This is a preprint of: “Global dynamics of the integrable Armbruster-Guckenheimer-Kim galactic potential”, Jaume Llibre, Clàudia Valls, Astrophys. Space Sci., vol. 364(130), 1–6, 2019. DOI: [10.1007/s10509-019-3624-y] GLOBAL DYNAMICS OF THE INTEGRABLE ARMBRUSTER-GUCKENHEIMER-KIM GALACTIC POTENTIAL JAUME LLIBRE AND CLAUDIA VALLS Abstract. We study the global dynamics of the completely inte- grable Armbruster-Guckenheimer-Kim galactic potential. In these cases this system has two first integrals H1 and H2 independent and in involution. Let Ih1 and Ih2 be the set of points of the phase space on which H1 and H2 take the values h1 and h2 , respectively. The sets Ih1 h2 = Ih1 ∩ Ih2 are invariant by the dynamics. We char- acterize the global flow on these sets and we describe the foliation of the phase space by the invariant sets Ih1 h2 . 1. Introduction The Armsbruster-Guckenheimer-Kim potential is a galactic potential introduced in [2] that studies the dynamics for the interchanging of nearly nondegenerate modes with square symmetry. They derived the model starting with a normal form given by a system of differential equations which represented the codimension two bifurcation problem. More precisely, the Hamiltonian function that they provided is 1 1 a b H(x, px , y, py ) = (p2x + p2y ) + (x2 + y 2 ) − (x2 + y 2 )2 − x2 y 2 , 2 2 4 2 where a, b are arbitrary constants. If we add the term −ω(xpy − ypx ) then the system describes the dynamics of rotation of a nearly ax- isymetric galaxy rotating with a constant velocity ω around a fixed axis. The existence of such ω denotes that the rotation of the galaxy must be taken into account when we study the stellar orbits (see [8]). Many studies concerning the integrability and non-integrability of such systems have been done (see for instance [1, 4, 5]) using different tech- niques such as the Painlevé analysis and the Morales-Ramis theory as well as the study of the existence of periodic orbits which was done in [7]. In particular, it was proved in [5] that if b = 2a or b = −a the Key words and phrases. Armbruster-Guckenheimer-Kim galactic potential, in- variant manifolds, complete integrability. 1 2 JAUME LLIBRE AND CLAUDIA VALLS system is completely integrable but the authors do not describe com- pletely the dynamics of the integrable systems form the point of view of the Liouville-Arnold theorem (see section 2). This is the main aim of this paper. When b = 2a the Hamiltonian has the form 1 1 a H = (p2x + p2y ) + (x2 + y 2 ) − (x2 + y 2 )2 − ax2 y 2 . 