ON THE FAMILIES OF PERIODIC ORBITS OF THE SITNIKOV PROBLEM

Transcription

1 ON THE FAMILIES OF PERIODIC ORBITS OF THE SITNIKOV PROBLEM JAUME LLIBRE AND RAFAEL ORTEGA Abstract. The main goal of this paper is to study analytically the families of symmetric periodic orbits of the elliptic Sitnikov problem for all values of the eccentricity in the interval [0, ). The basic tool for proving our results is the global continuation method of the zeros of a function depending on one parameter provided by Leray and Schauder and based in the Brouwer degree.. Introduction The Sitnikov problem is a special case of restricted three body problems where the two primaries with equal masses are moving in a circular or an elliptic orbit of the two body problem, and the infinitesimal mass is moving on the straight line orthogonal to the plane of motion of the primaries which passes through their center of mass. When the orbit described by the primaries is circular the Sitnikov problem is known as the circular Sitnikov problem. In 907 G. Pavanini [29] expressed its solutions by means of Weierstrassian elliptic functions. Four years later MacMillan [22] expressed the solutions in terms of Jacobian elliptic functions (a detailed description of this work can be found in Stumpff [34]). Some other analytical expressions for the solutions of this problem can be found for instance in [35], [3] and [38]. The precise definition of the Sitnikov problem is given in Section 2. The elliptic Sitnikov problem is the case when the orbit describing the primaries is elliptic. This problem became important when Sitnikov [33] in 960 used it to show, for the first time, the possibility of the existence of oscillatory motions in the three body problem. The existence of this kind of motions was predicted by Chazy [6, 7, 8] in , when he classified the final evolutions of the 3 body problem. Later on Alekseev [2] in proved that, in the special case of the 3 body problem studied by Sitnikov, all of the possible combinations of final motions in the sense of Chazy were realized. Moser [28] in 973 gave alternative proofs of the results of Alekseev which are simpler than those in [2]. Since then many other authors have studied the circular or elliptic Sitnikov problem. For instance Llibre and Simó [20], Perdios and Markellos [30], Jie Liu and Yi Sui Sun [7], Hagel [5] and [6], Martinez Alfaro and Chiralt [24], Dvorak [4], [9], Wodnar [36, 37, 38], Dankowicz and Holmes [3],... The families of symmetric periodic orbits of the elliptic Sitnikov problem, for sufficiently small values of the eccentricity e, by continuing the known periodic orbits of the circular Sitnikov problem have been studied by several authors, analytically by Corbera and Llibre [0, ] using the analytical continuation method of Poincaré, and 99 Mathematics Subject Classification. Primary 70F5; Secondary 37N05. Key words and phrases. 3 body problem, Sitnikov problem, periodic orbits, global continuation. The first author is partially supported by a MCYT grant BFM C02 02 and by a CIRIT grant number 200SGR 0073, and the second one is partially supported by BFM

2 2 J. LLIBRE AND R. ORTEGA Cabral and Xia [5] by applying the subharmonic Melnikov method,...; numerically by Belbruno, Llibre and Ollé [3],andJiménez and Escalona [8]. In this last paper the authors describe numerically some families of symmetric periodic orbits for almost all values of the eccentricity e in [0, ). The main objective of this paper is to study analytically the families of symmetric periodic orbits of the elliptic Sitnikov problem for non necessarily small values of the eccentricity e. More precisely, we will show that some periodic orbits for e =0can be continued to all values of e in [0, ). In Theorems, 2 and 3 of Section 3 are the statements of our main results. The main tool for proving our results is the global continuation of the zeros of a function depending on one parameter provided by Leray and Schauder and based in the Brouwer degree, see Section 4. In Section 5 we show that, with the convenient formulation, the Sitnikov problem satisfies the basic assumptions of the global continuation theorem. In Section 6 we study the dynamics around the unique equilibrium point of the Sitnikov problem. This equilibrium point corresponds to one of the three collinear relative equilibrium solution of Euler for the general 3 body problem, see for more details Section 2. Finally, in Section 7 we provide the last steps in the proof of Theorems, 2 and 3. The use of the global continuation techniques in the study of nonlinear boundary value problems is classical. We refer to [32] for recent results applicable to general classes of nonlinearities. In another context we should also mention the paper [25] by Mathlouthi. He studies the Sitnikov problem with variational techniques and obtains results about the existence of periodic solutions which are global in the sense that they are valid for arbitrary eccentricity. The use of continuation methods will allow us to obtain many continuous families and to be more precise about the oscillatory properties of the solutions. 2. The Sitnikov problem Let m = m 2 be two punctual masses (called primaries) describing a circular or an elliptic orbit of the two body problem. We consider an infinitesimal mass m 3 that moves on the straight line ρ orthogonal to the plane of motion of the primaries that passes through their center of mass. The Sitnikov problem will consist of describing the motion of the infinitesimal mass. In particular, if the primaries are moving in circular (respectively elliptic) orbits, we have the circular (respectively elliptic) Sitnikov problem. We choose the units of mass, length and time so that m = m 2 =/2, the gravitational constant G =, and the period of the orbit described by the primaries be 2π. If z denotes the position of the particle m 3 in a coordinate system on ρ with origin at the center of mass of the primaries (see Figure ), then the equation of motion of the Sitnikov problem becomes z () z = (z 2 + r 2 (t, e)) 3/2, where r(t, e) is the distance of the primaries to their center of mass and it is given by (2) r(t, e) = ( e cos u(t)), 2

