −ẋ(t − 0 ), if x(t0) = 0, where n ∈ N, pi(t+ 1) = pi(t) and pi(t) ∈ C(R/Z).


Introduction and main result
In Ref. [1], Dieckerhoff and Zehnder proved the boundedness of solutions for a time-dependent nonlinear differential equation: with p i (t + 1) = p i (t) and p i (t) ∈ C ∞ and asked whether or not the boundedness phenomenon is related to the smoothness of p i (t). Then Laederich and Levi [2] relaxed the smoothness requirement of p i (t) to C 5+ε with ε > 0, Yuan [3,4] relaxed the smoothness requirement of p i (t). to C 2 , etc.
The system is included in the system of impact oscillators given by which serve as models of dynamical systems with discontinuities [5]. From the viewpoint of mechanics, this equation models the motion of particle attached to a nonlinear spring and bouncing elastically against the fixed barrier. The system of this form also relate to the research of the Fermi accelerator [6], dual billiards [7] and celestial mechanics [8].
The nonsmoothness caused by the impact limit the applications of many powerful mathematical tools. However there are also many interesting papers on the impact oscillators, see [9]- [16] and their references.
To overcome the problem of nonsmoothness, Qian and Torres [9] and Qian and Sun [10] used success mapping in their papers and proved the existence of invariant tori by a variant version of Moser's small twist theorem by Ortega [17]. In Zharnitsky [11], Wang [12,13], they exchanged the position of angle and time variables of the Hamiltonian function which only C 0 in angle variable, and finally obtained the boundedness of solutions by Moser's small twist theorem.
In [12], Z.Wang and Y.Wang studied the following impact oscillator: where n ∈ N + , p(t) is a C 5 periodic function with 1, and the term x 2n+1 models a hard spring. They proved the Lagrangian stability for system (1.4) by exchanging the roles of angle and time variables and using Moser's small twist theorem. In this paper we will extend (1.4) to the case of time-periodic polynomial potentials with constant leading coefficient as (1.2) and use similar method to prove the boundedness of solutions for (1.2). More exactly, we obtain the following conclusion: Theorem 1 Every solution of (1.2) is bounded, i.e., x(t) exits for t ∈ R and sup(|x(t)| + |ẋ(t)|) < +∞.
The idea of proving the boundedness of solutions of (1.2) is as follows. First of all, we describe (1.2) by a Hamiltonian function H 3 (ρ, φ, t) (see (2.6)) in action-angle variables defined on the whole space R + × S 1 × S 1 by means of transformation theory. Due to the existence of impact, H 3 (ρ, φ, t) is only continuous in φ, but is C 5 smooth in ρ and t. Exchanging the roles of the variables φ and t outside of a large disk D r = {(ρ, φ), ρ < r} in (ρ, φ)-plane, (2.9) is transformed into a perturbation of an integrable Hamiltonian system H 4 (I, θ, τ ) (see (3.3)) which is sufficiently smooth in I and θ. The Poincaré mapping with respect to the new time τ is closed to a so-called twist mapping in (R + × S 1 ) \ D r , and satisfies Moser's invariant curve theorem after a scaling transformation. Then Moser's theorem guarantees the existence of arbitrarily large invariant curves diffeomorphic to υ = const. over υ = 0 in (υ, θ)-plane. Go back to the equivalent system (2.1), every such curve is a base of a time-periodic and flow-invariant cylinder in the extended phase space (x, y, t) ∈ R + ×R×R, which confines the solutions in the interior and which leads to a bound of these solutions if the uniqueness of initial value problem holds. Note that, from the proof below, system (2.1) is equivalent to a smooth system (2.9) in (R + × S 1 ) \ D r which has uniqueness. Hence system (2.1) has uniqueness.

Remark 1.1
The time-dependence of the stiffness coefficient p i (t) of the spring can be produced by periodic changes of the temperature or other physical variables. Following Laederich and Levi [2] and Z.Wang and Y. Wang [12], we assume p i (t) ∈ C 5 . Remark 1.2 Compared with (1.4), the potentials of (1.1) become more complicated which will make more difficulties in the proof of Theorem 1.