2 2 4 Introducing the new variables 1 1 1 1 u = √ (x − y), v = √ (x + y), pu = √ (px − py ), pv = √ (px + py ), 2 2 2 2 it can be written as 1 a 1 H(x, px , y, py ) = (p2x + p2y ) + (x4 + y 4 ) − (x2 + y 2 ) 2 4 2 =H̃1 (x, px ) + H̃2 (y, py ), where a ∈ R, we have renamed the variables (u, v) again as (x, y) and 1 a 1 1 a 1 H̃1 (x, px ) = p2x + x4 − x2 , H̃2 (y, py ) = p2y + y 4 − y 2 . 2 4 2 2 4 2 Note that H̃i : R2 → R while H : R4 → R. In all the paper we will de- note by H the Hamiltonian associated to a system with two degrees of freedom and so H = H(x, px , y, py ) : R4 → R, Hi = Hi (x, px , y, py ) : R4 → R for i = 1, . . . , 4, and we will denote by H̃ the Hamiltonian associated to a system with one degree of freedom and so H̃1 = H̃1 (x, px ) : R2 → R and H̃2 = H̃2 (y, py ) : R2 → R. We observe thta H1 and H2 are two first integrals, independent and in involution. Hence, the Hamiltonian system associated to the Hamil- tonian H is (1) ẋ = px , ẏ = py , ṗx = −ax3 + x, ṗy = −ay 3 + y and it is completely integrable. We recall that H1 and H2 are indepen- dent if the matrix   H1x H1px H1y H1px H2x H2px H2y H2px has rank 2 in any point of R4 except, perhaps in a zero Lebesgue- measure set. As usual Hiy = ∂Hi /∂y. Moreover, we say that H1 and H2 are in involution if their Poisson bracket is zero. Finally, a Hamiltonian system with two degrees of freedom is completely integrable if it has two independent first integrals in involution. ARMBRUSTER-GUCKENHEIMER-KIM GALACTIC POTENTIAL 3 Note that the phase space of system (1) is R4 . Since H1 and H2 are fist integrals the sets Ih1 ={(x, px , y, py ) ∈ R4 : H1 = h1 } = {(x, px ) ∈ R2 : H̃1 = h̃1 } × R2 =Ih̃1 × R2 , Ih2 ={(x, px , y, py ) ∈ R4 : H2 = h2 } = {(y, py ) ∈ R2 : H̃2 = h̃2 } × R2 =R2 × Ih̃2 , as well as Ih1 h2 = {(x, px , y, py ) ∈ R4 : H1 = h1 , H2 = h2 } = {(x, px , y, py ) ∈ R4 : H1 = h1 } ∩ {(x, px , y, py ) ∈ R4 : H2 = h2 }   = Ih1 ∩ Ih2 = Ih̃1 × R2 ∩ R2 × Ih̃2 = Ih̃1 × Ih̃2 are invariant by the flow of the Hamiltonian system (1). The first objective of this paper is to describe the foliations of the phase space R4 by the invariant sets Ihi for i = 1, 2 as well as by Ih1 h2 . The foliations provide a good description of the phase portraits of the Hamiltonian flow (1) when a varies. When b = −a the Hamiltonian has the form 1 a 1 H(x, px , y, py ) = (p2x + p2y ) − (x4 + y 4 ) + (x2 + y 2 ) 2 4 2 =H̃3 (x, px ) + H̃4 (y, py ), where a ∈ R with 1 a 1 1 a 1 H̃3 (x, px ) = p2x − x4 + x2 , H̃4 (y, py ) = p2y − y 4 + y 2 . 