3 ON THE FAMILIES OF PERIODIC ORBITS OF THE SITNIKOV PROBLEM 3 m 3 =0 F 3 F 32 F z m 2 =/2 r(t) z =0 3 m =/2 Π ρ Figure. The Sitnikov problem. which is a circular or an elliptic solution of the Kepler problem (3) r = e2 6 r 3 8 r 2, with eccentricity e = 0 or 0 <e<, respectively. Here u(t) is the eccentric anomaly which is a function of time via the Kepler s equation (4) u e sin u = t l, with l the time at the pericenter passage. Without loss of generality when 0 <e< we usually take the origin of time in such a way that at t = 0 the primaries are at the pericenter of the ellipse (i.e. l =0). We note that system () depends on one parameter, the eccentricity e [0, ). When the eccentricity e is zero (that is, the primaries move on the circular orbit r(t) = /2 of the Kepler problem (3)), equation () becomes the equation of motion z (5) z = (z 2 +/4) 3/2, for the circular Sitnikov problem. This equation defines an integrable Hamiltonian system of one degree of freedom with Hamiltonian (6) H = ( 2 v2 z 2 + /2, 4) where v =ż. The orbits for the circular Sitnikov problem in the energy level h are described by the curve H = h, where h varies in [ 2, ). Then depending on the value of h we have different types of orbits in the phase space (z,v) (see Figure 2): () when h< 2 wehavenoorbits;

4 4 J. LLIBRE AND R. ORTEGA v 2 3 z Figure 2. The circular Sitnikov phase portrait. (2) when h = 2 we have the equilibrium point (z =0,v = 0), or equivalently the trivial solution z(t) 0, which correspond to one of the well known collinear relative equilibrium solution of Euler for the 3 body problem (see for instance []); (3) when 2 <h<0 we have periodic orbits; (4) when h = 0 we have two parabolic orbits (i.e. two orbits that leave and reach infinity with zero velocity); (5) when h>0 we have two hyperbolic orbits (i.e. two orbits that leave and reach infinity with positive velocity). If the eccentricity e (0, ), then differential equation () corresponds to the elliptic Sitnikov problem. We note that this differential equation is non autonomous, i.e. the time appears explicitely in the right hand side of () through r(t, e). Moreover, r(t, e) is a periodic function in t of minimal period 2π. Consequently all periodic solutions (z(t), ż(t)) of () with e (0, ) must have period a multiple of 2π. Hence, all periodic orbits of the infinitesimal mass m 3 for the elliptic Sitnikov problem are also periodic orbits involving the three masses. Of course, in general, this was not the case for the circular Sitnikov problem. 3. Statement of the main results Given an integer N wedefine ν = ν N =[2 2N], where [ ] denotes the integer part function. Our main result is the following one. Theorem. For each p =,...,ν and ε>0 there exists a family (or a branch) of solutions {(z s (t),e s )} s [0,) of the equation () satisfying: () The map (s, t) [0, ) R (z s (t), ż s (t),e s ) is continuous. (2) The solutions z s (t) are even and 2Nπ periodic; i.e. for all s [0, ) we have z s ( t) =z s (t), z s (t +2Nπ)=z s (t). (3) For each s [0, ) we have that z s (0) > 0 (hence, the solution z s (t) is non trivial; i.e. z s (t) 0), and z s (t) has exactly p zeros in the interval [0,Nπ]. (4) e 0 =0, e s [0, ε] for each s and one of the following alternatives holds.

