1 PERIODIC ORBITS OF A HAMILTONIAN SYSTEM RELATED WITH THE FRIEDMANN-ROBERTSON-WALKER SYSTEM IN ROTATING COORDINATES CLAUDIO BUZZI1, JAUME LLIBRE2 AND PAULO SANTANA3 Abstract. We provide sufficient conditions on the four parameters of a Hamil- tonian system, related with the Friedmann-Robertson-Walker Hamiltonian system in a rotating reference frame, which guarantee the existence of 12 con- tinuous families of periodic orbits, parameterized by the values of the Hamil- tonian, which born at the equilibrium point localized at the origin of coordi- nates. The main tool for finding analytically these families of periodic orbits is the averaging theory for computing periodic orbits adapted to the Hamilton- ian systems. The technique here used can be applied to arbitrary Hamiltonian systems. 1. Introduction and statement of the main results In astrophysics the study of the dynamics of the universe is an area where the application of the techniques of the dynamical systems provide good results, mainly in galactic dynamics, see the articles [2,13,15,16,21] and the references cited therein. Recently chaotic motion has been detected in the following simplified version of the Friedmann-Robertson-Walker Hamiltonian (1) H = 1 2 (p2Y − p2X) + 1 2 (Y 2 −X2) + b 2 X2Y 2, introduced by Calzeta and Hasi in [6]. In fact this model is too simplified in order to be considered realistic, but it is interesting due to its simplicity and for showing the existence of chaos in cosmology, look for more details in [6]. Hawking [7] and Page [14] used analogous models to analyze the relation between the thermodynamic arrow of time and the cosmology. A large number of potentials in galactic dynamics are of the form V (x2, y2), see the article [17] and the previous mentioned articles on galactic dynamics. These potentials show a reflection symmetry with respect to both axes. Then in [11] was studied the following generalized version of the Calzeta–Hasi’s model (2) H = 1 2 (p2Y − p2X) + 1 2 (Y 2 −X2) + a 4 X4 + b 2 X2Y 2 + c 4 Y 4. Following the classical restricted circular three-body problem in which its dynam- ics is better understand in a rotating frame that in a sideral frame of coordinates, 1Paulo Santana is the corresponding author. 2010 Mathematics Subject Classification. Primary 70H12. Secondary 34C25. Key words and phrases. families of periodic orbits, Hamiltonain systems, generalized Friedmann-Robertson-Walker Hamiltonian, averaging theory. 1 2 C. BUZZI, J. LLIBRE AND P. SANTANA our objective is to study the dynamics of the generalized version of the Calzeta– Hasi’s model (2) in rotating coordinates. More precisely, we consider the following generalized version of the Calzeta–Hasi’s model in rotating coordinates that itself is a simplified version of the Friedmann-Robertson-Walker Hamiltonian (3) H = 1 2 ( y2 − x2 + p2y − p2x ) + 1 4 ( ax4 + 2bx2y2 + cy4 ) − ω (xpy − ypx) , where a, b, c, ω ∈ R and ω > 0. Therefore the corresponding Hamiltonian system is (4) ẋ = ωy − px, ẏ = −ωx+ py, ṗx = x+ ωpy − ax3 − bxy2, ṗy = −y − ωpx − bx2y − cy3. In the qualitative theory of differential equations any orbit or trajectory is home- omorphic either to a straight line, or to a circle, or to a point. The equilibrium points are the orbits homeomorphic