Linearized Stability and Instability of Nonconstant Periodic Solutions of Lagrangian Equations

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1 Linearized Stability and Instability of Nonconstant Periodic Solutions of Lagrangian Equations Meirong Zhang Department of Mathematical Sciences, Tsinghua University, Beijing 184, China Xiuli Cen School of Mathematics Zhuhai, Sun Yat-sen University, Zhuhai, Guangdong 51982, China Xuhua Cheng Department of Applied Mathematics, Hebei University of Technology, Tianjin 313, China Abstract This paper is motivated by the stability problem of nonconstant periodic solutions of time-periodic Lagrangian equations, like the swing and the elliptic Sitnikov problem. As a beginning step, we will study the linearized stability and instability of nonconstant periodic solutions which are bifurcated from those of autonomous Lagrangian equations. Applying the theory for Hill s equations, we will establish a criterion for linearized stability. The criterion shows that the linearized stability depends on the temporal frequencies of the perturbed systems in a delicate way. Keywords: Nonconstant periodic solution; linearized stability; linearized instability; Hill s equation; Lagrangian equation; Sitnikov problem. Mathematics Subject Classification 21: 34D2; 34C25; 34C23 1 Introduction This paper is motivated by the study for the stability and instability of nonconstant periodic solutions of the elliptic Sitnikov problem [24, 8, 1] and the swing, which contain parameters like the eccentricity in the elliptic Sitnikov problem. There are many analytic and topological methods to establish the existence and the continuation of periodic solutions. However, as Lagrangian equations are conservative, and, moreover, we are considering nonconstant periodic solutions, the classical Lyapunov s methods cannot give too much for the Lyapunov stability/instability. In fact, for conservative systems, the Lyapunov stability/instability is involved of deep theories like the Birkhoff normal forms and Moser s twist theorem, even for systems of lower degrees of freedom [14, 23]. Moreover, even for the equilibria of nonautonomous systems, it is also a difficult problem to study the Lyapunov stability/instability. Correspondence author. This author is supported by the National Natural Science Foundation of China Grant No , No and No This author is supported by the National Natural Science Foundation of China Grant No , No and No This author is supported by the National Natural Science Foundation of China Grant No

2 For example, in a series of papers [17, 18, 19], Ortega has successfully characterized the Lyapunov stability for the equilibrium xt of the swing or the pendulum of variable length ẍ + lt sin x =, 1.1 where the variable length lt > is 2π-periodic. Roughly speaking, if the linearized equation of 1.1 along xt = ẍ + ltx = is stable, then, as a 2π-periodic solution, the equilibrium xt of 1.1 is also Lyapunov stable. In doing so, Ortega has systematically developed the third-order approximation method, which has further been profound later as in [7, 25, 27]. See also [1, 3, 4, 5, 15, 16]. As a beginning step towards nonconstant periodic solutions, in this paper we will study the linearized stability/instability. The choice for the problem to be studied is very initiative, i.e., we are considering periodic solutions which are bifurcated from nonconstant parabolic periodic solutions of autonomous Lagrangian equations. To simplify the argument and notation, let us start with an autonomous Lagrange equation of degree-one-of-freedom ẍ + fx =. 1.2 Here = d dt and fx is a smooth, odd function such that xfx > x. Hence x = is the unique equilibrium of Eq Then we consider the bifurcation problem ẍ + F x, t, e =. 1.3 Here the parameter e [, + is assumed to be one-sided. Motivated by the symmetries in the Sitnikov problem, we assume that F x, t, e is a smooth function of x, t, e R R [, + such that F x, t, fx and F x, t + 2kπ, e F x, t, e k Z, 1.4 F x, t, e F x, t, e and F x, t, e F x, t, e. 1.5 For autonomous equation 1.2, there are two conventional methods to obtain nonconstant periodic solutions. The first one is based on the period function. Define Ex := x fu du, x R, an even function. Because of the oddness of fx, we choose initial values x = ξ, +, ẋ =. 1.6 Let us use x = Xt, ξ to denote the unique solution of problem Then x = Xt, ξ satisfies ẋ 2 /2 + Ex Eξ. 1.7 For each ξ, +, Xt, ξ is a nonconstant periodic solution of Eq. 1.2 with the minimal period ξ du T ξ = 2 = 2 ξ du ξ 2Eξ Eu Eξ Eu 2

