This is a preprint of: “Periodic motion in non-axially symmetric galaxies”, Jaume Llibre, Claudio Vidal, J. Geom. Phys., vol. 140, 1–9, 2019. DOI: [10.1016/j.geomphys.2018.09.023] PERIODIC MOTION IN NON–AXIALLY SYMMETRIC GALAXIES JAUME LLIBRE1 AND CLAUDIO VIDAL2 Abstract. We apply the averaging theory for proving the existence of twelve families of periodic orbits in a three-dimensional galactic Hamiltonian systems. 1. Introduction and statements of main results In order to study the dynamics of the galaxies we need to have a model describing the properties of their motion. Of course the information for obtaining the dynamical models of the motion of the galaxies comes from observations. In order to simplify the dynamics of their orbits these galactic models usually are axially symmetric or spherical. Spherical models for galaxies were studied in [8, 12, 18]. Moreover, interesting axially symmetric galaxy models were presented and studied in [1, 3, 4, 7, 10, 15, 16, 17, 19]. The potential of our model is 1 1 V (X, Y, Z) = − √ √ + ,. a+b X 2 + Y 2 + (a + b2 + Z 2 )2 In this potential the gravitation constant and the mass of the galaxy is one taking conveniently the unit of mass and of the distance, while a and b are parameters connected to the geometry of the galaxy. When and b ≫ a the model describes an elliptical galaxy with a and b being the scale–lengths of its semiaxes. See for more details about this potential the articles [5, 6, 11, 20]. The main reason for choosing this potential is that despite of the major part of the galaxies are not exactly axially symmetric, the axial symmetry is a good approximation which simplifies the computations for studying the motion of the galaxy. Our main goal is to study analytically the periodic motion of the Hamiltonian system asso- ciated to the Hamiltonian 1 2 1 (1) H= (p + p2Y + p2Z ) − √ √ , 2 X X 2 + Y 2 + (a + b2 + Z 2 ) 2 where a > 0 and b > 0. We omit the term of the potential 1/(a + b) because it does not play any role in the motion. http://www.gsd.uab.cat 2000 Mathematics Subject Classification. Primary 34C10, 34C25. Key words and phrases. periodic orbits, galactic bar model, averaging theory. 1 2 J. LLIBRE AND C.VIDAL The Hamiltonian system associated to the Hamiltonian (1) is Ẋ = pX , Ẏ = pY , Ż = pZ , X (2) p˙X = − √ , [X 2 + Y 2 + (a + b2 + Z 2 )2 ]3/2 Y p˙Y = − √ , [X 2 + Y + (a + b2 + Z 2 )2 ]3/2 2 √ Z(a + b2 + Z 2 )2 p˙Z = − √ √ . b2 + Z 2 [X 2 + Y 2 + (a + b2 + Z 2 )2 ]3/2 As usual the dot denotes derivative with respect to the independent variable t ∈ R, the time. In general the periodic orbits are the most simple orbits after the equilibrium points of a differential system. Their study is interesting because the motion in their neighborhood can be determined by their type of stability. For a given Hamiltonian system usually it is very difficult to study analytically the existence of its periodic orbits and their kind of stability. In order to apply the averaging theory to differential system (2), for studying analytically its periodic √ √ solutions, √ √ we introduce √ √ a small parameter ε by scaling the variables (X, Y, Z, pX , pY , pZ ) = ( εx, εy, εz, εpx , εpy , εpz ). This