2 4 2 2 4 2 Note that H3 and H4 are two first integrals, independent and in involution. Hence the Hamiltonian system (2) ẋ = px , ẏ = py , ṗx = ax3 − x, ṗy = ay 3 − y is completely integrable. The sets Ih3 ={(x, px , y, py ) ∈ R4 : H3 = h3 } = Ih̃3 × R2 , Ih4 ={(x, px , y, py ) ∈ R4 : H4 = h4 } = R2 × Ih̃3 , as well as Ih3 h4 = {(x, px , y, py ) ∈ R4 : H3 = h3 , H4 = h4 } = Ih3 ∩ Ih4 = Ih̃3 × Ih̃4 are invariant by the flow of the Hamiltonian system (2). The second main objective of the paper is to describe the foliations of R4 by the invariant sets Ihi for i = 3, 4 and by the invariant sets Ih3 h4 . Again, 4 JAUME LLIBRE AND CLAUDIA VALLS these foliations provide a good description of the phase portraits of the Hamiltonian flow (2) when a varies. The paper is organized as follows. In section 2 we recall the Liouville- Arnold theory for Hamiltonians systems with two degrees of freedom. In section 3 we describe the topology of the sets Ih1 (since the study for Ih2 is analogous). For doing that and taking into account that Ih1 = Ih̃1 × R2 we will only describe the topology of the sets Ih̃1 by computing the sets of singular points and critical values for H̃1 and the Hill regions according to the different values of a and h̃1 . In section 4 we study the topology of the sets Ih1 h2 . In section 5 we describe the topology of the sets Ih3 (again because the study for Ih4 is analogous) and recalling that Ih3 = Ih̃3 × R2 we will only describe the topology of the sets Ih̃3 by computing the sets of singular points and critical values for H̃3 and the Hill regions according to the different values of a and h̃3 . In section 6 we study the topology of the sets Ih3 h4 . 2. Integrable Hamiltonian systems In this section we recall the Liouville-Arnold theorem for the inte- grable Hamiltonian systems with two degrees of freedom. We recall that a flow defined on the phase space R4 is complete if its solutions are defined for all time t in R. Theorem 1. The Hamiltonian system (1) (resp. system (2)) defined on the phase space R4 has the Hamiltonians H1 and H2 (resp. H3 and H4 ) as two independent first integrals in involution. If Ih1 h2 6= ∅ (resp. Ih3 h4 6= ∅) and (h1 , h2 ) (resp. (h3 , h4 )) is a regular value of the map (H1 , H2 ) (resp. (H3 , H4 )) then the following statements hold. (a) Ih1 h2 (resp. Ih3 h4 ) is a two-dimensional submanifold of R4 in- variant under the flow of system (1) (resp. system (2)). (b) If the flow on a connected component Ih∗1 h2 (resp. Ih∗3 h4 ) of Ih1 h2 (resp. Ih3 h4 ) is complete, then Ih∗1 h2 (resp. Ih∗3 h4 ) is diffeomor- phic either to the torus S1 × S1 , to the cylinder S1 × R, or to the plane R2 (c) Under the assumption of statement (b), the flow on Ih∗1 h2 (resp. on Ih∗3 h4 ) is conjugated to a linear flow either on S1 × S1 , or on S1 × R, or on R2 . Note that Theorem 1 does not provide information on the topology of the invariant sets Ih1 h2 (resp. Ih3 h4 ) when (h1 h2 ) (resp. (h3 h4 )) is ARMBRUSTER-GUCKENHEIMER-KIM GALACTIC POTENTIAL 5 not a regular value of the map (H1 , H2 ) (resp. (H3 , H4 )), or how the energy levels Ih1 or Ih2 (resp. Ih3 or Ih4 ) foliate R4 . In this paper we solve these problems for systems (1) and (2). 