5 ON THE FAMILIES OF PERIODIC ORBITS OF THE SITNIKOV PROBLEM 5 z(0) A A A 2 A 3 0 E 3 E 2 ɛ e Figure 3. Families of periodic orbits. (4.a) e s ε and z s (0) ξ>0 as s ; (4.b) lim s e s = E exists with E< ε, z s (t) converges to 0 as s, and the linear differential equation ÿ + r(t, E) 3 y =0 has a non trivial, even, 2Nπ periodic solution with exactly p zeros in the interval [0,Nπ]. Just to illustrate the theorem we sketch a hypothetical situation, see Figure 3. For a fixed N 2 we take three numbers p <p 2 <p 3 ν and draw the sets A, A 2, A 3 defined by A i = {(z(0),e): z(t) isaneven2nπ periodic solution with p zeros in [0,Nπ], z(0) > 0,e [0, ε]} For p [resp. p 3 ] one finds a family satisfying (4.a) [resp. (4.b)]. For p 2 both alternatives are possible. The following result provides additional information about Theorem. Theorem 2. The following statements hold. () If p<n then statement (4.a) of Theorem holds. (2) If p N, ρ N < ε and statement (4.b) of Theorem holds, then E>ρ N with { ( ) N 2/3 ( ) } N 2/3 ρ N =min 2, 2. ν ν +

6 6 J. LLIBRE AND R. ORTEGA We note that statement () of Theorem 2 provides families which can be globally continued to the whole interval of eccentricities [0, ε]. In the case N/p wedonot know if the continued family is defined or not in the whole interval [0, ) but statement (2) of Theorem 2 gives us an estimation of the size of the interval of eccentricities where it can be extended. Another consequence of the previous result is the existence of even solutions with minimal period 2Nπ, N 2, for arbitrary eccentricity. It is sufficient to consider the global family associated to p =. A similar result was obtained in [25]. After adapting Theorem 2 and Corollary 3 of that paper to our notation, one obtains the existence of odd periodic solutions with minimal period 2Nπ, N 2. Finally, we are going to compare two different families of solutions. Given positive integers M, N, p, q, p ν N and q ν M, we denote by {(z s,e s )} and {(z s,e s)} the families given by Theorem for the couples (p, N) and(q, M), respectively. Theorem 3. Using the previous notation we assume that M/q > N/p. () The sets {(z s (0),e s ):s [0, )} and {(zs(0),e s):s [0, )} do not intersect. (2) If statement (4.a) of Theorem holds for {(z s,e s )}, then the same is true for {(zs,e s)}. (3) If statement (4.b) of Theorem holds for {(zs,e s)} with E = lim s e s, then thesameistruefor{(z s,e s )} with E = lim s e s E. 4. Global continuation in the sense of Leray Schauder Given an open and bounded subset Ω of R d and a function f : Ω R d which is continuous and does not vanish on the boundary of Ω (i.e. f(x) 0forallx Ω), we can define the Brouwer degree deg(f,ω), sometimes denoted by deg(f,ω, 0); for more details see [2, 26]. As usual Ω denotes the closure of Ω in R d. Given x 0 Ω a zero of f (i.e. f(x 0 ) = 0), if it is isolated in the set of zeros, then we can define the Brouwer index of the zero by ind(f,x 0 )=deg(f,u), where U is a small neighborhood of x 0. This definition is correct because x 0 is isolated and the degree has the property of excision. An important property of the degree is its invariance by homotopy. We now state a generalized version where the domain changes with the parameter. It can be found in several papers and books on degree theory but it is already in [2] (see Lemma 3 of that paper). Let A be a subset of R d [a, b]. If we denote by (x, λ) the points of R d [a, b], then we define A λ = {λ = λ } = {(x, λ) A : λ = λ }. Lemma 4. Let U be an open and bounded subset of R d [a, b], and f : U R d be continuous and such that f(x, λ) 0 for all (x, λ) U. Then deg(f λ,u λ ) is independent of λ, where f λ (x) =f(x, λ). We remark that U λ canbeemptyforsomeλ, in such a case, by Lemma 4, the degree is 0. We conclude these preliminary remarks stating some properties of continua.