3 To study Eq. 1.3, we start with a nonconstant 2π-periodic solution of Eq However, the minimal period will not be restricted to be 2π, but 2π/p for some p N, i.e., by assuming that T ξ = 2π/p 1.9 has a solution ξ = ξ p, +, ϕt = ϕ p t := Xt, ξ p is a periodic solution of Eq. 1.2 of the minimal period 2π/p. Moreover, ϕt possesses the following properties ϕ t ϕt and ϕt + kπ/p 1 k ϕt k Z, 1.1 and ϕt has precisely p zeros in each interval [t, t + π. As ϕ p t looks like cospt, these will be called even 2π-periodic solutions of Eq These periodic solutions can also be obtained using the shooting method, as did in [9]. Using the solutions Xt, ξ of initial value problems, the velocity at t = π defines a function X t π,ξ of ξ, which yields the shooting equation X t =, ξ, π,ξ Then solutions of 1.9 and 1.11 are the same. At any positive solution ξ = ξ p of 1.11, we also assume that system 1.2 satisfies the following non-degeneracy condition č p := 2 X ξ t π,ξp For Eq. 1.3, we use Xt, ξ, e to denote the unique solution with the initial condition 1.6. Thus Xt, ξ, Xt, ξ. Let us consider the shooting equation for 1.3 X t =, ξ, π,ξ,e When e =, 1.13 returns to Eq Under the non-degeneracy condition 1.12, the Implicit Function Theorem implies that there exists some e p > and a smooth function Ξ p e of e [, e p such that Ξ p = ξ p and ξ = Ξ p e satisfies Eq for all e [, e p. Using these Ξ p e as initial values, for each e [, e p, ϕt, e = ϕ p t, e := Xt, Ξ p e, e is an even 2π-periodic solution of Eq. 1.3 such that it has precisely p zeros in each interval [t, t + π. These periodic solutions ϕt, e are bifurcated from ϕt = ϕt,. Usually, when e >, the minimal period ϕt, e is 2π, not 2π/p. The main concern of this paper is the linearized stability/instability of these nonconstant periodic solutions ϕt, e for < e 1, i.e., whether the corresponding linearization equation ÿ + Qt, ey =, Qt, e := F x, 1.14 ϕt,e,t,e is Lyapunov stable. Since 1.14 is a Hill s equation, the stability is determined by the trace τe = τ p e of the corresponding 2π-periodic Poincaré matrix. For more details, see the next section. For e =, as 1.14 is the linearization of autonomous equation 1.2, one 3

4 has τ = 2. In Theorem 3.1, we will derive the formula for τ. In case τ, the sign of τ will yield the linearized stability/instability of these nonconstant periodic solutions ϕt, e for < e 1. See Theorem 3.2. These can be considered as an extension of the stability criterion of the Hopf bifurcation [2] to nonconstant periodic solutions. However, the content is much complicated. In Section 4, we will analyze the sign of τ for some typical examples. From these examples, it will be found that τ is dependent on the temporal frequencies of the perturbed systems in a delicate way. In particular, after analyzing the temporal frequencies of the elliptic Sitnikov problem, we find that τ = for those 2πperiodic solutions constructed in [9]. In order to obtain linearized stability/instability for the elliptic Sitnikov problem, further analysis on higher-order derivatives of τe is needed and will be undertaken in a future work. Besides the contents above, in Section 2, we will give an extensive study on the linearization equation of 1.2. In particular, the relation between the non-degeneracy condition 1.12, the period function T ξ in 1.8 and the unstable-parabolicity of solution ϕt will be established. In Section 3, we will study Eq and the main result Theorem 3.1 will be proved. We end the introduction by the following remarks. In the monograph [11], Loud has systematically studied the linearized stability/instability of nonconstant periodic solutions of second-order ordinary differential equations by finding some formulas similar to τ. However, his main concern is on equations with damping. On the other hand, our formula of τ for conservative Lagrangian equations is involved of relatively less information on the nonconstant periodic solutions of ϕt of the autonomous equations. Hence the formula is more convenient in applications, as seen from the examples in Section 4. 2 Hill s Equations and Linearization Equations 2.1 Hill s equations Let us consider general Hill s equation [12] ÿ + qty =, 2.1 where qt is T -periodic and has some integrability, say that qt is continuous. The fundamental solutions y = ψ i t = ψ i t, q, i = 1, 2, of Eq. 2.1 are those solutions of 2.1 satisfying initial conditions ψ 1, ψ 1 = 1, and ψ 2, ψ 2 =, 1, respectively. The fundamental matrix solution is ψ1 t ψ 2 t Mt :=. ψ 1 t ψ 2 t The Liouville law for Eq. 2.1 is The T -periodic Poincaré matrix of Eq. 2.1 is det Mt = ψ 1 t ψ 2 t ψ 2 t ψ 1 t a b P = P T := = c d 4 ψ1 T ψ 2 T ψ 1 T ψ 2 T.