3. The topology of the invariant sets Ih1 As explained in the introduction, taking into account that Ih1 = Ih̃1 × R2 we will restrict all the study to Ih̃1 . A point (x, px ) ∈ R2 is a singular point for the map H̃1 if it is a solution of ∂ H̃1 ∂ H̃1 = 0, = 0. ∂px ∂x The value h̃1 ∈ R is a critical value for the map H̃1 if there is some singular point belonging to H̃1−1 (h̃1 ) = Ih̃1 . If h̃1 is not critical value it is said a regular value. It is well-known that if h̃1 is a regular value of the map H̃1 then Ih̃1 is a one-dimensional manifold (see [6]). Note that the singular points for the map H̃1 are px = 0, x(ax2 − 1) = 0, and so √ the set of singular √ points of H̃1 is (0, 0) if a ≤ 0, and (0, 0) ∪ (0, −1/ a) ∪ (0, 1/ a) if a > 0. We define the Hill region as n a 4 x2 o Rh̃1 = x ∈ R : x − ≤ h̃1 4 2 This is the region of the configuration space {x ∈ R} where the motion of all orbits of the Hamiltonian system associated to H̃1 having energy h̃1 takes place. By Rh̃1 ≈ S, we denote that Rh̃1 is diffeomorphic to S. We will also denote by s p s p 1 − 1 + 4ah̃1 1 + 1 + 4ah̃1 P− = , P+ = a a have: (i) Rh̃1 ≈ R if a = 0 and h̃1 > 0, (ii) Rh̃1 ≈ R but here {0}, which is a singular point for H̃1 , is in the boundary ofpthe Hill region, p if a = 0 and h̃1 = 0, (iii) Rh̃1 ≈ (−∞, − −2h̃1 ] ∪ [ −2h̃1 , ∞) if a = 0 and h̃1 < 0, (iv) Rh̃1 ≈ R if a < 0 and h̃1 > 0, 6 JAUME LLIBRE AND CLAUDIA VALLS (v) Rh̃1 ≈ R but here {0}, which is a singular point for H̃1 , is in the boundary of the Hill region, if a < 0 and h̃1 = 0, (vi) Rh̃1 ≈ (−∞, −P− ] ∪ [P− , ∞) if a < 0 and h̃1 < 0, (vii) Rh̃1 ≈ ∅ if a > 0 and h̃1 < −1/(4a), q q (viii) Rh̃1 ≈ {− a } ∪ { a1 } which are two of the singular points for 1 the map H̃1 , if a > 0 and h̃1 = −1/(4a), (ix) Rh̃1 ≈ [−P+ , −P− ] ∪ [P− , P+ ], if a > 0 and h̃1 ∈ (−1/(4a), 0) , q q (x) Rh̃1 ≈ [− a2 , a2 ] but here {0}, which is a singular point for H̃1 , is in the boundary of the Hill region, if a > 0 and h̃1 = 0, (xi) Rh̃1 ≈ [−P+ , P+ ] if a > 0 and h̃1 > 0. Now we compute the energy levels Ih̃1 . From the definition of Ih̃1 we have [ (3) Ih̃1 = Ex x∈Rh̃ 1 where   p2x a 4 1 2 Ex = (x, px ) ∈ R : 2 + x − x = h̃1 . 