change of coordinates is canonical, and developing in these new variables the Hamiltonian (1) in Taylor series around ε = 0 we get ( ) 1 2 1 (a + b) 2 H= (px + p2y + p2z )ε + x 2 + y 2 + z − 2 2(a + b)3 b (3) [ ( )2 ] 1 2 2 (a + b) 2 a(a + b)2 4 3 x +y + z + z ε2 + O(ε3 ), 8(a + b)5 b b3 and the Hamiltonian system (2) becomes ẋ = px , ẏ = py , ż = pz , ( ) x 3 (a + b) 2 (4) ṗx = − + x x2 + y 2 + z ε + O(ε2 ), (a + b)3 2(a + b)5 b ( ) y 3 2 2 (a + b) 2 ṗy = − + y x + y + z ε + O(ε2 ), (a + b)3 2(a + b)5 b ( ) z 3b2 x2 + y 2 + (a + b)(a + 3b)z 2 ṗz = − +z ε + O(ε2 ). b(a + b)2 2b3 (a + b)4 We proceed to study of the existence of periodic solutions of system (4). First we analyze the solutions of system (4) with ε = 0, that is, the unperturbed Hamiltonian system x ẋ = pX , ṗX =− , (a + b)3 y (5) ẏ = pY , ṗY =− , (a + b)3 z ż = pZ , ṗZ =− , b(a + b)2 PERIODIC MOTION IN NON–AXIALLY SYMMETRIC GALAXIES 3 The solutions of this unperturbed system, which is a sum of three harmonic oscillators, with arbitrary initial conditions x(0) = x0 , y(0) = y0 , z(0) = z0 , px (0) = px0 , py (0) = py0 , pz (0) = pz0 are 1 x(t) = x0 cos ω1 t + px sin ω1 t, ω1 0 px (t) = px0 cos ω1 t − ω1 x0 sin ω1 t, 1 y(t) = y0 cos ω1 t + py sin ω1 t, (6) ω1 0 py (t) = py0 cos ω1 t − ω1 y0 sin ω1 t, 1 z(t) = z0 cos ω2 t + pz sin ω2 t, ω2 0 pz (t) = pz0 cos ω2 t − ω2 z0 sin ω2 t, √ ω2 a+b where ω1 = (a + b)−3/2 and ω2 = b−1/2 (a + b)−1 . Since = , then for the unperturbed ω1 b system the following statement hold: √ a+b (i) If is rational, then the set of all periodic solutions of system (5) has dimension b 6 contained √ in R6 . a+b (ii) If is irrational, then there are two sets of periodic solutions of system (5) one of b dimension 4 and the other of dimension 2. The first family is contained in the subspace (x, y, 0, px , py , 0) and the second is contained in (0, 0, z, 0, 0, pz ). Now we are in position to state our main results. The second result is generated using the periodic solutions on the subspace (x, y, 0, px , py , 0) and is the following. √ a+b Theorem 1. If is irrational, then for ε > 0 sufficiently small at every level H = b h > 0 the perturbed Hamiltonian systems (4) has at least one periodic solution γ(t, ε) = 3/2 (x(t, √ ε), y(t, ε), z(t, ε), px (t, ε), py (t, ε), pz (t, ε)) with period T = 2π(a + b) such that γ(0, ε) → 3 ( 2(a + b) h, 0, 0, 0, 0, 0) as ε → 0. Details of the proof of Theorem 1 will be given in section 2. √ a+b Theorem 2. If is irrational, then for ε > 0 sufficiently small at every level H = b h > 0 the perturbed Hamiltonian systems (4) has at least one periodic √ solution γ(t, ε) = (x(t, ε), y(t, ε), √ z(t, ε), p x (t, ε), p y (t, ε), p z (t, ε)) with period T = 2π(a + b) b such that γ(0, ε) → (0, 0, (a + b) 2bh, 0, 0, 0) as ε → 0. The proof of Theorem 2 will be given in section 3. Both theorems are proved using the averaging theory described in the appendix. Unfortu- nately this averaging theory does not provide any information about the periodic orbits (i) of the unperturbed system (5) which can be continued to system (4). 