7 ON THE FAMILIES OF PERIODIC ORBITS OF THE SITNIKOV PROBLEM 7 Lemma 5. Assume that X isametricspace,k X compact, and A, B K compact sets such that there is no subcontinuum of K connecting A and B. Then there is an open subset U of X satisfying A U, B U =, K U =. Lemma 5 or similar is employed very often in papers on global bifurcation. See, for instance, [3] and [26]. The next result is stated as a remark after the Théorème Fondamental in [2]. Actually, the result in [2] is more general because it works in infinite dimensions for a compact map. We will review the proof of [2]. See also [3] and [26]. Theorem 6. Let F : R d [a, b] R d be continuous, and Z = {(x, λ) :F (x, λ) =0} be the set of zeros of F. Assume that (H) Z is bounded. (H2) The set Z a is finite and there is (x 0,a) Z a with ind(f a,x 0 ) 0. Let C be the connected component of Z containing (x 0,a). Then one of the following alternatives holds: (a) C {λ = b}. (b) There exists (x,a) Z a, x x 0 such that (x,a) C. Proof: Consider the metric space X = R d [0, ] and the set K = Z. The assumption (H) implies that K is compact. Define A = {(x 0,a)}, B =(Z 0 \ A) {(x, b) : x M}, where M is a large constant so that Z is included in x <M. If neither (a) nor (b) holds, then there is no subcontinuum of Z meeting A and B. We find U open subset of X such that {(x 0,a)} = U a Z, U b =, Z U =. By Lemma 4, the deg(f λ,u λ ) is independent of λ. Since U b =, this degree must be zero. On the other hand, by (H2) we have deg(f a,u a )=ind(f a,x a ) 0. This contradiction shows that (a) or (b) must hold. In general the continuum C can be rather pathological, however there is a special case where one can guarantee that C is arcwise connected. In this case there are arcs joining all points of C and this corresponds to the usual idea of continuation. Theorem 7. Under the assumptions of Theorem 6 suppose that d =and F is real and analytic. Then, there is a continuum arc α :[0, ] Z, α(s) =(x(s),λ(s)) with x(0) = x 0, λ(0) = a such that either λ() = b, orλ() = a and x() x 0. Many results about the effect of analyticity on global continuation can be seen in [2] and [4]. The local structure of the set of zeros of F (x, λ) = 0 says that C is locally arcwise connected. Since C is, by definition, connected we conclude that C is arcwise connected.

8 8 J. LLIBRE AND R. ORTEGA 5. Periodic solutions of the Sitnikov problem Equation () is the equation of motion for the Sitnikov problem, where e [0, ) is the eccentricity and r(t, e) is the distance of the primaries to its center of mass. The eccentric anomaly u(t) satisfies u(t +2π) =u(t) +2π, u( t) = u(t), and so, by (2) and (4), r(t, e) isanevenand2π periodic function. Given an integer N we shall be interested in even, 2Nπ periodic solutions of (). They satisfy the boundary conditions (7) ż(0) = ż(nπ) = 0. Let ϕ(t; ξ,e) be the solution of () satisfying z(0) = ξ, ż(0) = 0. This is a real analytic function in the arguments (t; ξ,e) R R [0, ). Notice that these solutions are globally defined in (, + ) because the nonlinearity in () is bounded. We define F N : R [0, ) R, F N (ξ,e) = ϕ(nπ; ξ,e). The research of even 2Nπ periodic solutions of () satisfying (7) is equivalent to the study of the equation F N (ξ,e) =0. We want to apply Theorem 7 to F N and so, we must verify its assumptions. Towards assumption (H). We first consider the circular Sitnikov problem z z = (z 2 + R 2 ) 3/2, for some R>0. Let ψ(t; ξ) be its solution satisfying z(0) = ξ, ż(0) = 0, (ξ >0). Then, ψ(t; ξ) is periodic with minimal period T (ξ) > 0, lim ξ + T (ξ) =+ and ( 0 <ψ(t; ξ) <ξ, ψ(t, ξ) < 0ift 0, T (ξ) ). 4 Fix ξ > 0 such that T (ξ) > 4Nπ if ξ ξ. Proposition 8. Assume that r(t, e) R for all t, andletϕ(t; ξ,e) be a solution of () satisfying (7). Then ξ ξ. Proof: The equation of motion () is invariant under the symmetry (z,t) ( z, t). Therefore, we can assume that ϕ(0; ξ,e) =ξ > 0. Note that one can deduce from the equation that ξ is a local maximum of ϕ(t; ξ,e). Also, by integrating the equation between 0 and Nπ, one deduces that ϕ(t; ξ,e) changes sign in this interval. Let τ>0 be the first zero of ϕ(t; ξ,e) in[0,nπ]. We must have ϕ(t; ξ,e) < 0, for t (0,τ). Otherwise two consecutive critical points of ϕ(t; ξ,e) should be maxima. Set ϕ = ϕ(t) =ϕ(t; ξ,e). For t (0,τ)wehave ( ) ( ) d ϕ dt 2 ϕ2 (ϕ 2 + R 2 ) /2 = ϕ (ϕ 2 + r 2 ) 3/2 + ϕ (ϕ 2 + R 2 ) 3/2 0,

