PREVALENCE OF STABLE PERIODIC SOLUTIONS IN THE FORCED RELATIVISTIC PENDULUM EQUATION

. We study the prevalence of stable periodic solutions of the forced relativistic pendulum equation for external force which guarantees the existence of periodic solutions. We prove the results for a general planar system.

in the literature. Most results are related to the existence of its periodic solutions. The common tools include Leray-Schauder degree, the method of upper and lower solutions and variational methods. For example, Brezis and Mawhin proved in [4] that (1) has at least one periodic solution for any µ = 0 and any forcing term f ∈ F by finding a minimum for the corresponding functional. Throughout this paper, we shall use X = C(R/T Z) to denote the set of continuous T -periodic functions and use Obviously, F becomes a Banach space with the norm f ∞ = max t∈R |f (t)|. Later, the second periodic solution of (1) was found by Bereanu and Torres in [3] by using Szulkin's critical point theory [21]. Equation (1) can be seen as a special case of the more general periodic φ-problem where φ is a suitable increasing homeomorphism with φ(0) = 0. It was proved by Cid and Torres in [8] that (2) has at least two periodic solutions not differing by a multiple of ω for any forcing f ∈ F and the continuous function g : [0, T ] × R → R with its primitive G(t, x) satisfying G(t, x) = G(t, x + ω) for all (t, x) ∈ [0, T ] × R.
We refer the reader to [1,2,4,8,10,14,22] for more results on the existence of periodic solutions for equation (1) and the general problem (2). We also refer to [14] for a relatively complete introduction to the global results for the non-relativistic forced pendulum equations. Compared to the existence of periodic solutions, the study of the dynamical behavior of periodic solutions is more difficult but there are also some interesting results. The Lyapunov stability of the equilibrium of (1) without the external force was proved in [5]. The existence and stability of periodic solutions of relativistic singular differential equations were established in [6] based on a known connection between the index of a periodic solution and its stability. In [13], by applying a version of the Poincaré-Birkhoff theorem due to Franks, Marò proved the existence of at least two geometrically different periodic solution with winding number N for (1). The instability of a solution and the existence of twist dynamics are also given in [13]. The stability of periodic solutions for the more general problem (2) was studied in [8].
In this paper, we continue to study the dynamical behavior of periodic solutions for equation (1). To be precise, we shall focus on the prevalence of its stable periodic solutions. A prevalent set can be seen as the analogue of a set of full measure in infinite dimension. For more information on prevalence, we refer to the paper by Ott and York [20]. We are motivated by several recent works [7,16,17,18]. In [16,17], Ortega proved that the forced non-relativistic pendulum equations has at least one stable T -periodic solution for almost every forcing f ∈ F when 0 < β ≤ (π/T ) 2 , which is equivalent to say that the set {f ∈ F : (3) has a stable T -periodic solution} is prevalent in F. Two examples were constructed in [18] to explain that the upper bound (π/T ) 2 for β is optimal. The prevalence of stable periodic solutions of the dissipative case and the conservative Duffing equations has been studied by the first two authors in [7]. Note that equation (1) is equivalent to the planar system where φ −1 (y) = y 1 + y 2 .
In order to make our results more applicable, we prove the prevalence of stable periodic solutions for the following general planar system where f ∈ F and g 1 , g 2 satisfy some additional conditions (see Section 3). Under reasonable conditions, we shall prove that S is prevalent in E, where E = {f ∈ F : (5) has at least one T -periodic solution}, and S = {f ∈ F : (5) has at least one stable T -periodic solution}. As an application, we are able to show that if µ is less than some constant, (1) has at least one stable T -periodic solution for almost every forcing f ∈ F.
The paper is organized as follows. In Section 2, we present some preliminary results and some basic facts on planar linear Hamiltonian system. In Sections 3, we state and prove the main results of this paper.