2 4 2 Clearly for each x ∈ R the set Ex is either two points, or one point or the emptyset, if the point x is in the interior of the Hill region Rh̃1 , in its boundary, or it does not belong to Rh̃1 , respectively. Therefore, from (3) and using the Hill region, the topology of Ih̃1 is: (i) Ih̃1 ≈ R ∪ R if a < 0 and h̃1 6= 0, (ii) Ih̃1 ≈ X if a ≤ 0 and h̃1 = 0. Here X denotes two straight lines intersecting the origin of the two straight lines, (iii) Ih̃1 ≈ ∅ if a > 0 and h̃1 < −1/(4a), q (iv) Ih̃1 ≈ (± a1 , 0) which are the two equilibrium points of H̃1 if a > 0 and h̃1 = −1/(4a), (v) Ih̃1 ≈ S1 ∪ S1 if a > 0 and h̃1 ∈ (−1/(4a), 0), (vi) Ih̃1 ≈ ∞ if a > 0 and h̃1 = 0. Here ∞ denotes two homoclinic orbits at the origin. (vii) Ih̃1 ≈ S1 if a > 0 and h̃1 < 0. See in Figure 1 the phase portraits associated to the Hamiltonian system with Hamiltonian H̃1 depending on whether a > 0, a = 0, and a < 0. The phase portraits in Figure 1 are drawn in the Poincaré disc, which essentially is a unit closed disc centered at the origin of ARMBRUSTER-GUCKENHEIMER-KIM GALACTIC POTENTIAL 7 coordinates with its interior identified to R2 and with its boundary (the circle S1 ) identified with the infinity of R2 , for more details on the Poincaré disc see Chapter 5 of [3]. (a) a > 0 (b) a = 0 (c) a < 0 Figure 1. Phase portraits associated to the Hamilton- ian system with Hamiltonian H̃1 depending on whether a > 0, a = 0 and a < 0. 4. The topology of the invariant sets Ih1 h2 To obtain Ih1 h2 we recall that Ih2 is exactly the same as Ih1 and that Ih1 h2 = Ih1 ∩ Ih2 = Ih̃1 × Ih̃2 . Hence, in Table 1 we have given the description of the invariant sets Ih1 h2 for the different values of h1 , h2 and a 5. The topology of the invariant sets Ih3 As we did for the case H1 , we recall that Ih3 = Ih̃3 × R2 and so we will study only Ih̃3 . The singular points for the map H̃3 satisfy px = 0, x(1 − ax2 ) = 0 √ √ and so they are (0, 0) if a ≤ 0 and (0, 0) ∪ (0, −1/ a) ∪ (0, 1/ a) if a > 0. The Hill region is n a y2 o Rh̃3 = y ∈ R : − y 4 + ≤ h̃3 4 2 and so taking the notation s p s p 1 − 1 − 4ah̃3 1 + 1 − 4ah̃3 Q− = , Q+ = a a we have (i) Rh̃3 ≈ ∅ if a = 0 and h̃3 < 0, 8 JAUME LLIBRE AND CLAUDIA VALLS a h1 h2 Ih1 h2 ≤0 6= 0 6= 0 (R ∪ R) × (R ∪ R) ≤0 6= 0 =0 (R ∪ R) × X ≤0 =0 6= 0 X × (R ∪ R) ≤0 =0 =0 X ×X >0 < −1/(4a) ∈R ∅ >0 = −1/(4a) < −1/(4a) q ∅ q 1 1 >0 = −1/(4a) = −1/(4a) (± a , 0) × (± a , 0) q >0 = −1/(4a) ∈ (−1/(4a), 0) (± a1 , 0) × (S1 ∪ S1 ) q >0 = −1/(4a) =0 (± a1 , 0) × ∞ q >0 = −1/(4a) <0 (± a1 , 0) × S1 > 0 ∈ (−1/(4a), 0) < −1/(4a) ∅ q >0 ∈ (−1/(4a), 0) = −1/(4a) (S1 ∪ S1 ) × (± a1 , 0) >0 ∈ (−1/(4a), 0) ∈ (−1/(4a), 0) (S1 ∪ S1 ) × (S1 ∪ S1 ) >0 ∈ (−1/(4a), 0) =0 (S1 ∪ S1 ) × ∞ >0 ∈ (−1/(4a), 0) <0 (S1 ∪ S1 ) × S1 >0 =0 < −1/(4a) ∅q >0 =0 = −1/(4a) ∞ × (± a1 , 0) >0 =0 ∈ (−1/(4a), 0) ∞ × (S1 ∪ S1 ) >0 =0 =0 ∞×∞ >0 =0 <0 ∞ × S1 >0 <0 < −1/(4a) ∅q >0 <0 = −1/(4a) S1 × (± a1 , 0) >0 <0 ∈ (−1/(4a), 0) S1 × (S1 ∪ S1 ) >0 <0 =0 S1 × ∞ >0 <0 <0 S1 × S1 Table 1. The invariant sets Ih1 h2 