to a point and the periodic orbits are the ones homeomorphic to a circle. These two types of orbits are relevant in the study of the dynamics of a differential system, and usually their study is simpler than the study of the orbits homeomorphic to straight lines, that in general exhibit more com- plicate dynamics. Therefore in order to understand the dynamics of a differential system we must start analyzing its equilibrium points and its periodic solutions. The objective of this paper is to study analytically the periodic orbits of the Hamiltonian system (4) in each Hamiltonian level H = h varying h ∈ R. For obtaining the results we shall use the averaging theory for computing periodic so- lutions. We shall give sufficient conditions on the parameters of the Hamiltonian system (4) implying the existence of continuous families of periodic orbits parame- terized by h, and the expression of these families are provided explicitly up to first order in a small parameter. Our main result is the following one. Theorem 1. In section 3 we provide sufficient conditions for the existence of twelve families of periodic orbits of the Hamiltonian system (4) parametrized by the values of the Hamiltonian (3). Six of these families only exist for positive values of the Hamiltonian, two only exist for negative values of the Hamiltonian, and the remain- ing four families can exist either for positive or negative values of the Hamiltonian depending on the values of the parameters a, b and c. All these twelve families are born at the equilibrium point localized at the origin of coordinates of the Hamiltonain system (4). 2. The averaging theory In this section we recall the averaging theory of first order for finding periodic solutions. The averaging theory up to third order specifically for studying periodic orbit was developed in [5]. See this paper for a proof of the result stated in this section. The averaging theory of higher order can be found in [10]. Other versions of the averaging theory can also be found in [4] and in Theorems 11.5 and 11.6 of [20]. For a general view on the averaging theory see the book [18]. PERIODIC ORBITS OF A HAMILTONIAN SYSTEM 3 Theorem 2. Consider the differential system (5) ẋ(t) = εF (t, x) + ε2R(t, x, ε), where F : R×D → Rn, R : R×D × (−εf , εf )→ Rn are continuous functions, T - periodic in the first variable and D is an open subset of Rn. We define f : D → Rn as f(z) = 1 T ∫ T 0 F (s, z) ds, and assume that (i) F and R are locally Lipschitz with respect to x; (ii) for all a ∈ D with f(a) = 0, there exists a neighborhood V of a such that f(z) 6= 0 for all z ∈ V \{a} and dB(f, V, 0) 6= 0 (see its definition later on). Then for |ε| > 0 small enough there exists a T -periodic solution ϕ(·, ε) of system (5) such that ϕ(·, ε)→ a as ε→ 0. We denoted by dB(f, V, 0) the Brouwer degree at the triple (f, V, 0). A sufficient condition for showing that the Brouwer degree is non-zero is that the Jacobian of the function f at a (when it is defined) is non-zero, for a proof see [12]. For more details about the Brouwer degree see [3]. 