Global dynamics of the integrable Armbruster-Guckenheimer-Kim galactic potential

We study the global dynamics of the completely integrable Armbruster-Guckenheimer-Kim galactic potential. In these cases this system has two first integrals H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H_{1}$\end{document} and H2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H_{2}$\end{document} independent and in involution. Let Ih1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$I_{h_{1}}$\end{document} and Ih2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$I_{h_{2}}$\end{document} be the set of points of the phase space on which H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H_{1}$\end{document} and H2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H_{2}$\end{document} take the values h1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$h_{1}$\end{document} and h2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$h_{2}$\end{document}, respectively. The sets Ih1h2=Ih1∩Ih2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$I_{h_{1} h_{2}}=I_{h_{1}} \cap I_{h _{2}}$\end{document} are invariant by the dynamics. We characterize the global flow on these sets and we describe the foliation of the phase space by the invariant sets Ih1h2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$I_{h_{1} h_{2}}$\end{document}.


Introduction
The Armbruster-Guckenheimer-Kim potential is a galactic potential introduced in Armbruster et al. (1989) 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 where a, b are arbitrary constants. If we add the term −ω(xp y − yp x ) then the system describes the dynamics of rotation of a nearly axisymetric 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 Zeeuw and Merritt 1983). Many studies concerning the integrability and non-integrability of such systems have been done (see for instance Acosta-Humánez et al. 2018;Elmandouh 2016;El-Sabaa et al. 2019) using different techniques 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 Llibre and Roberto (2012). In particular, it was proved in El-Sabaa et al. (2019) that if b = 2a or b = −a the system is completely integrable but the authors do not describe completely the dynamics of the integrable systems form the point of view of the Liouville-Arnold theorem (see Sect. 2). This is the main aim of this paper. When b = 2a the Hamiltonian has the form Introducing the new variables it can be written as where a ∈ R, we have renamed the variables (u, v) again as (x, y) and In all the paper we will denote by H the Hamiltonian associated to a system with two degrees of freedom and so H = H (x, p x , y, p y ) : . . , 4, and we will denote byH the Hamiltonian associated to a system with one degree of freedom and sõ We observe that H 1 and H 2 are two first integrals, independent and in involution. Hence, the Hamiltonian system associated to the Hamiltonian H iṡ and it is completely integrable. We recall that H 1 and H 2 are independent if the matrix has rank 2 in any point of R 4 except, perhaps in a zero Lebesgue-measure set. As usual H iy = ∂H i /∂y. Moreover, we say that H 1 and H 2 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.
Note that the phase space of system (1) is R 4 . Since H 1 and H 2 are fist integrals the sets as well as 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 R 4 by the invariant sets I h i for i = 1, 2 as well as by I h 1 h 2 . 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 Note that H 3 and H 4 are two first integrals, independent and in involution. Hence the Hamiltonian systeṁ is completely integrable. The sets as well as are invariant by the flow of the Hamiltonian system (2). The second main objective of the paper is to describe the foliations of R 4 by the invariant sets I h i for i = 3, 4 and by the invariant sets I h 3 h 4 . Again, 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 Sect. 2 we recall the Liouville-Arnold theory for Hamiltonians systems with two degrees of freedom. In Sect. 3 we describe the topology of the sets I h 1 (since the study for I h 2 is analogous). For doing that and taking into account that I h 1 = Ih 1 × R 2 we will only describe the topology of the sets Ih 1 by computing the sets of singular points and critical values forH 1 and the Hill regions according to the different values of a andh 1 . In Sect. 4 we study the topology of the sets I h 1 h 2 . In Sect. 5 we describe the topology of the sets I h 3 (again because the study for I h 4 is analogous) and recalling that I h 3 = Ih 3 × R 2 we will only describe the topology of the sets Ih 3 by computing the sets of singular points and critical values forH 3 and the Hill regions according to the different values of a andh 3 . In Sect. 6 we study the topology of the sets I h 3 h 4 .

