Recurrent Motions in a Piecewise Linear Oscillator

We study the oscillator x¨+n2x+h(x)=p(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ddot{x} + n^2 x + h(x) = p(t)$$\end{document}, where h is a piecewise linear saturation function and p is a continuous 2π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\pi $$\end{document}-periodic forcing. It is shown that there is recurrence if and only if p satisfies the Lazer–Leach condition. This condition relates the n-th Fourier coefficient of p(t) with the maximum of h and was first introduced to characterize the existence of periodic solutions.


Introduction
For n ∈ N = {1, 2, . . .}, consider the forced oscillator model where h, p ∈ C(R) are bounded and p is 2π-periodic. The n-th Fourier coefficient of p is given by:p In the linear case h = 0, it is a well-established fact that solutions of (1.1) are 2π-periodic (and hence bounded) ifp n = 0, and otherwise unbounded and nonrecurrent due to resonance phenomena. In [10], Lazer  Later, it was proven in [2] that the negation of this inequality implies that all solutions x(t) satisfy lim |t|→∞ x(t) 2 +ẋ(t) 2 = ∞. (1.3) See also [17] for a previous related work. Results with respect to boundedness were obtained in [14]. There, it was shown that the same condition (1.2) leads to the boundedness of all solutions in the special case where h = h L with L > 0 is the piecewise linear function given by provided that p ∈ C 5 (R) is 2π-periodic (see also [6] for a related result with a discontinuous h). Moreover, this led to the insight that almost every solution x(t) is Poisson stable, which can be understood as follows in the context of 2π-periodic systems. There is a sequence of integers {σ n } n∈Z with σ n → ±∞ as n → ±∞ such that uniformly with respect to t ∈ [0, 2π]. In the same year, Liu obtained a similar result for general h ∈ C 6 (R) such that lim |x|→∞ x k h (k) (x) = 0 for 1 ≤ k ≤ 6, if p ∈ C 7 (R) is 2π-periodic [9]. Recent results for more general nonlinearities can be found in [15,18]. All latter results were obtained by using variants of Moser's small twist theorem. However, the application of any such invariant curve theorem requires a considerable degree of smoothness of either h(x) or p(t). It is an interesting question if any of the nice features of solutions survive if only mild regularity assumptions are made. In the present paper, we investigate this question for the piecewise linear equation: x + n 2 x + h 1 (x) = p(t).
(1.4) By rescalingẍ + n 2 x + h L (x) = p(t), one obtains the functionh L (x) = sign(x) for |x| ≥ 1 L andh L (x) = Lx for |x| < 1 L . Since the slope L has basically no effect on the dynamics, we have normalized the equation by setting L = 1. Besides giving a good starting point for more general nonlinearities, such piecewise linear oscillators are also known in the engineering literature. For example, (1.4) can be considered as a model for an oscillator with stops (see [5] and also [14] for the derivation of (1.4)). Our main result is the following. Theorem 1.1. Suppose p ∈ C(R) is 2π-periodic and satisfies the Lazer-Leach condition π|p n | < 2. If x(t) = x(t;x,ṽ) denotes the solution of (1.4) with the initial condition x(0) =x andẋ(0) =ṽ; then, x(t;x,ṽ) is Poisson stable for almost every (x,ṽ) ∈ R 2 .
This theorem is an improvement of Corollary 2.1 in [14]. The statement is also true for the discontinuous limit case.  [6]. Note that one first has to define a proper notion of solutions to (1.5) (see Definition 5.1).
We also refer the reader to [11] for a discussion of chaos in second-order equations with signum nonlinearities. (c) Theorems 1.1 and 1.2 show that the set of initial conditions such that (1.3) holds has measure zero. The authors know of no example exhibiting unbounded orbits, provided the assumptions of these theorems are satisfied.