5 The Floquet multipliers µ 1,2 are eigenvalues of the Poincaré matrix P. As a consequence of the Liouville law 2.2, one has µ 1 µ 2 = 1. According to the distribution of µ 1,2, Hill s equation 2.1 is classified as hyperbolic, if µ 1,2 = 1, elliptic, if µ 1,2 = 1 and µ 1,2 ±1, or parabolic, if µ 1 = µ 2 = ±1. It is well known that 2.1 is Lyapunov stable Lyapunov unstable respectively if it is elliptic hyperbolic respectively. In case 2.1 is parabolic, it is Lyapunov stable if P = ±I and Lyapunov unstable if P ±I, where I is the identity matrix. For the former case, 2.1 is called stable-parabolic, and for the latter, it is called unstable-parabolic. The trace of the Poincaré matrix is τ := trp = a + d = ψ 1 T + ψ 2 T. It is well known that 2.1 is hyperbolic, elliptic or parabolic if τ > 2, τ < 2 or τ = 2 respectively. In order to emphasize their dependence on potential q, we will write these as P q, aq, bq, µ i q, τq etc, which can be considered as nonlinear functionals of q. We need the following formulas in [22]. Lemma 2.1 [22] Given t [, T ], by considering the fundamental solutions ψ i t; q and their derivatives ψ i t; q as nonlinear functionals of potentials q L 1 R/T Z, the Lebesgue space endowed with the L 1 norm 1 = L 1 R/T Z, they are continuously Frechét differentiable in q. Moreover, the Frechét derivatives are given by q ψ i t; q h = q ψ i t; q h = t t ψ 1 t; qψ 2 s; q ψ 2 t; qψ 1 s; qψ i s; qhs ds, 2.3 ψ 1 t; qψ 2 s; q ψ 2 t; qψ 1 s; qψ i s; qhs ds, 2.4 for h L 1 R/T Z and i = 1, 2. Here the right-hand sides of 2.3 and 2.4 are understood as bounded linear functionals on the space L 1 R/T Z, 1. Besides the Poincaré matrix P, let us introduce the following symmetric matrix 1 b N = Nq := 2 a d, a d c and the following kernel function 1 2 Ks = Ks; q := ψ 1 s, ψ 2 snψ 1 s, ψ 2 s = bψ 2 1s + a dψ 1 sψ 2 s + cψ 2 2s. 2.5 Such a matrix N is related with the Green function for the Hill s equations. Moreover, it has been proved that Ks is T -periodic in s. See [26, Lemma 3.4]. Lemma 2.2 As a nonlinear functional of potential q L 1 R/T Z, 1, the trace τq of the Hill s equation 2.1 is continuously Frechét differentiable in q. Moreover, the Frechét derivative is q τq h = T Ks; qhs ds, h L 1 R/T Z