9 ON THE FAMILIES OF PERIODIC ORBITS OF THE SITNIKOV PROBLEM 9 and consequently 2 ϕ2 (ϕ 2 + R 2 ) /2 (ξ 2 + R 2 ) /2. For t (0,τ), ϕ is a solution of the differential inequality ϕ ( 2 ( For t 0, T (ξ) 4 ψ(t) is the minimal solution of ẋ = ( 2 (ϕ 2 + R 2 ) /2 (ξ 2 + R 2 ) /2 ) /2. ), ψ(t) = ψ(t; ξ) satisfies the associated differential equation. In fact, (x 2 + R 2 ) /2 (ξ 2 + R 2 ) /2 ) /2, x(0) = ξ. Therefore, the theory of differential inequalities implies that { ϕ(t) ψ(t) if 0 t min τ, T (ξ) }. 4 From here we conclude that τ T (ξ)/4. Thus Nπ > τ T (ξ)/4, and so ξ<ξ. The previous proposition allows us to verify (H) in each strip of the type R [0,E] with E<. In fact, if e [0,E], then r(t, e) ( E)/2 and we can apply the previous result with R =( E)/2. Towards assumption (H2). We want to study the zeros of F N (, 0). This is equivalent to study the solutions of z (8) z = (z 2 +, ż(0) = ż(nπ)=0. 4 )3/2 As we already mentioned, the solutions ϕ(t; ξ,0) are periodic with minimal period T (ξ), and T (ξ) is an increasing function in ξ, see [3]. By the symmetry, ϕ(t; ξ,0) with ξ 0 is a solution of the boundary problem (8) if and only if there is an integer p such that T (ξ)/2 =Nπ/p. Since inf T (ξ) =π/ 2 (see [3]), we have 2Nπ/p > π/ 2, and consequently p< 2 2N. Define ν = ν N = [2 2N] andletξ >... > ξ ν > 0 be the solutions of T (ξ)/2 =Nπ/p with p =,...,ν. Then (9) Z 0 = { ξ,..., ξ ν, 0,ξ ν,...,ξ }. We compute the indices. Since T = T (ξ) is increasing, we have that ϕ(nπ; ξ,0) > 0 if ξ<ξ close to ξ, ϕ(nπ; ξ,0) < 0 if ξ>ξ close to ξ. From here, ind(f N (, 0),ξ )=. In general, (0) ind(f N (, 0),ξ p )=( ) p. The indices for ξ p can be computed using the symmetry. We also compute the index at 0, although this information will not be employed in the rest of the paper. We do it by linearization, i.e. ( ) FN ind(f N (, 0), 0) = sign ξ (0; 0).

10 0 J. LLIBRE AND R. ORTEGA We note that F N ξ (0; 0) = ẏ(nπ), where y(t) is the solution of the variational problem So, we have ÿ +8y =0, y(0) =, ẏ(0) = 0. ( ind(f N (, 0), 0) = sign sin(2 ) 2Nπ) =( ) ν+. To sum up: for e = 0 there exist ν = ν N =[2 2N] nontrivial, even, and 2Nπ periodic solutions of (8) with z(0) > 0. They can be labelled by the number of zeros of z(t) in[0,nπ], for p =,...,ν,and ϕ (0) = ξ >...>ϕ ν (0) = ξ ν. Moreover, the index of each of these solutions, ind(f N (, 0),ξ p )is±. We summarize our knowledge of the set Z. We have the trivial continuum z =0, e [0, ). Also, we have the solutions (ξ i, 0). Since the index is different from zero there is at least a local branch emanating from them. Finally, we know that Z can only blow up as e. We want to study the possible collisions of the branches emanating from ξ i with the trivial continuum. To this end we linearize around z =0. 6. Linearization around the equilibrium The main objective of this section is to prove the next result. Theorem 9. Consider the boundary value problem () ÿ + r(t, e) 3 y =0, ẏ(0) = ẏ(nπ)=0. Then, there exists a sequence {E n,n } n, satisfying 0 <E,N <... < E n,n <... <, converging to, and such that there is a nontrivial solution of () if and only if e = E n,n. Moreover, E,N >ρ N (ρ N was defined in statement (2) of Theorem 2), and the solution y n corresponding to E n,n has a number of zeros in [0,Nπ] which becomes arbitrarily large as n. The linear differential equation () has been studied in detail in [24] and most of the statements above follow from their results. There is an alternative way of studying this equation. In fact, the change of the independent variable u e sin u = t transforms () into ( e cos u) d2 y dy e sin u +8y =0. du2 du Here we have used equation (2). This is a particular case of the so called Ince s equation (see [23]) which has been studied by several authors. In particular, the techniques of [27] can probably be employed for the effective computation of the numbers E n,n. Let y(t; e) be the solution of the equation appearing in () which satisfies y(0) =, ẏ(0) = 0. We study the zeros of ẏ(nπ,e)=0. Proposition 0. If e ρ N, then ẏ(nπ,e) 0. To prove Proposition 0 we need the following result, which is a well known consequence of Sturm comparison theory (see, for more details [9]). Lemma. Assume that a(t) is continuous, 2Nπ periodic, for some n 0 satisfies ( ) n 2 ( ) n + 2 a(t), for all t R, N N