for the different values of h1 , h2 and a (ii) Rh̃3 ≈ {0} if a = 0 and h̃3 = 0,then p p (iii) Rh̃3 ≈ [− 2h̃3 , 2h̃3 ] if a = 0 and h̃3 > 0 then (iv) Rh̃3 ≈ ∅ if a < 0 and h̃3 < 0, (v) Rh̃3 ≈ {0} if a < 0 and h̃3 = 0, (vi) Rh̃3 ≈ [−Q+ , Q− ] if a < 0 and h̃3 > 0, (vii) Rh̃3 ≈ R if a > 0 and h3 > 1/(4a), ARMBRUSTER-GUCKENHEIMER-KIM GALACTIC POTENTIAL 9 q (viii) Rh̃3 ≈ R, but here {± a1 }, which are singular points for H̃1 , are in the boundary of the Hill region, if a > 0 and h̃3 = 1/(4a), (ix) Rh̃3 ≈ (−∞, −Q+ ] ∪ [−Q− , Q− ] ∪ [Q+ , +∞) if a > 0 and h̃3 ∈ (0, 1/(4a)), (x) Rh̃3 ≈ R, but here {0}, which is a singular point for H̃1 , is in the boundary of the Hill region, if a > 0 and h̃3 = 0, (xi) Rh̃3 ≈ − ∞, −Q+ ] ∪ [Q+ , +∞) if a > 0 and h̃3 < 0. Now we compute the energy levels Ih̃3 . From the definition of Ih̃3 we have (4) Ih̃3 = ∪y∈Rh̃ Ey 3 where   p2y a 4 1 2 Ey = (y, py ) ∈ R : 2 − y + y = h̃3 . 2 4 2 Clearly for each y ∈ R the set Ey is either two points, or one point or the emptyset, if the point y is in the interior of the Hill region Rh̃3 , in its boundary, or it does not belong to Rh̃3 , respectively. Therefore, from (4) and using the Hill region, the topology of Ih̃3 is: (i) Ih̃3 ≈ ∅ if a ≤ 0 and h̃3 < 0, (ii) Ih̃3 ≈ {(0, 0)} if a ≤ 0 and h̃3 = 0, (iii) Ih̃3 ≈ S1 a ≤ 0 and h̃3 > 0, (iv) Ih̃3 ≈ R ∪ R if a > 0 and h̃3 > 1/(4a), (v) Ih̃3 ≈ P if a > 0 and h̃3 = 1/(4a). Here P denotes two curves with the shape of a parabola intersecting in two different points (the points are the two singular points), (vi) Ih̃3 ≈ R ∪ S1 ∪ R if a > 0 and h̃3 ∈ (0, 1/(4a)), (vii) Ih̃3 ≈ R ∪ {(0, 0)} ∪ R if a > 0 and h̃3 = 0, (viii) Ih̃3 ≈ R ∪ R if a > 0 and h̃3 < 0. See the phase portrait associated to H̃3 depending on whether a > 0, a = 0, or a < 0. See in Figure 2 the phase portraits associated to the Hamiltonian system with Hamiltonian H̃3 depending on whether a > 0 and a ≤ 0. 6. The topology of the invariant sets Ih3 h4 To obtain Ih3 h4 we recall that Ih4 is exactly the same as Ih3 and that Ih3 h4 = Ih3 ∩ Ih4 = Ih̃3 × Ih̃4 . Hence, in Table 2 we have given the 10 JAUME LLIBRE AND CLAUDIA VALLS (a) a > 0 (b) a ≤ 0 Figure 2. Phase portraits associated to the Hamilton- ian system with Hamiltonian H̃3 depending on whether a > 0 or a ≤ 0. description of the invariant sets Ih3 h4 for the different values of h3 , h4 and a. Acknowledgements The first author is partially supported by the Ministerio de Economı́a, Industria y Competitividad, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Univer- sitaris i de Recerca grant 2017SGR1617, and the H2020 European Re- search Council grant MSCA-RISE-2017-777911. The second author is partially supported by FCT/Portugal