Preliminaries.
2.1. Two prevalent results. Let E be a separable Banach space of infinite dimension. A subset N of E is Haar-null if there exists a Borel set B and a Borel measure µ on E such that The above definition of Haar-null set is the same as in [16]. We refer to [20] for more information on the notion of Haar-null sets. A subset of E is prevalent if its complement is Haar-null. We remark that Lemma 2.1 and Lemma 2.2 below are proved in [16,17] under a very slightly different definition of Haar-null set but the results still hold.
Given a vector e ∈ E with norm e , the open ball of radius r centered at e is denoted by B(e, r) = {f ∈ E : f − e < r}. The norm of a vector ξ in the space of finite dimension R d will be denoted by |ξ|. We consider a map h : Lemma 2.1. [16,Theorem 2] Assume that the following conditions hold: for each e ∈ U α , α ∈ A. Let C be a Borel subset of R with zero measure. Then the set is prevalent in E.

2.2.
Discriminant of planar linear system. Consider the planar linear Hamiltonian system y = JB(t)y, where with α(t), β(t), γ(t) being smooth T -periodic functions. It was proved in [11] that there exists a smooth function t → ψ(t) such that the change of variables will transform (6) into a simpler linear Hamiltonian system and the function ψ(t) was constructed explicitly in [11].
is also T -periodic. After these spatial changes, without loss of generality, we only consider the following linear systems where a, b ∈ C(R/T Z) and b(t) > 0 for all t ∈ R. We remark that the condition on the sign of b will be used in the proof of Lemma 2.3 below. Moreover, such a condition holds in many applied models, for example, the forced relativistic pendulum equation (1). The Poincaré matrix of (7) is where (φ 1 (t), ψ 1 (t)) and (φ 2 (t), ψ 2 (t)) are real-valued solutions of (7) satisfying φ 1 (0) = 1, ψ 1 (0) = 0 and φ 2 (0) = 0, ψ 2 (0) = 1, respectively. The eigenvalues λ 1,2 of M T are called the Floquet multipliers of (7). Obviously λ 1 · λ 2 = 1. We can distinguish (7) in the following three cases: Proof. It is sufficient to prove that the Wronskian W = W (t) of functions φ 2 1 , φ 1 φ 2 and φ 2 2 never vanishes. A simple computation shows that In particular, First we assume that the functions a(t) and b(t) are of class C 1 . The three functions z ij = φ i φ j , 1 ≤ i ≤ j ≤ 2, are the solutions of the third order linear equation Liouville's formula can be applied to this equation and it follows that the Wronskian W (t) = 2b 3 (t) never vanishes because b(t) > 0 for all t ∈ R. If the functions a(t) and b(t) are only continuous, then they can be approximated in the uniform sense by C 1 functions a n (t) and b n (t). By continuous dependence, we are able to show that the corresponding Wronskians W n of the system The discriminant of (7) is defined as It is easy to verify that the equation (7) is elliptic if |∆| < 2, hyperbolic if |∆| > 2 and parabolic if |∆| = 2. See [12] for some properties of the discriminant. The existence of non-trivial T -periodic solutions for (7) is equivalent to λ 1 = λ 2 = 1 or ∆ = 2. In the elliptic case the monodromy matrix M T is similar to a rotation. More precisely, there exist a number θ ∈ (0, 2π), θ = π, and 2 × 2 matrix P with det P =1 such that M T = P R[θ]P −1 with