3. Proof of Theorem 1 To prove Theorem 1 we apply Theorem 2 to the Hamiltonian system (4). Gener- ically the periodic orbits of a Hamiltonian system with more than one degree of freedom are on cylinders fulfilled of periodic orbits, see [1]. Therefore, we cannot apply Theorem 2 directly to system (4), because the Jacobian will always be zero. Then we must apply Theorem 2 at each Hamiltonian fixed level where the periodic orbits generically are isolated. In order to apply Theorem 2 we need a small parameter ε > 0. So in the Hamiltonian system (4) we scaling the variables as follows (6) (x, y, px, py) = √ ε(X,Y, pX , pY ). In these new variables system (4) becomes (7) Ẋ = ωY − pX , Ẏ = −ωX + pY , ṗX = X + ωpY − ε ( aX3 + bXY 2 ) , ṗY = −Y − ωpX − ε ( bX2Y + cY 3 ) . This system is again Hamiltonian with Hamiltonian (8) Y 2 −X2 + p2Y − p2X 2 + ε(aX4 + 2bX2Y 2 + cY 4) 4 − ω (XpY − Y pX) . Therefore for all ε 6= 0 the original and the transformed systems (4) and (7) have essentially the same phase portrait. The linear part of system (7) at the origin of 4 C. BUZZI, J. LLIBRE AND P. SANTANA coordinates is L =  0 ω −1 0 −ω 0 0 1 1 0 0 ω 0 −1 −ω 0  . One can see that L has two eigenvalues of multiplicity two, given by ±i √ 1 + ω2. Therefore we can apply a linear change of variables (X,Y, pX , pY ) to (u, v, pu, pv) such that the new system has the linear part J =  0 √ 1 + ω2 0 0 − √ 1 + ω2 0 0 0 0 0 0 √ 1 + ω2 0 0 − √ 1 + ω2 0  at the origin of coordinates in the real Jordan normal form. A linear change of variables doing this is X = u, Y = pu + √ 1 + ω2v ω , pX = pu, pY = √ 1 + ω2pv − u ω . Therefore the new system becomes (9) u̇ = √ 1 + ω2v, v̇ = − √ 1 + ω2u+ ε aω2u3 + bu ( p2u + 2 √ 1 + ω2puv + ( 1 + ω2 ) v2 ) ω2 √ 1 + ω2 , ṗu = √ 1 + ω2pv − ε ( au3 + bu ( pu + √ 1 + ω2v )2 ω2 ) , ṗv = − √ 1 + ω2pu − ε ( pu + √ 1 + ω2v ) ( bω2u2 + c ( pu + √ 1 + ω2v )2) ω2 √ 1 + ω2 , and the old Hamiltonian becomes the first integral (10) 1 + ω2 2ω2 ( u2 + v2 + p2u + p2v + 2 √ 1 + ω2 (vpu − upv) ) +ε 1 4 ( au4 + 2bu2 ( pu + √ 1 + ω2v )2 ω2 + c ( pu + √ 1 + ω2v )4 ω4 ) . Now we apply a generalized polar change of coordinates given by u = r cos θ, v = r sin θ, pu = ρ cos(θ + φ), pv = ρ sin(θ + φ). We recall that this is a change of variables when r > 0 and ρ > 0. Moreover, doing this change of variables, the angular variables θ and φ appear in the system. Later on the variable θ will be used for obtaining the periodicity necessary for applying the averaging theory. After this change of variables the first integral writes (11) H = 1 + ω2 2ω2 ( r2 + ρ2 − 2rρ √ 1 + ω2 sinφ ) + εW1, where W1 = 1 4 ( ar4 cos4 θ + 2b ω2 r2 cos2 θ W 2 2 + c ω4 W 4 2 ) , W2 = ρ cos(θ + φ) + r √ 1 + ω2 sin θ. PERIODIC ORBITS OF A HAMILTONIAN SYSTEM 5 In the new variables system (9) writes (12) ṙ = ε r cos θ sin θ ω2 √ 1 + ω2 W3, θ̇ = − √ 1 + ω2 + ε cos2 θ ω2 √ 1 + ω2 W3, ρ̇ = −ε ( r cos θ cos(θ + φ)W5 + 1√ 1 + ω2 W6 ) , φ̇ = ε ( r cos θ ρ W7 − cos2 θ ω2 √ 1 + ω2 W3 − cos(θ + φ) ρ √ 1 + ω2 W8 ) , where W3 = bρ2 cos2(θ + φ) + aω2r2 cos2 θ + b r sin θ W4, W4 = 2ρ √ 1 + ω2 cos(θ + φ) + r(1 + ω2) sin θ, W5 = ar2 cos2 θ + b ω2 W 2 2 , W6 = W2 ( br2 cos2 θ + c ω2 W 2 2 ) sin(θ + φ), W7 = ( ar2 cos2 θ + b ω2 W 2 2 ) sin(θ + φ), W8 = W2 ( br2 cos2 θ + c ω2 W 2 2 ) . In order to apply the averaging theory we take θ as the new independent vari- able, and denote by a prime the derivative with respect to θ. With this change of independent variable system (12) goes over to (13) r′ = −εr cos θ sin θ ω2(1 + ω2) W3 +O(ε2), ρ′ = ε√ 1 + ω2 ( r cos θ cos(θ + φ)W5 + 1√ 1 + ω2 W6 ) +O(ε2), φ′ = −ε√ 1 + ω2 ( r cos θ ρ W7 − cos2 θ ω2 √ 1 + ω2 W3 − cos(θ + φ) ρ √ 1 + ω2 W8 ) +O(ε2). This system has only three equations because we do not need the θ̇ equation of (12). Observe that system (13) is 2π-periodic in the variable θ. To apply Theorem 2 we must fix the value of the first integral at h ∈ R. By solving equation (11) in ρ we obtain ρ = r √ 1 + ω2 sinφ+ √ 2hω2 − r2(1 + ω2) + r2(1 + ω2)2 sin2 φ 1 + ω2 +O(ε). Substituting ρ into equations (13) we obtain the two differential equations (14) r′ = −εr cos θ sin θ ω2(1 + ω2) W 3 +O(ε2), φ′ = −ε√ 1 + ω2 ( r cos θ ρ W 7 − cos2 θ ω2 √ 1 + ω2 W 3 − cos(θ + φ) ρ √ 1 + ω2 W 8 ) +O(ε2), 6 C. BUZZI, J. LLIBRE AND P. SANTANA where W i = Wi(θ, r, ρ(r, φ), φ) with (15) ρ(r, φ) = r √ 1 + ω2 sinφ+ √ 2hω2 − r2(1 + ω2) + r2(1 + ω2)2 sin2 φ 1 + ω2 . We observe that in order to apply the first order averaging theory it is not necessary to have information about the terms in O(ε2). One can now see that system (14) satisfies the assumptions of Theorem 2, and it has the form (5) with T = π and F = (F1, F2) analytic where F1 = −r cos θ sin θ ω2(1 + ω2) W 3, F2 = − 1√ 1 + ω2 ( r cos θ ρ W 7 − cos2 θ ω2 √ 1 + ω2 W 3 − cos(θ + φ) ρ √ 1 + ω2 W 8 ) . The averaging function of first order is f(r, φ) = (f1(r, φ), f2(r, φ)) = 1 π ∫ π 0 (F1(θ, r, φ), F2(θ, r, φ)) dθ, becomes (16) f1(r, φ) = br cosφ ρ(r, φ) ( sinφ ρ(r, φ)− r √ 1 + ω2 ) 4ω2(1 + ω2) , f2(r, φ) = −Ar 3 sinφ+Br2ρ(r, φ) + Cr sinφρ(r, φ)2 +Dρ(r, φ)3 8ω2 √ 1 + ω2ρ(r, φ) , where A = (1 + ω2) ( b+ 2bω2 + 3 ( c+ (a+ c)ω2 )) , B = − √ 1 + ω2(b+ 6c+ 3(a+ b+ 2c)ω2 − ( 2b+ 3c+ (b+ 3c)ω2 ) cos(2φ)), C = 3(1 + ω2)(b+ 3c), D = − √ 1 + ω2(2b+ 3c+ b cos(2φ)). According with Theorem 2 we must find the zeros (r0, φ0) of the function f = (f1, f2) and check that the Jacobian determinant of f at these points is not zero. From f1(r, φ) = 0 we obtain r = r(φ), and in order that ρ(r(φ), φ) 6= 0 (otherwise f2(r, φ) is not defined), we get that r(φ) =  R0 with h > 0, R1 with h > 0 and sinφ > 0, R2(φ) with h(1 + ω2 − (1 + 2ω2) sin2 φ) > 0, where R0 = 0, R1 = √ 2hω2 1 + ω2 , R2(φ) = √ 2hω2 (1 + ω2)(1 + ω2 − (1 + 2ω2) sin2 φ) sinφ. PERIODIC ORBITS OF A HAMILTONIAN SYSTEM 7 Substituting r = R0 into f2(r, φ) = 0, and solving with respect to φ we obtain the following two zeros of the averaged function f(r, φ) (r1, φ1) = ( 0, arccos √ −b+ 3c 2b ) , (r2, φ2) = ( 0,− arccos √ −b+ 3c 2b ) . Since the value of ρ(ri, φi) = ω √ 2h (ω2 + 1) 1 + ω2 for i = 1, 2, and the determinant of the Jacobian matrix of f at these two zeros is 3h2(b+ c)(b+ 3c) 8 (1 + ω2) 4 , it follows from the averaging theory (Theorem 2) that if h > 0, 0 < −b+ 3c 2b ≤ 1, and (b+ c)(b+ 3c) 6= 0, then the zeros (ri, φi) provide two periodic solutions of the differential system (14), and consequently of the Hamiltonian system (4) in every level H = h > 0. Substituting r = R1 into f2(r, φ) = 0, and solving with respect to φ we obtain the following six zeros of the averaged function f(r, φ) (r3, φ3) = (√ 2hω2 1+ω2 , π ) , (r4, φ4) = (√ 2hω2 1+ω2 , 0 ) , (r5, φ5) = (√ 2hω2 1+ω2 ,− arccos √ b(4ω2+3)− √ b(12(ω2+1)(ω2(a+c)+c)+b(24ω4+36ω2+13)) 8b(1+ω2) ) , (r6, φ6) = (√ 2hω2 1+ω2 , arccos √ b(4ω2+3)− √ b(12(ω2+1)(ω2(a+c)+c)+b(24ω4+36ω2+13)) 8b(1+ω2) ) , (r7, φ7) = (√ 2hω2 1+ω2 ,− arccos √ b(4ω2+3)+ √ b(12(ω2+1)(ω2(a+c)+c)+b(24ω4+36ω2+13)) 8b(1+ω2) ) , (r8, φ8) = (√ 2hω2 1+ω2 , arccos √ b(4ω2+3)+ √ b(12(ω2+1)(ω2(a+c)+c)+b(24ω4+36ω2+13)) 8b(1+ω2) ) . Since the value of ρ(r3, φ3) = ρ(r4, φ4) = 0, ρ(r5, φ5) = ρ(r6, φ6) = ω √ h(b(4ω2+5)+ √ b(12(ω2+1)(ω2(a+c)+c)+b(24ω4+36ω2+13))) b(1+ω2) , ρ(r7, φ7) = ρ(r8, φ8) = ω √ h(b(4ω2+5)− √ b(12(ω2+1)(ω2(a+c)+c)+b(24ω4+36ω2+13))) b(1+ω2) . 8 C. BUZZI, J. LLIBRE AND P. SANTANA Since ρ(ri, φi) cannot be zero, otherwise f2 is not defined, for the zeros ρ(ri, φi) with i = 3, 4 the averaging theory does not provide any information about if these zeros produce or not periodic solutions of the differential system (14). The determinant D(r, φ) of the Jacobian matrix of f at the other four zeros is D(r5, φ5) = D(r6, φ6) = h7/2ω7(ω2+1) 6 sin(2φ)(bF+eD)(2A √ hω(ω2+1)− √ 2BC) 8b2 , D(r7, φ7) = D(r8, φ8) = h7/2ω7(ω2+1) 6 sin(2φ)(bF−eD)(2A √ hω(ω2+1)− √ 2BC) 8b2 , where A = √ −2ω4(3a+ 2b+ 3c)− 2ω2(3a+ b+ 6c) + b− 6c−D b (ω2 + 1) 2 , B = √ hω2 ( 1 + ω2 ) ( 4bω2 + 5b+D ) b , C = √ b ( 4ω2 + 3 ) −D b (1 + ω2) , D = √ b (12 (ω2 + 1) (ω2(a+ c) + c) + b (24ω4 + 36ω2 + 13)), E = 3ω4 ( 15ab− 9ac+ 34b2 − 15bc− 9c2 ) + ω2 ( 42ab− 27ac+ 145b2 − 66bc− 54c2 ) +9(b− c)(5b+ 3c), F = 18ω6(3a+ 14b− 9c)(a+ 2b+ c) + ω4(54a2 + 579ab− 225ac+ 1082b2 −147bc− 495c2) + 3ω2 ( 72ab− 39ac+ 247b2 − 24bc− 168c2 ) +3(b− c)(55b+ 57c). From Theorem 2 if for i = 5, 6 we have that 0 ≤ b(4ω2 + 3)− √ b (12 (ω2 + 1) (ω2(a+ c) + c) + b (24ω4 + 36ω2 + 13)) b(1 + ω2) ≤ 1, b(4ω2 + 5) + √ b (12 (ω2 + 1) (ω2(a+ c) + c) + b (24ω4 + 36ω2 + 13)) b(1 + ω2) ≥ 0, h > 0, sinφi > 0, ρ(ri, φi) > 0 and D(ri, φi) 6= 0, then these two zeros (ri, φi) provide two periodic solutions of the differential system (14), and consequently of the Hamiltonian system (4) in every level H = h > 0. From Theorem 2 if for i = 7, 8 we have that 0 ≤ b(4ω2 + 3) + √ b (12 (ω2 + 1) (ω2(a+ c) + c) + b (24ω4 + 36ω2 + 13)) b(1 + ω2) ≤ 1, b(4ω2 + 5)− √ b (12 (ω2 + 1) (ω2(a+ c) + c) + b (24ω4 + 36ω2 + 13)) b(1 + ω2) ≥ 0, h > 0, sinφi > 0, ρ(ri, φi) > 0 and D(ri, φi) 6= 0, then these two zeros (ri, φi) provide two periodic solutions of the differential system (14), and consequently of the Hamiltonian system (4) in every level H = h > 0. PERIODIC ORBITS OF A HAMILTONIAN SYSTEM 9 Substituting r = R2 into f2(r, φ) = 0, and solving with respect to φ we obtain the following φ = ± arccos (√ −(a+ b)ω2 b+ c+ (c− a)ω2 ) . Substituting these values of φ into R2 we get the following two zeros of the averaged function f(r, φ) (r9, φ9) = (√ −2(b+ c)h (a+ 2b+ c)(1 + ω2) ,− arccos √ −(a+ b)ω2 b+ c+ (c− a)ω2 ) , (r10, φ10) = (√ −2(b+ c)h (a+ 2b+ c)(1 + ω2) , arccos √ −(a+ b)ω2 b+ c+ (c− a)ω2 ) . We cannot guarantee that these last two solutions are all the solutions for r = R2, these are the ones that we can obtain explicitly. Since the value of ρ(ri, φi) = ( ω2|a+ b|+ (1 + ω2)(b+ c) )√ −2h (1 + ω2) (a+ 2b+ c)(b+ c+ (c− a)ω2) , for i = 9, 10, and we denote the determinant of the Jacobian matrix of f at these two zeros by D(ri, φi), we do not give its huge expression here. It follows from the averaging theory (Theorem 2) that if 0 ≤ −(a+ b)ω2 b+ c+ (c− a)ω2 ≤ 1, −2(b+ c)h a+ 2b+ c > 0, ρ(ri, φi) > 0, and D(ri, φi) 6= 0, then the zeros (ri, φi) for i = 9, 10 provide two periodic solutions of the differential system (14), and consequently of the Hamiltonian system (4) in every level H = h. From f1(r, φ) = 0 we obtain φ = φ(r), and in order that ρ(r(φ), φ) 6= 0 (otherwise f2(r, φ) is not defined), we get that φ(r) =  Φ1 with h < 0, Φ2(φ) with (b+ 3c)h (3a+ 2b+ 3c) < 0, where Φ1 = ±π 2 , Φ2 = ± arcsin ( r(1 + ω2)√ r2(1 + 3ω2 + 2ω4) + 2hω2 ) . 