6 Proof As τq = ψ 1 T ; q + ψ 2 T ; q, the continuous differentiability of τq in q follows immediately from Lemma 2.1. By 2.3 and 2.4, the directional derivative along any h L 1 R/T Z is q τq h = q ψ 1 T ; q h + q ψ 2 T ; q h = T T + ψ 1 T ; qψ 2 s; q ψ 2 T ; qψ 1 s; qψ 1 s; qhs ds ψ 1 T ; qψ 2 s; q ψ 2 T ; qψ 1 s; qψ 2 s; qhs ds, which can be rewritten as in 2.5 and Linearization of autonomous equations We consider autonomous equation 1.2 with the even 2π-periodic solution ϕt = ϕ p t = Xt, ξ p. The linearization equation of 1.2 along ϕt is a 2π-periodic Hill s equation ÿ + Qty =, Qt := f ϕt. 2.7 This corresponds to Eq with e =. As before, we use ψ i t and Mt to denote the fundamental solutions and the fundamental matrix solution of Eq Since Eq. 1.2 is autonomous, we have the following important observations. Lemma 2.3 Using Xt, ξ, the fundamental solutions of Eq. 2.7 are ψ 1 t = X ξ and ψ 1 t = 2 X t,ξp t ξ, 2.8 t,ξp ψ 2 t = ϕt ϕ = 1 X and ψ 2 t = ϕt fξ p t ϕ = fxt, ξ p. 2.9 fξ p t,ξp Proof Recall that Xt, ξ is the solution of Eq. 1.2 with the initial condition 1.6. By considering ξ as a parameter, it is well known that the variational equation for yt := X ξ is just Eq. 2.7 and the initial value is just y, ẏ = 1,. Thus ψ 1t t,ξp. Hence we have 2.8. X ξ t,ξp On the other hand, ϕt = Xt, ξ p satisfies Eq. 1.2, i.e., ϕt + fϕt =. Differentiating it with respect to t, we know that yt := ϕt is a solution of Eq. 2.7, with y = ϕ = and ẏ = ϕ = fϕ = fξ p. Hence ψ 2 t ϕt/fξ p. As ϕt = Xt, ξ p, we have the relations in 2.9. Remark 2.4 Formulas 2.8 and 2.9 show that the fundamental solutions of 2.7 can be deduced from solutions Xt, ξ of initial value problems of 1.2. As ϕt is 2π-periodic, we conclude from 2.9 that Eq. 2.7 is necessarily parabolic with the Floquet multipliers +1. 6

7 As f x is even, we know from properties in 1.1 that Qt of 2.7 is actually π/pperiodic. Hence 2.7 can be considered as Hill s equations of different periods. In particular, we are interested in the periods π/p, π and 2π, and their Poincaré matrixes ˆP := Mπ/p, ˇP := Mπ and P := M2π. One has relations ˇP = ˆP p, P = ˇP 2 and P = ˆP 2p. 2.1 Let us define ĉ p := ψ 1 π/p, č p := ψ 1 π and c p := ψ 1 2π Here, by 2.8, č p is the same as in the non-degeneracy condition Lemma 2.5 There hold č p = p 1 p+1 ĉ p and c p = 2 1 p č p = 2pĉ p Moreover, the 2π-periodic Poincaré matrix of Eq. 2.7 is P = 1 c p In particular, under assumption 1.12, Eq. 2.7 is unstable-parabolic and admits only one linearly independent 2π-periodic solution, say y = ψ 2 t or y = ϕt. Proof Note that ϕt satisfies ϕ, ϕ = ξ p, and by Eq. 1.2 itself, ϕ = fξ p. Due to 1.1, one has ϕπ/p = ξ p, ϕπ/p = and ϕπ/p = f ξp = fξ p. By 2.9, we have ψ 2 π/p = and ψ 2 π/p = 1. Moreover, it follows from 2.2 that ψ 1 π/p = 1. Hence 1 ˆP = ĉ p 1 where ĉ p is as in Now 2.1 shows that 1 ˇP = ĉ p 1 p =, 1 p p 1 p+1 ĉ p 1 p This gives the first relation of By using 2.1 again, 1 p 2 P = 1 p = č p p č p 1 Hence we have the second relation of 2.12 and result 2.13 is obtained. Next we give a relation between non-degeneracy condition 1.12 and the period function T ξ. Lemma 2.6 Suppose that the period function T ξ is differentiable at ξ = ξ p. Then ĉ p = fξ p T ξ p /2 and č p = p 1 p fξ p T ξ p /