2. Proof of Theorem 1 We consider the unperturbed periodic solutions defined in (6) with z = pz0 = 0, that is, we take the periodic solution with four parameters 4 J. LLIBRE AND C.VIDAL ( 1 1 γ0 (t) = x0 cos(ω1 t) + px0 sin(ω1 t), y0 cos(ω1 t) + py sin(ω1 t), 0, ω1 ω1 0 ) px0 cos(ω1 t) − ω1 x0 sin(ω1 t), py0 cos(ω1 t) − ω1 y0 sin(ω1 t), 0 , with period T = 2π/ω1 = 2π(a + b)3/2 which are in the subspace (x, y, 0, px , py , 0). Note that these periodic solutions are not in resonance with the other periodic solutions of system (5) living in the subspace (0, 0, z, 0, 0, pz ) due to the fact that ω1 /ω2 is not a rational number by assumptions. It is clear that the energy level of this family of periodic solutions with initial conditions (x0 , y0 , 0, px0 , py0 , 0) and ε = 0 is 1( 2 2 ) h= ω (x + y02 ) + p2x0 + p2y0 . 2 1 0 We are going to apply Theorem 3 of the appendix at every energy level H = h > 0, this will allow to eliminate one variable, in our case it will be px , and to reduce the study in a space of dimension five. Using the expression of H given in (3) we get √ ( ( ) ) b (a + b)3 p2y + p2z + x2 + y 2 + z 2 (a + b) px = 2h − + b(a + b)3 ( ) ( )2 6b2 z 2 (a + b) x2 + y 2 + z 4 (a + b)2 (a + 3b) + 3b3 x2 + y 2 √ ( ( ) ) ε + O(ε2 ). 3 2 2 2 2 2 b (a + b) 2h − py − p z − x − y − z (a + b) 8b3 (a + b)5 b(a + b)3 We consider the following order of the variables (x, y, py , z, pz ), it is verified that the equations of motion (4) on the energy level H = h > 0 are given by √ ( ( ) ) b (a + b)3 p2y + p2z + x2 + y 2 + z 2 (a + b) ẋ = 2h − + b(a + b)3 ( ) ( )2 6b2 (a + b)z 2 x2 + y 2 + (a + b)2 (a + 3b)z 4 + 3b3 x2 + y 2 √ ( ( ) ) ε + O(ε2 ), 3 2 2 2 2 2 b (a + b) 2h − p y − p z − x − y − z (a + b) 8b3 (a + b)5 b(a + b)3 ẏ = py , ( 2 ) z (a + b) 2 2 3y +x +y y b ṗy = − + ε + O(ε2 ), (a + b)3 2(a + b)5 ż = pz , ( ) z (a + b)(a + 3b)z 3 + 3b2 z x2 + y 2 ṗz = − + ε + O(ε2 ). b(a + b)2 2b3 (a + b)4 This differential system has the normal form of system (8), so it is ready for applying to it the averaging theory described in the appendix, i.e. the function F0 (x, y, py , z, pz ) is (√ ( ( ) ) b (a + b)3 p2y + p2z + x2 + y 2 + z 2 (a + b) y 2h − , py , − + b(a + b)3 (a + b)3 ( 2 ) z (a + b) ( )) 3y + x2 + y 2 b z (a + b)(a + 3b)z 3 + 3b2 z x2 + y 2 , pz , − + , 2(a + b)5 b(a + b)2 2b3 (a + b)4 PERIODIC MOTION IN NON–AXIALLY SYMMETRIC GALAXIES 5 and the function F1 (x, y, py , z, pz ) is ( ( ) ( )2 6b2 (a + b)z 2 x2 + y 2 + (a + b)2 (a + 3b)z 4 + 3b3 x2 + y 2 √ ( ( ) ) , 0, 3 2 2 2 2 2 b (a + b) 2h − py − p z − x − y − z (a + b) 8b3 (a + b)5 b(a + b)3 ( 2 ) z (a + b) ( )) 3y + x2 + y 2 b (a + b)(a + 3b)z 3 + 3b2 z x2 + y 2 , 0, . 2(a + b)5 2b3 (a + b)4 In what follows we will use the notation of Theorem 3 given in the appendix, that is, k = 3, n = 5, ξ(x, y, py , z, pz ) = (x, y, py ), α = (x0 , y0 , py0 ), β(α) = (0, 0). So the initial condition of the unperturbed periodic orbits of system (5) in the energy level H = h > 0 is xα = (x0 , y0 , py0 , 0, 0). For each xα the solution x(t, xα ) equal to ( 1 1 ) x0 cos(ω1 t) + px0 sin(ω1 t), y0 cos(ω1 t) + py0 sin(ω1 t), py0 cos(ω1 t) − ω1 y0 sin(ω1 t), 0, 0 ω1 ω1 2π is T = = 2π(a + b)3/2 -periodic. Let Mxα (t) be the fundamental matrix satisfying Mxα (0) is ω1 the identity matrix, solution of the variational equation (10) along the periodic solution x(t, xα ). Then, Mxα (t) is   x0 y0 1 py0  cos(ω1 t) − ω1 px0 sin(ω1 t) −ω1 px0 sin(ω1 t) − ω1 px0 sin(ω1 t) 0 0     1   0 cos(ω1 t) sin(ω1 t) 0 0   ω1   0 −ω1 sin(ω1 t) cos(ω1 t) 0 0 .    