11 ON THE FAMILIES OF PERIODIC ORBITS OF THE SITNIKOV PROBLEM and both inequalities are strict somewhere. Then, ÿ + a(t)y =0has no 2Nπ periodic solutions (excepting y 0). In our case, if e ρ N,since( e)/2 r(t, e) ( + e)/2, we have ( ) ν 2 ( ) 8 N ( + e) 3 r(t, e) 3 8 ν + 2 ( e) 3. N Now, from Lemma it follows Proposition 0. Proposition 2. The number of zeros of y(t; e) in (0,Nπ) tends to infinity as e. Proof: First, we notice that r(t, e) can be extended to e =. This extension is continuous in both variables and, in particular, r(, e) converges uniformly to r(, ) as e. Also, from u sin u = t (e = ), we deduce that u(t, ) = (6t) /3 a(t), where a is continuous and a(0) =. Next r(t, ) = u(t, ) ( cos u(t, )) = sin2 = 62/ t2/3 b(t), with b continuous and b(0) =. Thus, r(t, ) 3 = 6 9t 2 b(t) 3. Let us fix a number γ in the interval (/4, 6/9). For > 0 small enough, r(t, ) 3 > γ t 2, t (0, ]. Consider the Euler equation ÿ + γ y = 0. The solutions have infinitely many zeros t 2 accumulating at t = 0 because γ > /4. This can be checked by direct integration. Now, given an arbitrary m, we can find δ (0, ) such that the solutions of this Euler equation have at least m + zeros in the interval (δ, ). Since r(,e) converges uniformly to r(, ), it is possible to find e < such that r(t, e) 3 > γ t 2, t [δ, ], e [e, ]. By Sturm comparison theory, y(t, e) will have at least m zeros in [δ, ]. Proof of Theorem 9: Changing equation () to polar coordinates y = ρ cos θ, ẏ = ρ sin θ, ẏ(nπ,e) = 0 becomes equivalent to θ(nπ,e) πz. The angle θ(t, e) satisfies θ = r(t, e) 3 cos2 θ sin 2 θ. Thus, θ(t, e) is decreasing in t. When e is close to, θ(t, e) π/2+zπ for more and more positive t s in [0,Nπ]. This implies that θ(nπ,e)= inf θ(t, e), t [0,Nπ] as e. The function e [0, ) θ(nπ,e) is analytic and lim e θ(nπ,e)=. The numbers E n,n are the solutions of θ(nπ,e) Zπ.

12 2 J. LLIBRE AND R. ORTEGA We need the following two lemmas. 7. The conclusion Lemma 3. Let (z n (t),e n ) be a sequence of solutions of () satisfying ż n (0) = ż n (Nπ)= 0, z n (0) 0, z n (0) 0, e n e 0 <. Then, the number of zeros of z n (t) in [0,Nπ] coincides, for large n, with the number of zeros in the same interval of the nontrivial solutions of ÿ + r(t, e 0 ) 3 y =0. In particular, e 0 = E n,n for some n. Proof: By continuous dependence, z n (t) 0 uniformly in [0,Nπ]. Define v n (t) = z n (t)/z n (0). It satisfies v n + (z n (t) 2 + r(t, e n ) 2 ) 3/2 v n =0, v n (0) =, v n (0) = 0. Again, by continuous dependence v n (t) converges in C [0,Nπ] to the solution y(t) of ÿ + y =0, r(t, e 0 ) 3 y(0) =, ẏ(0) = 0. Since v n (0) = v n (Nπ) = 0, we have ẏ(0) = ẏ(nπ) = 0. Thus, all the zeros of y(t) in [0,Nπ] are in its interior. Since v n y, v n ẏ uniformly, and all the zeros of y(t) are nondegenerate (i.e. y(τ) =0andẏ(τ) 0), for large n we deduce that v n (t) andy(t) have the same number of zeros. Lemma 4. Let {x λ (t)} λ [0,] be a family of functions in C [0,T] satisfying (i) x λ (0) 0, x λ (T ) 0for all λ [0, ]. (ii) The zeros of x λ are nondegenerate (i.e. x λ (t) 2 +ẋ λ (t) 2 > 0 everywhere). (iii) The map (t, λ) [0,T] [0, ] (x λ (t), ẋ λ (t)) is continuous. Then, the number of zeros of x λ in [0,T] is independent of λ. Proof: The lemma follows from the fact that the number of zeros is locally constant. This is easy because functions which are C close and have nondegenerate zeros have the same number of zeros. Proof of Theorem : From the discussions of Section 5 we know that searching for z(t), even and 2Nπ-periodic solution of (), is equivalent to finding a root of the equation F N (z 0,e)=0 (z 0 = z(0)). In this way we are lead to the framework of Section 4 and we shall apply Theorem 7withF = F N. The parameter λ is now the eccentricity and [a, b] =[0, ɛ]. Let us check that (H) and (H2) hold. The first condition follows from Proposition 8 and the discussion after its proof. To deal with (H2) we recall the information obtained in Section 5 about the set Z 0, as given by (9). There is a unique ξ p Z 0 (0, ) with T (ξ p )/2 =Nπ/p. Here we are using p ν. We know from (0) that the index of F N (, 0) at ξ p is different from zero and so (H2) holds for x 0 = ξ p. At this moment it is important to observe that the solution of (5) with z(0) = ξ p, ż(0) = 0, is even, 2Nπ-periodic and has exactly p zeros in [0,Nπ]. From Theorem 7 we infer the existence of a continuous family {(ξ(s),e s )} s [0,] in R [0, ɛ] such that F N (ξ(s),e s )=0, ξ(0) = ξ p,e 0 =0