8 Proof By using the solutions Xt, ξ and the periodic function T ξ, one has X t. T ξ/2,ξ Differentiating with respect to ξ, we have = d X dξ t = 2 X T ξ T ξ/2,ξ t X T ξ/2,ξ 2 ξ t T ξ/2,ξ = f XT ξ/2, ξ T ξ + 2 X 2 ξ t T ξ/2,ξ At ξ = ξ p, one has T ξ p /2 = π/p, XT ξ p /2, ξ p = Xπ/p, ξ p = ξ p. Moreover, by 2.8, 2 X = 2 X ξ t t ξ = ψ 1 π/p = ĉ p. π/p,ξp π/p,ξp Now the first equality of 2.14 follows from 2.15, while the second can be deduced from relation Remark 2.7 Due to relation 2.12, the non-degeneracy 1.12 for ϕt is equivalent to the linearized instability of ϕt. On the other hand, by assuming that the period function T ξ has a non-zero derivative at ξ p, relation 2.14 implies that the non-degeneracy 1.12 for ϕt is satisfied, and, moreover, T ξ will be non-constant near ξ = ξ p and the solution ϕt is also necessarily Lyapunov unstable. 3 A Criterion for Linearized Stability/Instability Let us consider the family ϕt, e = ϕ p t, e of nonconstant even 2π-periodic solutions of Eq. 1.3 bifurcated from ϕt = ϕ p t of Eq We will give a criterion for the linearized stability/instability of ϕt, e for < e 1. For e [, e p, the linearization equation of Eq. 1.3 along x = ϕt, e is the Hill s equation The fundamental solutions of Eq are denoted by ψ i t, e, which yield the corresponding trace τe = τ p e := ψ 1 2π, e + ψ 2 2π, e. By 2.13, one has τ = 2. Due to the smoothness of F x, t, e, all objects above are smooth in e. Theorem 3.1 Consider the 2π-periodic solutions ϕt and ϕt, e. Let F 23 t be F 23 t := 2 F t e. 3.1 ϕt,t, Then τ = τ p := dτ p de = ψ 1 2π e= fξ p F 23, ψ 2 = c p fξ p 2 Here ψ i t s are fundamental solutions of Eq. 2.7 and u, v := utvt dt. F 23, ϕ

9 Proof With the potential q = Q, one has from 2.13 and 2.5 N = =, Kt = c p ψ c p ψ 2 1 2π 2t = ψ 1 2πψ2t. 2 By considering e as a parameter, potentials Q, e of 1.14 can be considered as a smooth curve in the space L 1 R/2πZ. Hence we obtain from 2.6 τ = c p ψ2tht 2 dt. 3.3 Here, following from 1.14, ht = Q e = 2 F t, x 2 φt + F 13 t = f ϕtφt + F 13 t, 3.4 ϕt,t, where φt := ϕ e and F 13 t := 2 F t, e x. 3.5 ϕt,t, The function φt can be determined as follows. On one hand, as ϕt, e is 2π-periodic in t, so is φt. On the other hand, ϕt, e = Xt, Ξe, e is the solutions of initial value problems of Eq By considering ξ, e as parameters for problem , φt satisfies the variational equation of Eq. 1.3 at ξ, e = ξ p, φ + Qtφ + F 3 t =, 3.6 where Qt is as in 2.7 and F 3 t := F e. 3.7 ϕt,t, To obtain τ, let us first compute τ := = 1 ϕ = 1 ϕ = 1 ϕ φtψ 2 2tf ϕt dt 3.8 φtψ 2 t f ϕt ϕt dt by 2.9 φtψ 2 t dqt Qtψ 2 t φt Qtφt ψ 2 t dt, as f ϕt ϕt dt = df ϕt = dqt by integrating by parts and noticing the condition ψ 2 = ψ 2 2π =. Using Eq. 2.7 for ψ i and Eq. 3.6 for φ, one has Qψ 2 φ Qφ ψ 2 = ψ 2 φ + φ + F 3 ψ 2 = ψ 2 φ + F3 ψ 2. As both ψ 2 and φ are 2π-periodic, we obtain τ = 1 ϕ F 3 t ψ 2 t dt = 1 F 3, ψ 2. ϕ 9