10 C. BUZZI, J. LLIBRE AND P. SANTANA Substituting φ = Φ1 into f2(r, φ) = 0, and solving with respect to r we obtain the following four zeros of the averaged function f(r, φ) (r11, φ11) = (√ − 2h 1 + ω2 ,−π 2 ) , (r12, φ12) = (√ − 2h 1 + ω2 , π 2 ) , (r13, φ13) = (√ − 2(b+ 3c)h (3a+ 2b+ 3c)(1 + ω2) ,−π 2 ) , (r14, φ14) = (√ − 2(b+ 3c)h (3a+ 2b+ 3c)(1 + ω2) ,−π 2 ) . Since the value of ρ(ri, φi) = √ −2h for i = 11, 12, ρ(r13, φ13) = √ 2h(3a+ b)ω2 (1 + ω2)(3a+ 2b+ 3c) + √ − 2h(b+ 3c) 3a+ 2b+ 3c , ρ(r14, φ14) = √ 2h(3a+ b)ω2 (1 + ω2)(3a+ 2b+ 3c) − √ − 2h(b+ 3c) 3a+ 2b+ 3c , and the determinant of the Jacobian matrix of f at these four zeros is D(ri, φi) = bh2(3a+ b) 8 (1 + ω2) 4 for i = 11, 12, D(ri, φi) = − bh2(3a+ b)(b+ 3c) 4(1 + ω2)4(3a+ 2b+ 3c) for i = 13, 14. Again from Theorem 2 we obtain that h < 0, and D(ri, φi) 6= 0, then the two zeros (ri, φi) for i = 11, 12 provide two periodic solutions of the differential system (14), and consequently of the Hamiltonian system (4) in every level H = h < 0. Also from Theorem 2 we get that ri > 0, ρi > 0, D(ri, φi) 6= 0, then the two zeros (ri, φi) for i = 13, 14 provide two periodic solutions of the differential system (14), and consequently of the Hamiltonian system (4) in every level H = h. Substituting φ = Φ2 into f2(r, φ) = 0, and solving with respect to r we obtain again the solutions (ri, φi) for i = 9, 10, 11, 12. Again we cannot guarantee that these last four solutions are all the solutions for φ = Φ2, because these four solutions are the ones that we can obtain explicitly. For i = 1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 and according with Theorem 2 the zero (ri, φi) provides a periodic solution (r̄i(θ, ε), φ̄i(θ, ε)) of the differential system (14) PERIODIC ORBITS OF A HAMILTONIAN SYSTEM 11 such that (r̄i(0, ε), φ̄i(0, ε))→ (ri, φi) when ε→ 0. Going back to the differential system (13) we obtain for this system a periodic solution (r̄i(θ, ε), ρ̄i(θ, ε), φ̄i(θ, ε)) such that (r̄i(0, ε), ρ̄i(0, ε), φ̄i(0, ε))→ (ri, ρi, φi) when ε→ 0, where ρi = ρ(ri, φi). Now going back to the differential system (12) we get for this system a periodic solution (r̄i(t, ε), θ̄(t, ε), ρ̄i(t, ε), φ̄i(t, ε)) such that (r̄i(0, ε), θ̄(0, ε), ρ̄i(0, ε), φ̄i(0, ε))→ (ri, 0, ρi, φi) when ε→ 0. Again going back to the differential system (9) we have for this system a periodic solution (ū(t, ε), v̄(t, ε), p̄u(t, ε), p̄v(t, ε)) such that (ū(0, ε), v̄(0, ε), p̄u(0, ε), p̄v(0, ε))→ (ri, 0, ρi cosφi, ρi sinφi) when ε→ 0. Going back to the Hamiltonian system (7) we have for this system a periodic solu- tion (X̄(t, ε), Ȳ (t, ε), p̄X(t, ε), p̄Y (t, ε)) such that (X̄(0, ε), Ȳ (0, ε), p̄X(0, ε), p̄Y (0, ε))→ ( ri, ρi cosφi ω , ρi cosφi, √ 1 + ω2ρi sinφi − ri ω ) when ε → 0. Finally going back to the Hamiltonian system (4) we have for this system a periodic solution (x̄(t, ε), ȳ(t, ε), p̄x(t, ε), p̄y(t, ε)) such that (x̄(0, ε), ȳ(0, ε), p̄x(0, ε), p̄y(0, ε))→ √ ε ( ri, ρi cosφi ω , ρi cosφi, √ 1 + ω2ρi sinφi − ri ω ) → (0, 0, 0, 0), when ε → 0. In summary, these 12 families of periodic orbits of the Hamiltonian system (4) born at the equilibrium localized at the origin of coordinates. This completes the proof of Theorem 1. 