13 ON THE FAMILIES OF PERIODIC ORBITS OF THE SITNIKOV PROBLEM 3 and either (2) e = ɛ or (3) e =0, ξ() ξ p. Let z s (t) denote the solution of () for e = e s which satisfies z s (0) = ξ(s), ż s (0) = 0. The family {(z s (t),e s )} s [0,] satisfies the conditions () and (2) of Theorem but there are no a priori reasons to suppose that it also satisfies (3). We distinguish two cases: Case : z s (0) > 0 for all s [0, ]. Lemma 4 implies that z s (t) hasp zeros in [0,Nπ]foreachs [0, ]. This proves that also (3) holds. Assuming now that the first alternative (2) holds, we arrive at the searched family satisfying (4.a). The second alternative (3) cannot occur, for otherwise e = 0 and ξ() ξ p. Then, since ξ() belongs to Z 0 (0, ), we deduce that ξ() = ξ q for some q p. This would imply that z (t) hasq zeros in [0,Nπ], a situation which would be incompatible with (3). Case 2: z s (0) vanishes for some s [0, ]. Let σ (0, ] be the first zero of z s (0), so that z s (0) > 0 if s [0,σ), z σ (0) = 0. The family {(ẑ s, ê s )} s [0,),withẑ s = z sσ and ê s = e sσ, satisfies (3). Again Lemma 4 has been used. The definition of σ implies that lim s ê s = e σ [0, ɛ] andẑ s (t) converges to 0 as s. Finally we apply Lemma 3 to conclude that e σ = E n,n for some n. Moreover, non-trivial solutions of () for e = e σ must have p zeros in [0,Nπ]. In this way we have constructed a family satisfying (4.b) and the Theorem is proven. Proof of Theorem 2: Sincer(t, e s ) < everywhere, we can apply the Sturm comparison theory to the equation appearing in () and to ÿ + y = 0, to deduce that the solutions of () must have at least N zeros in the interval [0,Nπ]. So, if p<n statement (4.b) does not hold. Therefore, statement () of Theorem 2 is proved. If e ρ N, Proposition 0 says that equation () has no periodic solutions different from the trivial one. So, statement (2) of Theorem 2 follows. Proof of Theorem 3: We shall prove statement () by contradiction. Assume the existence of σ [0, ) such that z σ (0) = zσ(0) and e σ = e σ. Since we know that ż σ (0) = żσ(0) = 0, by uniqueness, z σ and zσ are the same periodic solution. This solution has periods 2πM and 2πN, havingq zeros in [0,Mπ]andpzeros in [0,Nπ]. Let 2πs be the minimal period of this solution. Then, there are integers m and m 2 such that sm = M and sm 2 = N. Let r be the number of zeros of z σ in [0,sπ]. Therefore, m r = q and m 2 r = p. So, M/q = N/p, a contradiction. Consequently, statement () is proved. Statements (2) and (3) follow from () and the fact that z0 (0) >z 0(0). References [] R. Abraham and J.E. Marsden, Foundations of Mechanics, Benjamin, 978. [2] V.M. Alekseev, Quasirandom dynamical systems I, II, III, Math. USSR Sbornik 5 (968), 73 28; 6 (968), ; 7 (969), 43. [3] E. Belbruno, J. Llibre and M. Ollé, On the families of periodic orbits which bifurcate from the circular Sitnikov motions, Cel. Mech. & Dyn. Sys. 60 (994),