10 It can be simplified further. Note from 3.7 that F 3 t and ψ 2 t are 2π-periodic. Moreover, F 3 t = d F = 2 F dt e x e ϕt 2 F + ϕt,t, t e ϕt,t, = ϕf 13 tψ 2 t + F 23 t, ϕt,t, where F 23 t and F 13 t are as in 3.1 and 3.5. Hence, integrating by parts, we obtain τ = 1 F 3, ψ 2 = 1 ϕ ϕ = F 13, ψ 2 2 It follows from 3.3, 3.4 and 3.8, 3.9 that F 3, ψ 2 = 1 ϕf13 ψ 2 + F 23, ψ 2 ϕ 1 ϕ F 23, ψ τ = c p τ + F13, ψ2 2 c p = ϕ F 23, ψ 2 = This is the desired formula 3.2. c p fξ p F 23, ψ 2. As a corollary of Theorem 3.1, we have the following criterion on linearized stability/instability of the bifurcated periodic solutions ϕt, e. Theorem 3.2 If τ <, then ϕt, e is elliptic and is linearized stable for < e 1, and, if τ >, then ϕt, e is hyperbolic and is Lyapunov unstable for < e 1. We end this section with two remarks on formula 3.2. Formula 3.2 is only involved of the 2π-periodic solution ϕt of the autonomous equation 1.2 and the fundamental solutions ψ i t of the linearization equation 2.7 for ϕt. Formula 3.2 is reasonable. From 3.1, if F x, t, e satisfies 2 F t e, x,t,e we have F x, t, e = ˆF x, e + ˇF x, t. As F x, t, = fx, we have ˆF x, + ˇF x, t = fx. Therefore F x, t, e fx + ˆF x, e ˆF x, is independent of time t. As in Remark 2.4, one has τe 2 and τ =. This is consistent with formula 3.2 because F 23 t in this case. 4 Examples and Conclusions Consider the following Lagrangian equation ẍ + fx = ehtgx, 4.1 where fx is as in 1.2, gx is a smooth odd function, and ht is a smooth, even, 2πperiodic function. System 4.1 fulfills all requirements Here the parameter e can be considered as in R. 1

11 For later use, we introduce some notation for ht. As ht is even and 2π-periodic, it can be expanded as the Fourier series ht = a n cosnt. n= Then h := a is called the constant part of ht, while ht := ht h = a n cosnt n=1 is called the oscillatory part of ht. For any p N, let us define h p t := a 2pn cos2pnt, 4.2 n=1 called the π/p-periodic oscillatory part of ht. For the function cosnt, n N, one has 2p 1 1 cosnt + kπ/p = 1 2p 1 2p 2p R e e int inπ/p k k= k= { cosnt if n/p is even, = else. Thus the function h p t of 4.2 can also be characterized as h p t 1 2p 1 ht + kπ/p p k= Let p N and ϕ p t be an even 2π-periodic solution of Eq. 1.2 with ϕ p = ξ p >. For Eq. 4.1, one has F 23 t = gϕ p tḣt. Thus 3.2 is τ p = = c p fξ p 2 c p fξ p 2 = c p fξ p 2 = c p fξ p 2 where, by using equations 1.2 and 1.7 for ϕ p, gϕ p ϕ p ḣ dt gϕ p ϕ p h dt g ϕ p ϕ 2 p + gϕ p ϕ p h dt Φ p t ht dt, 4.4 Φ p = 2Eξ p g ϕ p 2Eϕ p g ϕ p fϕ p gϕ p. 4.5 Lemma 4.1 By using Φ p and h p, we can rewrite τ p as τ p = τ p = 4p2 ĉ p fξ p 2 8p2 ĉ p fξ p 2 π/p π/2p Φ p t h p t dt, 4.6 Φ p t h p t dt