FENG WANG, JIFENG CHU AND ZAITAO LIANG
In this case the discriminant is ∆ = 2 cos θ.
The discriminant of the linear system (7) can be interpreted as a functional depending on the coefficients a(t) and b(t). More precisely, we consider the functional ] of the linear system (7) has continuous partial derivatives and has exactly two critical values ∆ = ±2.
Proof. Given ξ, η ∈ C(R/T Z). The partial derivatives along ξ, η are given by Using the method of variation of constants, we deduce that where . From the above facts, using standard analysis, we can readily show that the discriminant ∆[a, b] has continuous partial derivatives.
Next we prove that all values in R\{−2, 2} are regular. Assume that a, b ∈ C(R/T Z) are such that ∆[a, b] = ±2. Going back to the formula (8), we notice that we must prove that both χ a and χ b are not identically zero. Assume by contradiction that χ a (s) = 0 or χ b (s) = 0 for any s ∈ R. Here we state only when χ a (s) = 0 because similar contradiction is obtained when χ b (s) = 0. In particular, χ a (0) = 0 and χ a (0) = 0. This implies that φ 2 (T ) = 0 and φ 1 (T ) − ψ 2 (T ) = 0 and so the function χ a takes the simplified form χ a (s) = ψ 1 (T )φ 2 2 (s). Now we conclude that ψ 1 (T ) = 0. Then the monodromy matrix M T takes the form 2.3. Ellipticity and index. We say that a T -periodic solution (x, y) of the system (5) is non-degenerate if its variational system has only the trivial T -periodic solution. Let us consider the following two sets  Theorem 2.6. Let (x, y) be a non-degenerate T -periodic solution of the system (5). In addition assume that Then γ(x, y) = 1 (resp. γ(x, y) = −1) if and only if (x, y) is elliptic (resp. hyperbolic).
Proof. Denote by λ 1 , λ 2 (|λ 1 | ≥ |λ 2 |) the Floquet multipliers of (9). By Lemma 2.5 the multipliers are either conjugate complex numbers or positive real numbers. The elliptic case λ 1 = λ 2 holds if and only if The hyperbolic case 0 < λ 1 < 1 < λ 2 holds if and only if The parabolic case is excluded because 1 cannot be a Floquet multipler since (x, y) is non-degenerate. 3.1. Prevalence of non-degenerate periodic solutions. let us assume that (σ 1 ) g 1 , g 2 ∈ C 1 are bounded, g 1 (0) = 0, g 1 (y) > 0 for all y ∈ R; g 2 is not locally trivial and g 2 (s + 2π) = g 2 (s) for each s ∈ R. Here g 2 is not locally trivial means that for every open and non-empty interval I ⊂ R there exists some x ∈ I such that g 2 (x) = 0.
Theorem 3.1. Assume that (σ 1 ) holds. Then the set E 1 is prevalent in E.

with
The theorem on continuous dependence can be applied to (11)- (12). It implies that the map (t; ξ, f ) ∈ R × R 2 × F → Φ(t; ξ, f ) ∈ R 2×2 is continuous. In particular it is uniformly continuous on compact sets. This implies that if ξ n → ξ and f n − f ∞ → 0, then We also consider the map and observe that the zeros of h(·, f ) are the initial conditions producing T -periodic solutions. This map is continuous and the theorem on differentiability with respect to initial conditions and parameters implies that it is Gateaux differentiable with partial derivatives given by The formula of variation of constants implies that where The continuity of Φ and the formulas (13) and (14) can be employed to prove the continuity of the partial derivatives of h. In particular the continuity of is a consequence of the estimate The previous discussions show that h is Fréchet differentiable and [(C 1 ), Lemma 2.1] holds. The condition (σ 1 ) implies that x(t; T (ξ), f ) = x(t; ξ, f ) + 2π. Then we can deduce that h satisfies the periodicity condition. Moreover, given (ξ, f ) ∈ Z we know that (x(t; ξ, f ), y(t; ξ, f )) is a T -periodic solution of (5). Hence The periodicity of (x(t; ξ, f ), y(t; ξ, f )) and the equation (5) imply that for some τ , we have y(τ ; ξ, f ) = 0. Then The condition (C 3 ) holds with B = ( g 2 ∞ + f ∞ )T .