The proof to Theorem 1.1 starts out similar to those of the boundedness results stated above. A twist mapP : (τ , v) → (τ 1 , v 1 ) on the annulus S 1 × [0, ∞[ is constructed so that its orbits correspond to large amplitude solutions of (1.4). The construction is done in such a way thatP is recurrent if and only if the time-2π map associated with (1.4) is recurrent. The lift ofP has the form where L ∈ C 2 is a positive 2π-periodic function and R, S ∈ C 1 . Our main abstract result (Theorem 3.1) states that any such map is recurrent if R and S satisfy certain bounds. This claim is established by first finding an adiabatic invariant of the system and then applying a refined version of Poincaré's recurrence theorem; a method recently introduced by Dolgopyat [4].
The paper is organized as follows: In Sect. 2, Maharam's recurrence theorem for measure-preserving maps is introduced. In Sect. 3, we find an adiabatic invariant for a family of exact symplectic twist maps. This leads to the proof of Theorem 3.1 stating that this family is recurrent. Section 4 contains the application to Eq. (1.4) and the proof of Theorem 1.1. Finally, the discontinuous equation (1.5) is discussed in the last section.

Recurrence
Let (X, A, μ) be a measure space and introduce the following useful notation. For A, B ∈ A, we write where N is a set of measure zero. Now, consider a map T : X → X which is bi-measurable, that is Such a map T is said to be measure-preserving, if As a consequence, such a measure-preserving transformation satisfies with equality if A ⊂ T (X) mod μ.
Remark 2.1. In the literature, there is no unique way of defining the two properties above. In particular, T is often called measure-preserving, if μ(T −1 (A)) = μ(A) for all A ∈ A. However, the definition in this work was chosen since it seems to be the most natural in the application to mechanical problems and suchlike.
As a simple example, consider the space X = [0, ∞[ equipped with the Lebesgue measure λ and the map T 1 (x) = x+1. Then, T 1 is measure-preserving in the above sense, but the strict inequality λ(T −1 (A)) < λ(A) holds, e.g., for Since T maps X into itself, the iterates T n = T n−1 • T , where T 0 = id, are well-defined for all n ∈ N. We call the map T recurrent, if for every A ∈ A for almost all x ∈ A there is n ∈ N such that T n (x) ∈ A, that is where T −n (A) denotes the pre-image under T n . In other words, the set of points in A not returning to A has measure zero. Since T is measure-preserving, also any (iterated) pre-image of this set has measure zero. Hence, T is even infinitely recurrent, i.e., for almost all x ∈ A there is an increasing sequence In the case of a finite measure-space, the famous Poincaré recurrence theorem characterizes the relation between measure-preserving and recurrent maps. We will use it in the following form.

Lemma 2.2.
Let (X, A, μ) be a measure space such that μ(X) < ∞ and suppose T : X → X is measure-preserving. Then, T is recurrent.
Unfortunately, the situation is less clear if the space has infinite measure. However, the statement of the recurrence theorem stays valid if there exists a set M of finite measure which acts as some kind of bottleneck. This is described in the following generalization of Lemma 2.2 due to Maharam, see [12], which also recently got some attention in the context of twist maps by Dolgopyat [3]. Proof. The "time of first return" r(x) = min{k ∈ N : T k (x) ∈ M} is welldefined for almost all x ∈ X by assumption. In particular, it can be shown that there is a set Γ of measure zero such that the induced map S : M\Γ → M given by S(x) = T r(x) (x) is well-defined and satisfies S(M\Γ) ⊂ M\Γ. Moreover, S is measure-preserving, and hence, one can apply the Poincaré recurrence theorem to see that S is also recurrent. Now, let A ∈ A be a measurable set in X and for k ∈ N consider the sets From this, it follows Finally, taking the union over all k ∈ N shows that almost every point in A returns to A.
There are two drawbacks to Lemma 2.3. On the one hand, such a set M does not exist for every recurrent measure-preserving transformation, as already a trivial example like the identity shows. On the other hand, even when it does exist, it can be hard to find. In the following section, we will introduce a class of measure-preserving transformations for which the construction of M can be done explicitly.