12 Proof Thus Let us notice from 1.5 that the function Φ p t of 4.5 is actually π/p-periodic. Φ p t ht dt = = 2p 1 k+1π/p k= kπ/p 2p 1 π/p = 2p k= π/p Φ p t ht dt Φ p t ht + kπ/p dt Φ p t h p t dt. 4.8 See 4.3. Now 4.6 can be obtained by exploiting 2.12 and 4.4. Note that Φ p t is even and π/p-periodic. Moreover, 4.2 shows that h p t is also even and π/p-periodic. Hence π/p Thus one has 4.7. Φ p t h p t dt = = 2 π/2p π/2p Φ p t h p t dt + Φ p t h p t dt. π/2p Φ p t h p t dt Formula 4.6 shows that the constant part and the non-π/p-periodic oscillatory parts of ht have no effect on τ p, i.e., the temporal frequencies of the perturbation play an important role in the linearized stability problem. 4.1 A spatial-linear perturbation In the following, we consider Eq. 4.1 by choosing fx = x 3, gx x and ht = a + cos2p t, where a R and p N, i.e., the equation ẍ + x 3 = ea + cos2p tx. 4.9 Here the temporal frequency of the perturbation is 2p. For system 4.9, one has Ex = x 4 /4 and, by 1.8, 4.2 and 4.5, 2B1/4, 1/2 T ξ = ξ ξ >, 4.1 Φ p t = ξp 4 3ϕ 4 pt/2, ht = cos2p t, Hence we conclude from 4.6 that h p t = { if p p, cos2p t if p p. τ p = if p p,

13 and Here τ p = 2p2 ĉ p ξ 6 p π/p ξp 4 3ϕ 4 pt cos2p t dt = 6p2 ĉ p ξp 2 C p,p if p p C p,p := π/p ϕ p t/ξ p 4 cos2p t dt. Note that 2.15 implies that ĉ p > and 4.1 can yield ξ p = pb1/4, 1/2 2π. Again, 4.11 and 4.12 show that the temporal frequencies of the perturbation have important effect on τ p. Numerically, with the choice of p = 1, 2, 3, 4, one has C p,p =, except the following cases C 1,1 =.7483, C 1,2 =.2597, C 1,3 =.45, C 2,2 =.3742, C 3,3 =.2494, C 1,4 =.77, C 2,4 =.13, C 4,4 =.1871 In conclusion, for these choices of p, p, the bifurcated families of nonconstant periodic solutions of Eq. 4.9 are elliptic and linearized stable hyperbolic and Lyapunov unstable, respectively for < e 1 for 1 e <, respectively. For other cases, the analysis for linearized stability/instability appeals for further discussion on τ p, τ p etc. 4.2 The swing Let us consider nonconstant periodic solutions of the swing 1.1. Suppose that the variable length is lt := a 2 + eht, where a > and ht is even 2π-periodic. Hence 1.1 is which is Eq. 4.1 with When e =, 4.13 is the pendulum ẍ + a 2 + eht sin x =, 4.13 fx = a 2 sin x, gx = sin x. ẍ + a 2 sin x = In order that 4.14 has nonconstant even 2π-periodic solution ϕ p t, it is necessary and sufficient that a > p. For 4.14, one has Ex a 2 1 cos x. By 4.5, Φ p = a cos ξ p cos ϕ p 3 cos 2 ϕ p. 13

14 Now 4.4 is c p τ p = a 2 sin 2 ξ p c p = a 2 sin 2 ξ p = 4p2 ĉ p a 2 sin 2 ξ p π/p cos ξp cos ϕ p t 3 cos 2 ϕ p t ht dt 2 cos ξp cos ϕ p t 3 cos 2 ϕ p t ht dt 3 cos 2 ϕ p t 2 cos ξ p cos ϕ p t hp t dt, 4.15 as did in 4.6. By relation 2.14, we know that ĉ p > in With the choice ht = cos2p t, p N, we have from 4.15 that τ p = if p p, and { τ p = 4p2 ĉ p C p,p := π/p a 2 sin 2 ξ p C p,p 3 cos 2 ϕ p t 2 cos ξ p cos ϕ p t cos2p t dt 4.16 if p p. Here one has ĉ p >. For example, let a = 5/2. Hence p = 1, 2. With the choice of p = 1, 2, the constants C p,p in 4.16 are C 1,1 =.7946, C 2,1 =, C 1,2 = , C 2,2 = 1.22 The linearized stability/instability for nonconstant periodic solutions of the swing 4.13 depends on p, p. 4.3 The Sitnikov problems The circular and the elliptic Sitnikov problems are respectively described by the second-order autonomous and non-autonomous Lagrangian equations where e [, 1 is the eccentricity and, with r := 1/2, ẍ + fx =, 4.17 ẍ + F x, t, e =, 4.18 fx = x x 2 + r 2 3/2, x F x, t, e = x 2 + r 2 t, e 3/2, rt, e = r 1 e cos ut, e, where, after some translation of time, u = ut, e is the solution of the Kepler s equation u e sin u = t. Systems 4.17 and 4.18 fulfill the requirements of this paper. There are extensive studies on this simplest three-body problem. A rigorous mathematical study on the existence and continuation of nonconstant periodic solutions of 4.18 can be found in [9, 2, 21]. In recent years, the stability problem has been also studied as in [6, 13]. 14