Integrable Hamiltonian systems
In this section we recall the Liouville-Arnold theorem for the integrable Hamiltonian systems with two degrees of freedom. We recall that a flow defined on the phase space R 4 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 R 4 has the Hamiltonians H 1 and H 2 (resp. H 3 and H 4 ) as two independent first integrals in involution. H 4 )) then the following statements hold.
(a) I h 1 h 2 (resp. I h 3 h 4 ) is a two-dimensional submanifold of R 4 invariant under the flow of system (1) (resp. system (2)). (b) If the flow on a connected component I * h 1 h 2 (resp. I * h 3 h 4 ) of I h 1 h 2 (resp. I h 3 h 4 ) is complete, then I * h 1 h 2 (resp. I * h 3 h 4 ) is diffeomorphic either to the torus S 1 × S 1 , to the cylinder S 1 × R, or to the plane R 2 . (c) Under the assumption of statement (b), the flow on I * h 1 h 2 (resp. on I * h 3 h 4 ) is conjugated to a linear flow either on S 1 × S 1 , or on S 1 × R, or on R 2 .
Note that Theorem 1 does not provide information on the topology of the invariant sets I h 1 h 2 (resp. I h 3 h 4 ) when (h 1 h 2 ) (resp. (h 3 h 4 )) is not a regular value of the map (H 1 , H 2 ) (resp. (H 3 , H 4 )), or how the energy levels I h 1 or I h 2 (resp. I h 3 or I h 4 ) foliate R 4 .
In this paper we solve these problems for systems (1) and (2).

The topology of the invariant sets I h 1
As explained in the introduction, taking into account that I h 1 = Ih 1 × R 2 we will restrict all the study to Ih 1 .
A point (x, p x ) ∈ R 2 is a singular point for the mapH 1 if it is a solution of The valueh 1 ∈ R is a critical value for the mapH 1 if there is some singular point belonging toH −1 1 (h 1 ) = Ih 1 . Ifh 1 is not critical value it is said a regular value. It is well-known that ifh 1 is a regular value of the mapH 1 then Ih 1 is a onedimensional manifold (see Hirsch 1976).
Note that the singular points for the mapH 1 are and so the set of singular points ofH 1 is (0, 0) if a ≤ 0, and if a > 0. We define the Hill region as This is the region of the configuration space {x ∈ R} where the motion of all orbits of the Hamiltonian system associated toH 1 having energyh 1 takes place. By Rh 1 ≈ S, we denote that Rh 1 is diffeomorphic to S. We will also denote by , which is a singular point forH 1 , is in the boundary of the Hill region, if a = 0 and which is a singular point forH 1 , is in the boundary of the Hill region, if a < 0 and  Now we compute the energy levels Ih 1 . From the definition of Ih 1 we have Clearly for each x ∈ R the set E x is either two points, or one point or the emptyset, if the point x is in the interior of the Hill region Rh 1 , in its boundary, or it does not belong to Rh 1 , respectively. Therefore, from (3) and using the Hill region, the topology of Ih 1 is: Here X denotes two straight lines intersecting the origin of the two straight lines, (iii) Ih 1 ≈ ∅ if a > 0 andh 1 < −1/(4a), (iv) Ih 1 ≈ (± 1 a , 0) which are the two equilibrium points ofH 1 if a > 0 andh 1 = −1/(4a), Here ∞ denotes two homoclinic orbits at the origin. (vii) Ih 1 ≈ S 1 if a > 0 andh 1 < 0. See in Fig. 1 the phase portraits associated to the Hamiltonian system with HamiltonianH 1 depending on whether a > 0, a = 0, and a < 0. The phase portraits in Fig. 1 are drawn in the Poincaré disc, which essentially is a unit closed disc centered at the origin of coordinates with its interior identified to R 2 and with its boundary (the circle S 1 ) identified with the infinity of R 2 , for more details on the Poincaré disc see Chap. 5 of Dumortier et al. (2006).

The topology of the invariant sets I h 1 h 2
To obtain I h 1 h 2 we recall that I h 2 is exactly the same as I h 1 and that I h 1 h 2 = I h 1 ∩ I h 2 = Ih 1 × Ih 2 . Hence, in Table 1 we have given the description of the invariant sets I h 1 h 2 for the different values of h 1 , h 2 and a.

The topology of the invariant sets I h 3
As we did for the case H 1 , we recall that I h 3 = Ih 3 × R 2 and so we will study only Ih 3 . The singular points for the map H 3 satisfy
Clearly for each y ∈ R the set E y is either two points, or one point or the emptyset, if the point y is in the interior of the Hill region Rh 3 , in its boundary, or it does not belong to Rh 3 , respectively. Therefore, from (4) and using the Hill region, the topology of Ih 3 is: Here P denotes two curves with the shape of a parabola intersecting in two different points (the points are the two singular points), See the phase portrait associated toH 3 depending on whether a > 0, a = 0, or a < 0.  Fig. 2 the phase portraits associated to the Hamiltonian system with HamiltonianH 3 depending on whether a > 0 and a ≤ 0.

The topology of the invariant sets I h 3 h 4
To obtain I h 3 h 4 we recall that I h 4 is exactly the same as I h 3 and that I h 3 h 4 = I h 3 ∩ I h 4 = Ih 3 × Ih 4 . Hence, in Table 2 we have given the description of the invariant sets I h 3 h 4 for the different values of h 3 , h 4 and a.