through UID/MAT/04459/2013. References [1] P. Acosta-Humánez, M. Alvarez-Ramirez and T.J. Stuchi, Nonin- tegrability of the Armbruster Guckenheimer Kim quartic Hamiltonian trough Morales Ramis theory, SIAM J. Appl. Dyn. Syst. 17 (2017), 78–96. [2] D. Armbruster, J. Guckenheimer and S. Kim, Chaotic dynamics in sys- tems with square symmetry, Phys. Lett. A 140 (1989), 416–420. [3] F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, UniversiText, Springer–Verlag, New York, 2006. [4] A.A. Elmandouh, On the dynamics of Armbruster Guckenheimer Kim galac- tic potential in a rotating reference frame, Astrophys. Space Sci. 361 (2016), 182–194. [5] F.M. El-Sabaa, M. Hosny and S.K. Zakria, Bifurcations of Armbruster Guckenheimer Kim galactic potential, Astrophys. Space Sci. 364 (2019), 34– 43. [6] M.W. Hirsch, Differential Topology, Graduate Texts in Mathematics, Springer, 1976. [7] J. Llibre and L. Roberto, Periodic orbits and non-integrability of Armbruster-Guckenheimer-Kim potential, Astrophys. Space Sci. 343 (2012), 69–74. ARMBRUSTER-GUCKENHEIMER-KIM GALACTIC POTENTIAL 11 a h1 h2 Ih1 h2 ≤0 <0 ∈R ∅ ≤0 =0 <0 ∅ ≤0 =0 =0 {(0, 0)} × {(0, 0)} ≤0 =0 >0 {(0, 0)} × S1 ≤0 >0 <0 ∅ ≤0 >0 =0 S1 × {(0, 0)} ≤0 >0 >0 S1 × S1 ≤0 > 1/(4a) > 1/(4a) (R ∪ R) × (R ∪ R) ≤0 > 1/(4a) = 1/(4a) (R ∪ R) × P ≤0 > 1/(4a) ∈ (0, 1/(4a)) (R ∪ R) × (R ∪ S1 ∪ R) ≤0 > 1/(4a) =0 (R ∪ R) × (R ∪ {(0, 0)} ∪ R) ≤0 > 1/(4a) <0 (R ∪ R) × (R ∪ R) ≤0 = 1/(4a) > 1/(4a) P × (R ∪ R) ≤0 = 1/(4a) = 1/(4a) P ×P ≤0 = 1/(4a) ∈ (0, 1/(4a)) P × (R ∪ S1 ∪ R) ≤0 = 1/(4a) =0 P × (R ∪ {(0, 0)} ∪ R) ≤0 = 1/(4a) <0 P × (R ∪ R) ≤0 ∈ (0, 1/(4a)) > 1/(4a) (R ∪ S1 ∪ R) × (R ∪ R) ≤0 ∈ (0, 1/(4a)) = 1/(4a) (R ∪ S1 ∪ R) × P ≤0 ∈ (0, 1/(4a)) ∈ (0, 1/(4a)) (R ∪ S1 ∪ R) × (R ∪ S1 ∪ R) ≤0 ∈ (0, 1/(4a)) =0 (R ∪ S1 ∪ R) × (R ∪ {(0, 0)} ∪ R) ≤0 ∈ (0, 1/(4a)) <0 (R ∪ S1 ∪ R) × (R ∪ R) ≤0 =0 > 1/(4a) (R ∪ {(0, 0)} ∪ R) × (R ∪ R) ≤0 =0 = 1/(4a) (R ∪ {(0, 0)} ∪ R) × P ≤0 =0 ∈ (0, 1/(4a)) (R ∪ {(0, 0)} ∪ R) × (R ∪ S1 ∪ R) ≤0 =0 =0 (R ∪ {(0, 0)} ∪ R) × (R ∪ {(0, 0)} ∪ R) ≤0 =0 <0 (R ∪ {(0, 0)} ∪ R) × (R ∪ R) ≤0 <0 > 1/(4a) (R ∪ R) × (R ∪ R) ≤0 <0 = 1/(4a) (R ∪ R) × P ≤0 <0 ∈ (0, 1/(4a)) (R ∪ R) × (R ∪ S1 ∪ R) ≤0 <0 =0 (R ∪ R) × (R ∪ {(0, 0)} ∪ R) ≤0 <0 <0 (R ∪ R) × (R ∪ R) Table 2. The invariant sets Ih3 h4 for the different values of h3 , h4 and a [8] T. Zeeuw and D. Merritt, Stellar orbits in a triaxial galaxy I. Orbits in the plane of rotation, Astrophys. 267 (1983), 571–595. 12 JAUME LLIBRE AND CLAUDIA VALLS Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat Departamento de Matemática, Instituto Superior Técnico, Univer- sidade de Lisboa, 1049-001 Lisboa, Portugal Email address: cvalls@math.tecnico.ulisboa.pt