15 Since we are only interested in even periodic solutions, let us describe some existence results in [9]. For Eq. 4.17, the period function T ξ is strictly increasing in ξ, + and T min = lim T ξ = 2π/ 8 and T max = lim T ξ = ξ + ξ + Given N N, the circular Sitnikov problem 4.17 has nonconstant 2N π-periodic solutions ϕ p,n t of the minimal period 2Nπ/p if and only if p {1, 2,, ν N }, where ν N := [ 8N] N. 4.2 See As before, we can take ϕ p,n t so that it is also even. Bifurcated from these solutions, one has the following results on the elliptic Sitnikov problem. Lemma 4.2 [9, Theorem 1] Let N N and p be as in 4.2. Then there exists some e p,n, 1] such that for any e [, e p,n, Eq has an even 2Nπ-periodic solution ϕ p,n t, e = Xt, ξ p,n e, e. Now we apply Theorem 3.1 to these 2π-periodic solutions ϕ p,1 t, p = 1, 2. From the defining equalities for fx and F x, t, e, one has 2 F 3r 2 t e = x x,t, x 2 + r 2 5/2 sin t This shows that the temporal frequency of the elliptic Sitnikov problem is 1. For p = 1, 2, we have from 3.2 and 4.21 τ p = By introducing the π/p-periodic function r2 c p 3ϕ p t ϕ p t fξ p 2 ϕ 2 p t + r 2 5/2 sin t dt Φ p t := 1/ ϕ 2 pt + r 2 3/2, one has Φ p t = 3ϕ p t ϕ p t/ ϕ 2 pt + r 2 5/2. Integrating 4.22 by parts, we obtain τ p = r2 c p fξ p 2 Φ p t cos t dt. By taking ht := cos t and arguing as in 4.8, we have τ p = 2pr2 c p fξ p 2 π/p Φ p t h p t dt =, p = 1, 2, 4.23 because h p t = cos t p, following simply from the defining equality 4.2. Result 4.23 shows that, in order to detect the linearized stability/instability of 2πperiodic solutions ϕ p,1 t of Eq. 4.18, one needs to find higher-order derivatives τ p etc. 15

16 4.4 Conclusions Let us go back to system 1.3. Note that 2 F is odd in t and can be expanded as the x,t, Fourier expansion 2 F t e = x,t, where f n x s are odd in x. By defining which is even in x, one has from 3.2 c p τ p = fξ p 2 = c p fξ p 2 E n x := n n=1 t e f n x sinnt, 4.24 n=1 x f n u du, ϕ p t f n ϕ p t sinnt dt n=1 E n ϕ p t cosnt dt. Note that E n ϕ p t is π/p-periodic in t. As did in 4.6, one has τ p = 8p3 ĉ p fξ p 2 n n=1 π/p E 2pn ϕ p t cos2pnt dt Now 4.25 shows that only the terms of frequencies 2pn, n N, in 4.24, have effect on τ p. However, the effects of terms of other frequencies on higher-order derivatives τ p k remain open. Acknowledgment The authors would like to thank the referee for his/her very careful reading of the original version of the paper and some important suggestions. Accordingly, some fault statement on stability of the elliptic Sitnikov problem has been corrected and some references have been added. References [1] A. Boscaggin and R. Ortega, Periodic solutions of a perturbed Kepler problem in the plane: from existence to stability, J. Differential Equations, , [2] S.-N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, Berlin, [3] J. Chu, J. Ding and Y. Jiang, Lyapunov stability of elliptic periodic solutions of nonlinear damped equations, J. Math. Anal. Appl., , [4] J. Chu, J. Lei and M. Zhang, Lyapunov stability for conservative systems with lower degree of freedom, Discrete Cont. Dynam. Syst. Ser. B, ,

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