Remark 2.4.
If X is σ-finite, the following observation can be made. A measurepreserving map T : X → X is recurrent if and only if there is a covering {X j } j∈N of X and a collection of sets Note that there are several more generalizations to the Poincaré recurrence theorem, and depending on the situation one might choose the appropriate version. For example, if {X j } j∈N is a covering of X with μ(X j ) < ∞ and for every fixed j ∈ N the measure-preserving map T satisfies then T is also recurrent (see [8]). For a more thorough discussion of maps preserving an infinite measure, we refer the reader to [1,3].

Exact Symplectic Twist Maps
We identify the circle S 1 with the quotient space R/2πZ. With a small abuse of notation, C n (S 1 ) denotes the space of n-times continuously differentiable functions F : R → R that are 2π-periodic. Sometimes, we will not differentiate between a mapF : S 1 → S 1 and its lift satisfying In this chapter, we will mostly study maps: and we will use the same convention as above regarding its lift on the universal cover assumef is a C 1 -diffeomorphism with respect to its image. We sayf satisfies the twist condition if where we use the usual abuse of notation dθ 1 = ∂θ1 ∂θ dθ + ∂θ1 ∂v dv. Then,f is called exact symplectic twist map. There are two direct consequences: First, the map is symplectic in the sense that And second, given any Let us also introduce the class F u (m) of continuous functions F : Since we want to analyze the recurrence properties off , we need to make sure that the forward iteratesf n are well-defined for n ∈ N. To this end, let is bi-measurable, sincef is a diffeomorphism and also preserves the measure μ, due to (3.2). Moreover, this restricted version off is a self-map, which is necessary for the application of Lemma 2.3. Note that possibly D = ∅. The so-called escaping set off is given by: and clearly f is non-recurrent on E. Its complement D\E on the other hand can be covered by the measurable sets: Since every orbit starting in B m eventually has to enter the set S 1 × [0, m + 1], Lemma 2.3 can be applied to the restricted mapf : B m → B m . It follows easily thatf is recurrent on D\E. Therefore, proving thatf : D → D is recurrent is equivalent to showing μ(E) = 0. This is the subject of our main abstract result. The strategy of the proof will be to apply Lemma 2.3 with X = E and T =f .
which the lift f is given by for every θ ∈ R and ν = (ν 1 , ν 2 ) with |ν| = 1. Moreover, assume thatf is oneto-one and exact symplectic in the sense that there is a function denotes the escaping set off .
Under the stronger assumptions L ∈ C 6 (S 1 ), R, S ∈ C 5 (M v * ) and for any multi-index ν = (ν 1 , ν 2 ) with |ν| ≤ 5, KAM-theory is applicable and shows the boundedness of all orbits. See [14] for a suitable invariant curve theorem and its application to a map of the type under consideration.
In the proof, we will need the following auxiliary lemma, which is basically a variant of Lemma 4.1 in [7].

A Piecewise Linear Oscillator
In this section, we prove our main result, Theorem 1.1. As indicated in the introduction, we start by constructing a twist map suitable for the application of Theorem 3.1, such that its orbits correspond to large-amplitude solutions ofẍ In a second step, we then show that the recurrence of this twist map implies Poisson stability of almost every solution.
