LINEAR TYPE CENTERS OF POLYNOMIAL HAMILTONIAN SYSTEMS WITH NONLINEARITIES OF DEGREE 4 SYMMETRIC WITH RESPECT TO THE Y-AXIS

. We provide the phase portraits in the Poincar´e disk for all the linear type centers of polynomial Hamiltonian systems with nonlinearities of degree 4 symmetric with respect to the y -axis given by the Hamiltonian function H ( x,y ) = 12 ( x 2 + y 2 ) + ax 4 y + bx 2 y 3 + cy 5 in function of its parameters.

1. Introduction and statement of the main result. In this work we deal with polynomial differential systems in R 2 of the forṁ x = P (x, y),ẏ = Q(x, y), where the dot denotes derivative with respect to an independent real variable t, usually called the time. Assume the origin O is an equilibrium point of system (1). When all the orbits of system (1) in a punctured neighborhood of the equilibrium point O are periodic, we say that the origin is a center. The study of the centers started with Poincaré [18] and Dulac [8], and in the present days many questions about them remain open. If a polynomial system (1) has a center at the origin, then after a linear change of variables and a scaling of the time variable, it can be written in one of the following three forms:ẋ = −y + P 2 (x, y),ẏ = x + Q 2 (x, y), then the origin is called a linear type center, x = y + P 2 (x, y),ẏ = Q 2 (x, y), then the origin is called a nilpotent center, x = P 2 (x, y),ẏ = Q 2 (x, y), then the origin is called a degenerate center, where P 2 (x, y) and Q 2 (x, y) are polynomials without constant and linear terms.
The classification of the centers of quadratic differential systems (which all of them are linear type centers) started with the works of Dulac [8], Kapteyn [11,12] and Bautin [3], and the characterization of their phase portraits in the Poincaré disk was due to Vulpe [21], see also Schlomiuk [20]. There are many partial results for the centers of polynomial differential systems of degree larger than 2. We must mention that Malkin [14], and Vulpe and Sibirsky [22] characterized the linear type centers of the polynomial differential systems with linear and homogeneous nonlinearities of degree 3. For polynomial differential systems of the form linear with homogeneous nonlinearities of degree greater than 3 the centers at the origin are not characterized, but there are partial results for degrees 4 and 5 for those linear type centers, see Chavarriga and Giné [4,5]. On the other hand it remains a lot of work for obtaining the complete classification of the linear type centers for all polynomial differential systems of degree 3. Some interesting results on some subclasses of those systems appeared in Rousseau and Schlomiuk [19], and the ones ofŻo ladek [23,24]. Recently, the linear type centers of polynomial Hamiltonian systems with nonlinearities of degree 3 were classified by Colak et al. in [7].
In this paper we study the phase portraits in the Poincaré disk of the linear type centers of polynomial Hamiltonian systems with nonlinearities of degree 4, but since this class has 6 parameters and degree 4 his study is huge, in a first approach for studying this class we restrict our attention to the subclass of such systems which are symmetric with respect to the y-axis. More precisely, we shall classify the phase portraits in the Poincaré disk of the Hamiltonian systeṁ with Hamiltonian H(x, y) = 1 2 (x 2 + y 2 ) + ax 4 y + bx 2 y 3 + cy 5 , and satisfying that a 2 +b 2 +c 2 = 0, i.e., that the polynomial system (2) has degree 4.
Since H is a first integral of system (2) and the origin of this system has eigenvalues ±i, the origin is a center. We note that system (2) is invariant under the symmetry (x, y, t) → (−x, y, −t), i.e., its orbits are symmetric with respect to the y-axis, and consequently the phase portrait of the system. We note that the centers of the Hamiltonian systems with Hamiltonian H(x, y) = ax 4 y + bx 2 y 3 + cy 5 , have been studied in [6], and that there are no nilpotent centers for the Hamiltonian systems with Hamiltonian H(x, y) = 1 2 y 2 + ax 4 y + bx 2 y 3 + cy 5 , as it can be proved used Theorem 3.5 of [9]. In order to classify the phase portraits of systems (2) we will use the Poincaré compactification of polynomial vector fields, see subsection 2.1, which roughly speaking is to add to the plane R 2 its boundary, the circle S 1 which corresponds to the infinity, identifying this compact space with a closed disk (the Poincaré disk), and extending the flow of system (2) to this closed disk in a unique analytic way. We say that two vector fields on the Poincaré disk are topologically equivalent if there exists a homeomorphism of this disk which sends orbits to orbits preserving or reversing the direction of the flow, see a more precise definition in subsection 2.1. Our main result is the following one. Theorem 1. The phase portrait in the Poincaré disk of a linear type center of a polynomial Hamiltonian system with nonlinearities of degree 4 symmetric with respect to the y-axis is topologically equivalent to one of the following 30 phase portraits of Figure 1.
In section 2 we provide the notations, basic definitions and results which will allow to do the topological classification of all phase portraits for the Poincaré disk of the linear type centers for the polynomial Hamiltonian systems with nonlinearities of degree 4 symmetric with respect to the y-axis, i.e., of systems (2). A second step in this work will be to perturbed the Hamiltonian centers of the phase portraits of Theorem 1 inside the class of all quartic polynomial differential systems and study how many limit cycles can bifurcate from their periodic orbits using the tools of [10], see for instance [13] where these tools are applied.