4. Conclusions The study of the families of periodic orbits of the Hamiltonian systems is a classical problem, see for instance the hundreds of papers dedicated to study the families of periodic orbirs of the restricted three-body problem in the references of the three books [8,9,19]. Usually these studies are done numerically. Here we study the families of periodic orbits of a Hamiltonian system related with the Friedmann- Robertson-Walker system. We show what changes of variables are necessary in order that we can apply to a Hamiltonian system the averaging theory for studying analytically their periodic orbits. Our main results can be summarized as follows. Under different conditions on the parameters of the Hamiltonian system (4) we have found analytically 12 families of periodic orbits. These families are associated to the given zeros (ri, φi) of the averaged function f(r, φ) = (f1(r, φ), f2(r, φ)) given in (16) for i = 1, 2, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14. The explicit expressions of these families are given in section 3. We remark that the twelve families of the periodic orbits associated to the zeros (ri, φi) of the averaged function with i = 1, 2, 5, 6, 7, 8 only exist in the levels H = h > 0, with i = 11, 12 only exist in the levels H = h < 0, while with i = 9, 10, 13, 14 12 C. BUZZI, J. LLIBRE AND P. SANTANA can exist either in the levels H = h > 0, or H = h < 0, depending on the values of the parameters a, b and c. Acknowledgements We thank to the reviewers their comments and suggestions which help us to improve the presentation of this paper. The first author is partially supported by São Paulo Research Foundation (FAPESP) grant 2019/10269-3, CAPES grant 88881.068462/2014-01 and CNPq grant 304798/2019- 3. 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Belmonte, Quantitative predictions with detuned normal forms, Celest. Mech. Dyn. Astr. 102 (2008), 163–176. [18] J. A. Sanders, F. Verhultst and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems, Applied Mathematical Sci. 59, Springer Verlag, New York, rev. ed. 2007. PERIODIC ORBITS OF A HAMILTONIAN SYSTEM 13 [19] V. Szebehely, Theory of orbits. The restricted problem of three bodies, Academic Press, New York, 1968. [20] F. Verhulst, Nonlinear differential equations and dynamical systems, Universitext, Springer Verlag (1996). [21] H.S. Zhao, C.M. Carollo and T. De Zeeuw, Can galactic nuclei be non-axisymmetric? The parameter space of power-law discs, Monthly Notices of the Royal Astronomical Society 304 (1999), 457–464. 1 IBILCE–UNESP, CEP 15054–000, S. J. Rio Preto, São Paulo, Brazil Email address: claudio.buzzi@unesp.br 2 Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain Email address: jllibre@mat.uab.cat 3 IBILCE–UNESP, CEP 15054–000, S. J. Rio Preto, São Paulo, Brazil Email address: paulo.santana@unesp.br