1   0 0 0 cos(ω2 t) sin(ω2 t)   ω2  0 0 0 −ω2 sin(ω1 t) cos(ω1 t) In order to apply Theorem 3 of the appendix we need to compute the matrix Mx−1 α (0) − Mx−1 α (T ), which is   0 0 0 0 0    0 0 0 0 0       0 0 0 0 0   (√ ) ( √ )   √   a+b 2 a+b .  0 0 0 2 sin2 √ π b(a + b) sin √ π   b b     ( √ )   2 a+b   sin √ π (√ )   b a+b  0 0 0 − √ 2 sin2 √ π b(a + b) b It is clear that the upper right corner the 3 × 2 is the ( zero √ matrix, ) and in the lower √ right 2 π a+b b corner we have a 2 × 2 matrix ∆α with det ∆α = 4 sin √ ̸= 0, because √ is an b a+b irrational number. Now we compute the averaged function defined in (11), i.e. (∫ 3/2 ) 2π(a+b) F(x0 , y0 , py0 ) = ξ Mx−1 α (t) F1 (t, x(t, xα )) dt 0 = (f11 , f12 , f13 ), 6 J. LLIBRE AND C.VIDAL and we get 3π ( ( ) ) f11 = √ [ −3px0 (a + b)3 p2x0 + p2y0 − 3px0 x20 − px0 y02 − 2py0 x0 y0 ] 8 a+b 3π ( ( ( ) ) ) f12 = − √ [ py0 3(a + b)3 p2x0 + p2y0 + x20 + 2px0 x0 y0 + 3py0 y02 ] 8 a+b 3π ( ( ( ) ) ) f13 = 7/2 [ y0 (a + b)3 p2x0 + 3p2y0 + 3x20 + 2(a + b)3 px0 py0 x0 + 3y03 ], 8(a + b) √ x2 + y02 where px0 = ± 2h − p2y0 − 0 . Solving F(x0 , y0 , py0 ) = 0, we obtain the two real solu- (a + b)3 √ tions x∗0 = ± 2h(a + b)3 , y0 = py0 = 0. It is verified that the differential DF(x0 , y0 , py0 ) = (dfij ), where    ( )  3π  py0 y0 − x0 3(a + b)3 h − y02  , df11 = − √ √ ( ) 4 a+b  2 3 2 2  − 2h − py0 (a + b) + x 0 + y 0  (a + b)3 − 3  (a + b) √ ( ) 3π − 2h − py0 (a + b)3 + x20 + y02 2 df12 = − √ py0 x0 − 2y0 − 4 a+b (a + b)3  ( )  y0 3(a + b)3 h − y02  , − √ ( ) 2 3 2 2   − 2h − py (a + b) + x 0 + y 0 (a + b)3 − 0 3  (a + b)   ( )  3π  x0 y0 − √ py0 3(a + b)3 h − y02  , df13 = − √  ( )  4 a+b − 2h − p2y0 (a + b)3 + x20 + y02  −  (a + b)3   √  3π  y0 x20 x20 + y02  df21 = − √ − √ − 2py0 x0 + y0 2h − − p2 , 4 a+b 3 y0   x2 + y02 (a + b)  (a + b)3−p2y0 + 2h − 0 (a + b)3 ( 2 2 ) x + y0 3πx0 2h − 0 − p2y0 (a + b)3 df22 = − √ , √ x20 + y02 4 a + b 2h − − p2y0 (a + b)3     3π  py0 x0 y0  df23 = − √ 3ha3 + 9bha2 + 9b2 ha − x20 + 3b3 h − √ ,  4 a+b  x20 + y02 2  2h − − py0 (a + b)3 PERIODIC MOTION IN NON–AXIALLY SYMMETRIC GALAXIES 7    √  3π  x2 + y02 2py0 x20  df31 = 2py0 2h − 0 − p2y0 (a + b)3 + 4x0 y0 − √ , 8(a + b)7/2  (a + b)3   x20 + y02  2h − − p2y0 (a + b)3     3π ( 2 ) py0 x0 y0  df32 =  py + h (a + b)3 + x20 + 3y02 − √ , 4(a + b)7/2  0   x20 + y02  2h − − p2y0 (a + b)3    √  3π  x0 p2y0 x20 + y02  df33 = √ − √ + 2y0 py0 + x0 2h − − p2 . 4 a+b 3 y0   x20 + y02 (a + b)  2h − − p2y0 (a + b)3 It is verified that the determinant (12) is not zero, i.e. 243π 3 h3 x0 (a + b)3/2 det(F(x0 , 0, 0)) = √ . x20 64 2h − (a + b)3 √ Thus the determinant evaluated at x∗0 = ± 2h(a + b)3 , which is different from zero, in fact, is ∞. Since both solutions generate the same periodic orbit, we conclude the proof. 