To this end, suppose x is a solution of (4.1) such that there are τ ∈ R and v > 0 with x(τ ) = 0 andẋ(τ ) = v. Then, x is also a solution of the integral equation sin n(t − s) n ds, (4.2) and the derivative is given bẏ Given any time span T > 0, it follows from these formulas that x(t)/v is arbitrary close to (sin n(t − τ ))/n in C 2 [τ, τ + T ] for large values of v. In particular, one can find v * > 0 with the following property. If v > v * , then x(t) has 2n consecutive non-degenerate zeros τ = τ 0 < τ 1 < · · · < τ 2n = τ and crosses the line x = (−1) i twice in each interval (τ i , τ i+1 ). We denote these crossings by τ * i < * τ i+1 and write v i =ẋ(τ i ), v * i =ẋ(τ * i ), * v i =ẋ( * τ i ), for the corresponding velocities. For i = 0, . . . , 2n − 1, each of the three maps , v i+1 ) can be described in terms of a forced linear oscillator. The arguments in Proposition 2.2 and Proposition 2.3 of [13] show that these maps are of class C 1 and exact symplectic in the sense of (3.1). Since the induced function can be decomposed into 6n such maps, alsoP ∈ C 1 (M v * ) is exact symplectic. The mapP is one-to-one due to the unique solvability of the corresponding initial value problem. Following the computations in Section 7 of [14], it can be seen that for p ∈ C(S 1 ) the associated lift P : and R 1 ∈ F 1 (2), R 2 ∈ F 0 (1). Here, F k (r) denotes the space of functions for every multi-index ν = (ν 1 , ν 2 ) with |ν| ≤ k. We define F k u (r) to be the subspace F k (r)∩F u (r), i.e., all F ∈ F k (r) such that {v r F (·, v)} v≥v * converges uniformly as v → ∞. Throughout the computations in [14], one can in fact replace the space F k (r) by F k u (r) with some obvious adjustments. This leads to the conclusion that R 1 ∈ F 1 u (2) and R 2 ∈ F u (1). The Poincaré map of the discontinuous oscillator discussed in the next section has an expansion of the same form. This is shown in full detail in [16]. Finally, note that for L 1 ∈ C 2 (S 1 ) we have L 1 = nL 2 and also the condition L 1 > 0 is guaranteed by (1.2). In total,P satisfies all assumptions of Theorem 3.1 and therefore the escaping set Going back to the question of Poisson stability, we denote by x(t) = x(t;x,ṽ) the solution of (4.1) satisfying the initial condition x(0) =x anḋ x(0) =ṽ. Thus, the time-2π map of (4.1) is given by It can be shown that Π preserves the 2-dimensional Lebesgue measure λ. We now prove that it is also recurrent. To this end, consider x(t) = x(t;x,ṽ) for some (x,ṽ) ∈ R 2 . The solution of the unperturbed linear systemz + n 2 z = 0 satisfying the same initial condition z(0) =x,ż(0) =ṽ is given by z(t) = r sin n(t−τ ) n for someτ ∈ R andr = √ n 2x2 +ṽ 2 . Furthermore, x(t) also solves the integral equation Again, x(t)/r is close to (sin n(t −τ ))/n in C 2 [0, 4π] for large values ofr. Let r(t) = n 2 x(t) 2 +ẋ(t) 2 . Then, one can infer from (4.5) that there is a constant C p > 0 (depending on p ∞ ) such that Thus, ifr = r(0) > v * +C p , then r(t) > v * holds for all t ∈ [0, 4π]. In particular, there is a unique first τ ≥ 0 such that x(τ ) = 0 and v =ẋ(τ ) > v * . Let S be the induced map S is a diffeomorphism with respect to its image and the inverse map can be obtained by plugging t = 0 into (4.2) and (4.3). For a given solution x(t) = 710 R. Ortega, H. Schliessauf Ann. Henri Poincaré x(t;x,ṽ) define r n = r(2πj) for j ∈ N. Then, the escaping set E Π of the map Π is given by It can be shown by the same argument as in Sect. 3 that the restricted map Π : R 2 \E Π → R 2 \E Π is recurrent. Thus, it remains to show that λ(E Π ) = 0. Suppose (x,ṽ) ∈ E Π . In view of (4.6), this means lim t→∞ r(t) = ∞. Let m ∈ N be such that Π j (x,ṽ) ∈ R 2 \E for all j ≥ m. Moreover, set (τ 0 , v 0 ) = S (Π m (x,ṽ)) and denote its corresponding orbit by (τ j , v j ) = P j (τ 0 , v 0 ). Then, clearly lim j→∞ v j = lim j→∞ r(τ j ) = ∞ so that ι (S (Π m (x,ṽ))) ∈ E P , where ι : v). This leads to the inclusion which in turn implies λ(E Π ) = 0. In summary, we have shown that Π is recurrent. Due to the symmetry of the problem, the same is true for the inverse map Π −1 . Now, the Poisson stability of almost every solution x(t;x,ṽ) follows from the fact that the corresponding flow is Lipschitz-continuous on R 2 .