Prevalence of elliptic periodic solutions.
Lemma 3.2. Let (x, y) be a T -periodic solution of (5). Then Proof. If (x, y) is a T -periodic solution, we integrate the first equation over a period and obtain  Proof. The planar system (5) has an associated Poincaré map defined as where (x(t; ξ), y(t; ξ)) is the solution of (5) satisfying x(0) = ξ 1 , y(0) = ξ 2 and ξ = (ξ 1 , ξ 2 ) ∈ R 2 . The fixed points of P T are the initial conditions of the periodic solutions with periodic T . Given a T -periodic solution (x(t), y(t)) and the fixed point ξ * = (x(0), y(0)), the theorem on differentiability with respect to initial conditions implies that the derivative of P T at the fixed point is precisely the monodromy matrix of (9), that is, P T (ξ * ) = M T . In consequence, if f ∈ E 2 , all fixed points of P T will satisfy det(I − P T (ξ * )) = 0. The implicit function theorem can be applied to deduce that all these fixed points are isolated. If we combine this fact with bound in Lemma 3.2 we can conclude that the set of fixed points Then deg[F, R, 0] = 0. Lemma 3.5. There exists ρ > 0 such that, for any (x 0 , y 0 ) ∈ R 2 verifying |y 0 | > ρ, one has Proof. The solution x(t; x 0 , y 0 ) of the system (5) with the initial condition x(0) = x 0 , y(0) = y 0 satisfies Then we obtain which implies that the result holds since g 1 (y) is increasing in y and g 1 (0) = 0.
Proof. Let α = x i (0), i = 1, 2, · · · , n. Consider the rectangle where ρ > 0 is given by Lemma 3.5. It follows from Lemma 3.4 that Since R contains all the fixed points of P T (after the identification x(t) ≡ x(t)+2nπ), now the proof is completed.
Theorem 3.7. Let the condition (σ 1 ) be satisfied. If (10) holds for any T -periodic solution (x, y), then the set E 2 is prevalent in E.
Following the main ideas come from [9,17], we establish the stable result which will be used later.
Proof. The Poincaré map Π associated to the differential system (5) is C ∞ areapreserving diffeomorphism. Note that (x, y) is a T -periodic solution of (5) is equivalent to say that the initial condition (x(0), y(0)) is a fixed point of Π. Recall that ∆ = 2 cos θ. Then the discriminant satisfies |∆| < 2 and is not in C and so (x, y) is elliptic and θ 2π ∈ R\{L ∪ Q}, namely θ 2π is Diophantine. Theorem 3.3 in [9] can be applied and therefore (x, y) is stable.
Given f ∈ F and a T -periodic solution (x, y) of (5), we define D = D[x, y] as the discriminant of the linearized system (9). To be precise on the domain of the functional, we introduce the set The rigorous definition of the functional is From the chain rule we deduce that for each (u, v) ∈ T (x,y) (M ), where χ 1 = χ a , χ 2 = χ b with a(t) = g 2 (x(t)) and b(t) = g 1 (y(t)). Consider the new domain for the functional where x n satisfies g 2 (x n ) = 0. Of course we also have that g 2 (x n ) = 0. This is an open subset of M and the restriction of the functional will be denoted by D * : M * → R.
Since the zeros of g 2 are isolated, there exists an interval I such that for every t ∈ I g 2 (x(t)) = 0.
Hence the periodic problem for (5) with f ∈ F is equivalent to the equation Proof. Assume first that A [x, y] is an isomorphism and let (u(t), v(t)) be a Tperiodic solution of the linearized system (9). Integrating over a period, we obtain T 0 g 1 (y(t))v(t)dt = 0, Assume now that (x, y) is non-degenerate. Then it is obvious that the kernel of A [x, y] is trivial. To prove that A [x, y] is onto, let p be a given function in F. By Fredholm's alternative we know that the non-homogeneous system

The map A is smooth with derivative
has a unique T -periodic solution. This solution satisfies Then (u, v) ∈ T (x,y) (M ) and A [x, y](u, v) = (0, p).
From Lemma 3.10, we know A [x, y] is an isomorphism because the solution (x, y) is non-degenerate. Applying the inverse function theorem, we can find open sets M f ⊂ M * , U f ⊂ Ω with (x, y) ∈ M f , f ∈ U f , and such that the restriction A : M f → F × U f is a diffeomorphism. After restricting the size of U f we can assume that the functional is prevalent in F. By the definition of G there exists a T -periodic solution (x, y) such that the discriminant of the linearized equation is given by ∆ = d f ( f ), for some f ∈ Ω with f ∈ U f . Moreover, |∆| < 2 and ∆ ∈ C. By using now Lemma 3.8, (x, y) is stable.
Thus for each f ∈ G, we have that the system ẋ = g 1 (y), y + g 2 (x) = f (t) has a stable T -periodic solution. Now the proof of Theorem 3.11 is finished because G ⊂ S.
Open problem. We believe that the constant 4/T 2 in the above Theorem can be improved. However, up to now, we do not know how to obtain the sharp bound for µ and leave it as an open problem.