14 4 J. LLIBRE AND R. ORTEGA [4] B. Buffoni and J. Toland, Analytic Theory of Global Bifurcation, Princeton Series in Applied Mathematics, [5] H. Cabral and Z. Xia, Subharmonic solutions in the restricted three body problem, DiscreteContin. Dynam. Systems (995), [6] J. Chazy, Sur l allure finale du mouvement dans le problème des trois corps quand le temps croit indefiniment, Annales de l Ecole Norm. Sup. 3 sr. 39 (922), [7] J. Chazy, Sur l allure finale du mouvement dans le problème des trois corps, J. Math. Pures Appl. 8 (929), [8] J. Chazy, Sur l allure finale du mouvement dans le problème des trois corps, Bull. Astron. 8 (932), [9] E.A. Coddington and N. Levinson, Theory of ordinary differential equations. McGraw-Hill Book Company, Inc., New York Toronto London, 955. [0] M. Corbera and J.Llibre, Periodic orbits of the Sitnikov problem via a Poincaré map, Cel. Mech. & Dyn. Sys. 77 (2000), [] M. Corbera and J.Llibre, On Symmetric Periodic Orbits of the Elliptic Sitnikov Problem Via the Analytic Continuation Method, Contemporary Math. 292 (2002), [2] E.N. Dancer, Global structure of the solutions of non linear real analytic eigenvalue problems, Proc. London Math. Soc. 27 (973), [3] H. Dankowicz and P. Holmes, The Existence of Transverse Homoclinic Points in the Sitnikov Problem, J. Differential Equations 6 (995), [4] R. Dvorak, Numerical results to the Sitnikov problem, Cel. Mech.& Dyn. Sys. 56 (993), [5] J. Hagel, A new analytical approach to the Sitnikov Problem, Cel. Mech. & Dyn. Sys. 53 (992), [6] J. Hagel and T. Trenkler, A computer aided analysis of the Sitnikov problem, Cel. Mech. & Dyn. Sys. 56 (993), [7] Jie Liu and Yi Sui Sun, On the Sitnikov Problem, Cel. Mech. & Dyn. Sys. 49 (990), [8] L. Jiménez Lara and A. Escalona Buendía, Symmetries and bifurcations in the Sitnikov problem, Cel. Mech. & Dyn. Sys. 79 (200), [9] J. Kallrath, R. Dvorak and J. Schlöder, Periodic orbits in the Sitnikov problem, The dynamical behaviour of our planetary system (Ramsau,996), , Kluwer Acad. Publ., Dordrecht, 997. [20] J. Llibre and C. Simó, Estudio cualitativo del problema de Sitnikov, Publicacions Matemàtiques U.A.B. 8 (980), [2] J. Leray and J. Schauder, Topologie et équations fonctionnelles, Ann. Sc. Ec. Nor. Sup. 5 (934), [22] W.D. MacMillan, An integrable case in the restricted problem of three bodies, Astron. J. 27 (93),. [23] W. Magnus and S. Winkler, Hill s equation, Dover, 979. [24] J. Martinez Alfaro and C. Chiralt, Invariant rotational curves in Sitnikov s problem, Cel. Mech. & Dyn. Sys. 55 (993), [25] S. Mathlouthi, Periodic orbits of the restricted three-body problem, Trans. Am. Math. Soc. 350 (998), [26] J. Mawhin, Continuation theorems and periodic solutions of ordinary differential equations, in Topological Methods in Differential Equations and Inclusions (editors, A. Granas, M. Frigon), pp , Kluwer, 995. [27] R. Mennicken, On Ince s equation, Archive for Rat. Mech. and Anal. 29 (968), [28] J. Moser, Stable and random motions in dynamical systems, Annals of Math. Studies 77, Princeton Univ. Press, New Jersey, 973. [29] G. Pavanini, Sopra una nuova categoria di soluzioni periodiche nel problema di tre corpi, Annali di Mathematica, Serie III, Tomo XIII (907). [30] E. Perdios and V.V. Markellos, Stability and bifurcations of Sitnikov motions, Cel. Mech. & Dyn. Sys. 42 (988), [3] P. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Functional Analysis 7 (97), [32] C. Rebelo, F. Zanolin, On the existence and multiplicity of branches of nodal solutions for a class of parameter-dependent Sturm-Liouville problems, Diff. and Integral Equs. 3 (2000),

15 ON THE FAMILIES OF PERIODIC ORBITS OF THE SITNIKOV PROBLEM 5 [33] K.A. Sitnikov, Existence of oscillating motion for the three body problem, Dokl. Akad. Nauk 33 (960), [34] K. Stumpff, Himmelsmeckanik, Band II, VEB, Berlin, 965, pp [35] V. Szebehely, Theory of orbits, Academic Press, New York, 967. [36] K. Wodnar, New formulations of the Sitnikov problem, Predictability, Stability, and Chaos in N Body Dynamical Systems, Edited by A.E. Roy, Plenum Press, New York, 99. [37] K. Wodnar, The original Sitnikov article New insights, Cel. Mech.& Dyn. Sys. 56 (993), [38] K. Wodnar, Analytical approximations for Sitnikov s problem, From Newton to Chaos Edited by A.E. Roy and B.A. Steves, Plenum Press, New York, 995. Departament de Matemàtiques, Universitat Autònoma de Barcelona, 0893 Bellaterra, Barcelona, Spain Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, 807 Granada, Spain. address: jllibre@mat.uab.es, rortega@ugr.es