The Discontinuous Case
Consider the piecewise linear oscillator where p ∈ C(S 1 ). Let N = {t ∈ R : |p(t)| = 1} and suppose the set ∂N of its boundary points is countable. The goal of this section is to proof Theorem 1.2, that is to show that almost every solution of (5.1) is Poisson stable. But first we have to give the following Definition 5.1. We say a function x ∈ C 1 (I) with I =]α, β[⊂ R is a solution of (5.1) if it satisfies the following conditions: Moreover, we say a solution is global if I = R.
We have shown that there is a set Γ of measure zero such that all initial condition in R 2 \Γ lead to global solutions of (5.1). In particular, the time-2π map Π : R 2 \Γ → R 2 \Γ is well-defined. We will demonstrate that this map is also measure-preserving. To this end, we keep the notation introduced in the proof of Lemma 5.2. Given (x,ṽ) ∈ (R ± × R)\(Ω r ∪ Γ), letS ± (x,ṽ) = (τ 0 ,v 0 ), then there is an infinite series of non-degenerate consecutive zeros (τ j ) j∈N0 of x(t;x,ṽ). Moreover, letτ 0 = ∞ if (x,ṽ) ∈ Ω r . We define the sets A ± j = (x,ṽ) ∈ (R ± × R)\Γ : j = min{i ∈ N 0 :τ i ≥ 2π} , where the index j counts the number of zeros in the interval [0, 2π]. Clearly, . Moreover, the sets Ω r and A ± j are measurable. For Ω r , this follows from the fact that (R ± × R)\(Ω r ∪Σ ± ) is open, so that Ω r differs from a Borel set only by a set of measure zero. In the case of A ± j with j ∈ N, consider the maps g j (x,ṽ) = (τ j ,v j ). These maps are well-defined and continuous almost everywhere on R 2 \Ω r and hence measurable. Thus also is measurable. The argument for j = 0 is similar. On A ± 0 the map Π is just the time-2π map of a linear oscillator and thus it preserves the 2-dimensional Lebesgue measure λ. For A ± j with j ∈ N we again consider the maps g j . Without loss of generality, let (x,ṽ) ∈ A + j , where j = 2k with k ∈ N. Then (τ j ,v j ) = (S + • S − ) k (S + (x,ṽ)).
S + restricted to (R + × R)\(Ω r ∪Σ + ) is a diffeomorphism with respect to its image and the inverse is given byS −1 + (τ ,v) = (y + (0;τ ,v),ẏ + (0;τ ,v)). Considering formula (5.2), one easily derives det DS −1 + (τ ,v) = −v for the Jacobian determinant. This implies that we have λ(B) = μ(S + (B)) for any measurable set B ⊂ (R + × R)\(Ω r ∪Σ + ), where μ = v dτ ⊗ dv. Furthermore, the maps S ± are exact symplectic in the sense of (3.1) on the relevant domain and therefore preserve the measure μ. It follows that the sets have measure zero. Finally, note that for (x,ṽ) ∈ A + j \B + j we have Π(x,ṽ) =S −1 + (τ j − 2π,v j ). In view of the argument above, the latter identity shows that Π preserves the two-dimensional Lebesgue measure also on A ± j with j ≥ 1 and hence on all of R 2 \Γ. Analogously to the continuous case, it now follows that Π is recurrent and that almost every solution x(t;x,ṽ) is Poisson stable.