Remark 1.3
The main difficulty in this paper is also the nonsmoothness caused by the impact. Similar to [12], we will also exchange the roles of angle and time variables of the Hamiltonian function H 3 (ρ, φ, t) (see below), and then obtain the boundedenes of solutions by Moser's small twist theorem.
The rest of this paper is organized as follows. In section 2, we will make some action-angle transformations to transform the system into an equivalent Hamiltonian H 4 (I, θ, τ ) (see below) which is defined in the whole plane R + × S 1 × S 1 and C 5 in I, θ, but only C 0 in τ . In section 3, we do more canonical transformations such that the Poincaré mapping of the new system is closed to the twist mapping and then use Moser's theorem to complete the proof of Theorem 1. Finally, we will give the proofs of some technical lemmas in section 4 and draw a conclusion in section 5.
Throughout this paper, we denote

Action-angle variables
In this section, we will make some action-angle transformations to transform the system into an equivalent Hamiltonian H 4 (I, θ, τ ) (see below) which is defined in the whole plane R + ×S 1 ×S 1 and C 5 in I, θ, but only C 0 in τ .
Without impact, Eq.(1.2) is just Eq.(1.1) which is of second order and equivalent to the one order system (2.1) is a system defined in the whole phase plane XOY , which has a Hamiltonian function with the symplectic form dx ∧ dy.
From Eq.(2.3), we can find that C(t) and S(t) satisfy: (1) C(t), S(t) ∈ C ∞ (R), and C(t Now we can make coordinate transformation by the mapping Ψ 1 : 3) and has T 0 as its minimal period, which concludes that Ψ 1 is one to one and onto. This proves the claim.
Under Ψ 1 , Hamiltonian H is transformed into which is C 5 in λ, t and C 0 in ϑ.
Define a sympletic transformation Ψ 2 : (φ, ρ) → (ϑ, λ), given by ϑ = 1 2 , λ = 2ρ. And H 1 is transformed into 2n+2 . To get a system which is equivalent to (1.2), we could define a new Hamiltonian function by where [φ] denotes the largest integer less than or equal to φ. And the corresponding Hamiltonian system is (2.7) Obviously, H 3 is periodic in φ with 1, and C 5 in ρ, φ when φ / ∈ Z, but only continuous in φ when φ ∈ Z. In fact, H 3 is right continuous when φ ∈ Z.
Remark 2.1 The systems mentioned above are equivalent, see [12] and [13], we omit the proof.

The proof of Theorem 1
Now we are concerned with the Hamiltonian system (2.7) with Hamiltonian function H 3 given by (2.6). To cope with the nonsmoothness in φ and use Moser's small twist theorem to prove the Lagrangian stability, we will exchange the positions of variables (ρ, φ) and (H 3 , t) below. This trick has been used in [18,19,20].
From (2.6), we have that lim if ρ ≥ 1. By the implicit theorem, we know that there is a function R = R(H 3 , t, φ) such that Then we make another transformation Ψ 3 : (ρ, φ, t) → (I, θ, τ ) given by This transformation also leads to a Hamiltonian system with new Hamiltonian function which is C 5 in θ, C ∞ in I and C 0 in τ.  To be convenient for readers, the proof is given in section 4. In order to use the Moser's small twist theorem, we introduce a new variable υ and a small positive ε by the formula I = υ ε , where υ ∈ [1, 2], ε > 0 and ε → 0, when I → ∞. Then the system (3.4) is equivalent to the following system
Proof. From (4.10), we have Using Leibnitz's rule, we can easily prove from (4.7) and the definition of p i (τ ). Finally, from (4.16), we have Proof. From (4.11), we have Using Leibnitz's rule, we have Then we can prove that from (4.7) and (4.15).

Conclusion
In this paper, we study the Lagrangian stability for a class of impact oscillators with time dependent polynomial potentials. By exchanging the roles of angle and time, we overcome the nonsmoothness on angle variable caused by the existence of impact. Then we obtain an equivalent Hamiltonian which is sufficiently smooth in new angle variable by some canonical transformations. Thus by direct application of Moser's twist theorem, we obtain the existence of invariant curves for the Poincaré mapping of the equations, which implies the Lagrangian stability.