3. Proof of Theorem 2 We consider the unperturbed periodic solutions defined in (6) with x0 = y0 = px0 = py0 = 0, that is, we take the periodic solution with two parameters 1 γ0 (t) = (0, 0, z0 cos ω2 t + pz sin ω2 t, 0, 0, pz0 cos ω2 t − ω2 z0 sin ω2 t), ω2 0 2π and with period T = which is in the plane (0, 0, z, 0, 0, pz ). Note that these periodic ω2 solutions are not in resonance with the other periodic solutions of system (5) living in the subspace (x, y, 0, px , py , 0) due to the fact that ω1 /ω2 is not a rational number. It is clear that the energy level of this family of periodic solutions with ε = 0 is 1( 2 2 ) h= ω2 z0 + p2z0 . 2 Again we apply Theorem 3 of appendix at every energy level H = h > 0, this allows to eliminate one variable, now the variable pz , and we reduce the space of motion to dimension five. Using the expression of H in (3) we get √ ( ( ) ) b (a + b)3 p2x + p2y + x2 + y 2 + z 2 (a + b) pz = 2h − + b(a + b)3 ( ) ( )2 6b2 z 2 (a + b) x2 + y 2 + z 4 (a + b)2 (a + 3b) + 3b3 x2 + y 2 √ ε + O(ε2 ). 8b5/2 (a + b)7/2 b ((a + b)3 (2h − px2 − py2 ) − x2 − y 2 ) − z 2 (a + b) 8 J. LLIBRE AND C.VIDAL We will consider the following order of the variables (z, x, px , y, py ), it is verified that the equations of motion (4) on the energy level H = h are given by √ ( ( ) ) b (a + b)3 p2x + p2y + x2 + y 2 + z 2 (a + b) ż = 2h − + b(a + b)3 ( ) ( )2 6b2 z 2 (a + b) x2 + y 2 + z 4 (a + b)2 (a + 3b) + 3b3 x2 + y 2 √ ( ( ) ) ε + O(ε2 ), 5/2 8b (a + b) 7/2 3 2 2 2 2 b (a + b) 2h − px − py − x − y − z (a + b) 2 (7) ẋ = px , ( ) x 3 2 2 (a + b) 2 ṗx = − + x x + y + z ε + O(ε2 ), (a + b)3 2(a + b)5 b ẏ = py , ( ) y 3 2 2 (a + b) 2 ṗy = − + y x + y + z ε + O(ε2 ). (a + b)3 2(a + b)5 b This differential system has the normal form of system (8) for applying to it the averaging theory of the appendix, where (√ ( ( ) ) b (a + b)3 p2x + p2y + x2 + y 2 + z 2 (a + b) F0 (z, x, px , y, py ) = 2h − 3 , b(a ) + b) x y px , − , py , − , (a + b)3 (a + b)3 and F1 (z, x, px , y, py ) is ( ) ( )2 ( 6b2 z 2 (a + b) x2 + y 2 + z 4 (a + b)2 (a + 3b) + 3b3 x2 + y 2 √ ( ( ) ) , 0, 8b5/2 (a + b)7/2 b (a + b)3 2h − p2x − p2y − x2 − y 2 − z 2 (a + b) ( ) ( ) 3 2 2 (a + b) 2 3 2 2 (a + b) 2 ) x x + y + z , 0, y x + y + z . 2(a + b)5 b 2(a + b)5 b In what follows we will use the notation of Theorem 3 given in the appendix, that is, k = 1, n = 5, ξ(z, x, px , y, py ) = z, α = z0 , β(α) = (0, 0, 0, 0), and zα = (z0 , 0, 0, 0, 0) is the initial condition of a periodic orbit of the unperturbed system (9) corresponding to system (7). For each zα the 1 2π √ solution x(t, zα ) = (z0 cos ω2 t + pz0 sin ω2 t, 0, 0, 0, 0) has period T = = 2π(a + b) b. Let ω2 ω2 Mxα (t) be the fundamental matrix of the variational equation (10) along the periodic solutions x(t, zα ) such that Mxα (0) is the identity matrix, . Then Mxz0 (t) is  z0  cos(ω2 t) − ω2 sin(ω2 t) 0 0 0 0  pz0   1   0 cos(ω1 t) sin(ω1 t) 0 0   ω1   .  0 −ω1 sin(ω1 t) cos(ω1 t) 0 0   1     0 0 0 cos(ω1 t) sin(ω1 t)  ω1 0 0 0 −ω1 sin(ω1 t) cos(ω1 t) PERIODIC MOTION IN NON–AXIALLY SYMMETRIC GALAXIES 9 Next in order to apply Theorem 3 of the appendix we compute the matrix Mx−1 z0 (0) − Mx−1z0 (T ), and we obtain   0 ( 0√ ) 0( 0 0 √ )  bπ 2 bπ   0 2 sin2 √ (a + b)3/2 sin √ 0 0     a+b a+b   ( √ )   2 bπ   sin √ ( √ )     a+b bπ   0 − 2 sin 2 √ 0 0   (a + b)3/2 a+b   ( ) ( √ ) .  √   bπ 2 bπ   0 0 0 2 sin 2 √ 3/2 (a + b) sin √   a+b a+b   ( √ )     2 bπ   sin √ ( √ )   a+b   2 bπ  0 0 0 − 2 sin √ (a + b)3/2 a+b It is clear that the upper right corner the 1 × 4 is the zero ( matrix, and in the lower right √ ) √ π b b corner we have a 4 × 4 matrix ∆z0 with det ∆α = 16 sin4 √ ̸= 0, because √ is a+b a+b an irrational number. Then we have (∫ √ ) 2π(a+b) b −1 F(z0 ) = ξ Mxz0 (t) F1 (t, x(t, xz0 )) dt 0 √ 2 z02 3πh(a + b) (a + 3b) 2h − b(a + b)2 = − √ . 4 b √ Solving F(z0 ) = 0 we obtain the two solutions z0 = z0∗ = ±(a + b) 2bh, for which we have 3πhz0 (a + 3b) pz0 = 0. It is verified that the derivative F ′ (z0 ) = √ , which is different 3/2 z02 4b 2h − b(a + b)2 ∗ from zero at z0 = z0 , in fact, it is ∞. Since both values of z0 generate the same periodic orbit, we conclude the proof. Appendix: Averaging theory of first and second order In this section we recall the averaging theory of first order to find periodic orbits, for more information on the general averaging theory see the book [14]. Consider the differential system (8) ẋ = F0 (t, x) + εF1 (t, x) + ε2 F2 (t, x, ε), with ε > 0 sufficiently small, x ∈ Ω, where Ω is an open subset of Rn , and t ≥ 0. Moreover, we assume that both F0 , F1 and F2 are C 2 functions and T -periodic in the first variable. Let x(t, z) be the solution of the unperturbed system (9) ẋ = F0 (t, x), such that x(0, z) = z. We write the linearization of the unperturbed system along the periodic solution x(t, z) as (10) ẏ(t) = Dx F0 (t, x(t, z))y. 10 J. LLIBRE AND C.VIDAL Now we denote by Mz (t) some fundamental matrix of the linear differential system (10), and by ξ : Rk × Rn−k −→ Rk , the projection of Rn onto its first k coordinates; i.e., ξ(x1 , . . . , xn ) = (x1 , . . . , xk ). Theorem 3 (First Order Averaging Theorem). Let V ⊂ Rk be open and bounded, and let β0 : CL(V ) −→ Rn−k be a C 2 function. We assume that (i) Z = {zα = (α, β0 (α)), α ∈ Cl(V )} ⊂ Ω and that for each zα ∈ Z the solution x(t, zα ) of (9) is T - periodic; (ii) for each zα ∈ Z there is a fundamental matrix Mzα (t) of (10) such that the matrix Mz−1 α (0) − Mz−1α (T ) has in the upper right corner the k × (n − k) zero matrix, and in the lower right corner a (n − k) × (n − k) matrix ∆α with det ∆α ̸= 0. We consider the function F(α) : Cl(V ) → Rk (∫ ) T −1 (11) F(α) = ξ Mzα (t)F1 (t, x(t, zα ))dt . 0 If there exists a ∈ V with F(a) = 0 and (12) det((dF/dα(a)) ̸= 0, then there is a T –periodic solution x(t, ε) of system (8) such that x(0, ε) → a as ε → 0. For an easy proof of Theorem 3 see [2]. In fact the result of Theorem 3 is a classical result due to Malkin [9] and Roseau [13]. Acknowledgments The first author is partially supported by the MINECO grants MTM2013-40998-P and MTM2016-77278-P (FEDER), the AGAUR grant 2014 SGR568, and the projecte MDM-2014- 0445 (BGSMath). The second author is partially supported by Fondecyt project 1130644. References [1] Binney, J.: More dynamical models of our Galaxy. Mon. Not. R. Astron. Soc. 426, 1328–1337 (2012) [2] Buică, A., Françoise, J.P., Llibre J.: Periodic solutions of nonlinear periodic differential systems with a small parameter, Comm. on Pure and Appl. 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