2.
Notations and basic results. In this section we recall the basic definitions and notations that we will need for the analysis of the local phase portraits of the finite and infinite equilibria of the polynomial Hamiltonian systems (2), and for doing their phase portraits in the Poincaré disk.
We denote by P 4 (R 2 ) the set of polynomial vector fields in R 2 of the form X (x, y) = (P (x, y), Q(x, y)) where P and Q are real polynomials in the variables x and y such that the maximal degree of P and Q is 4.
2.1. Poincaré compactification. Let X ∈ P 4 (R 2 ) be a vector field of degree 4 in R 2 . The Poincaré compactified vector field p(X ) of X is an analytic vector in S 2 defined in the following way, for more details see for instance Chapter 5 of [9]. Let S 2 = {y = (y 1 , y 2 , y 3 ) ∈ R 3 : y 2 1 + y 2 2 + y 2 3 = 1} be the Poincaré sphere and T y S 2 be the tangent space to S 2 at the point y. We identify the plane T (0,0,1) S 2 with the R 2 where we have our vector field X .
We define the central projection f : T (0,0,1) S 2 −→ S 2 in such a way that to each point q of the plane T (0,0,1) S 2 the map associates the two intersection points of the straight line, joining q with (0, 0, 0), with the sphere S 2 . This map provides two copies of X , one in the northern hemisphere and the other in the southern one. Denote by X the vector field Df • X on S 2 except on its equator S 1 = {y ∈ S 2 : y 3 = 0}. Clearly S 1 is identified to the infinity of R 2 .
In order to extend X to a vector field on the whole S 2 we consider the vector field p(X ) = y 4 3 X in S 2 . In short, on S 2 \ S 1 there are two symmetric copies of X , and knowing the behavior of p(X ) around S 1 , we know the behavior of X at infinity. The infinity S 1 is invariant under the flow of p(X ).
The projection of the closed northern hemisphere of S 2 on y 3 = 0 under (y 1 , y 2 , y 3 ) → (y 1 , y 2 ) is called the Poincaré disk, and it is denoted by D 2 .
Here two polynomial vector fields X and Y on R 2 are topologically equivalent if there exists a homeomorphism on S 2 preserving the infinity S 1 carrying orbits of the flow induced by p(X ) into orbits of the flow induced by p(Y); preserving or reversing simultaneously the sense of all orbits.
On the sphere S 2 we take for i = 1, 2, 3 the six local charts U i = {y ∈ S 2 : y i > 0}, and V i = {y ∈ S 2 : y i < 0}, with the diffeomorphisms F i : U i −→ R 2 and G i : V i −→ R 2 , which are the inverses of the central projections from the planes tangent at the points (1, 0, 0); (−1, 0, 0); (0, 1, 0); (0, −1, 0); (0, 0, 1) and (0, 0, −1), respectively. We denote by z = (z 1 , z 2 ) the value of F i (y) or G i (y) for any i = 1, 2, 3. Thus z means different coordinates in the distinct local charts. Easy computations (1) (  be the vector field associated to our Hamiltonian system (2) with a 2 + b 2 + c 2 = 0, and let p(X ) be its Poincaré compactification. Let q be an equilibrium point of p(X ). If some of the two eigenvalues λ 1 and λ 2 of the linear part of the vector field p(X ) at the equilibrium point q is not zero, then this equilibrium is elementary.
The local phase portrait of a finite elementary equilibrium q with both eigenvalues non-zero for a Hamiltonian system is well known, q is a saddle if λ 1 λ 2 = 0 (which in fact only can be λ 1 λ 2 < 0), and a center if Reλ 1 = Reλ 2 = 0, for more details see for instance [2]. We recall that the flow of a Hamiltonian system in the plane preserves the area, so their finite equilibria cannot have parabolic and elliptic sectors, only hyperbolic sectors, or they must be centers. We define energy levels of the vector field (6) as the curves on which its Hamiltonian (3) is constant.
The local phase portrait of an infinite elementary equilibrium q with both eigenvalues non-zero can studied using, for instance, Theorem 2.15 of [9].
Let q be an elementary equilibrium point with an eigenvalue zero. Then q is called a semi-hyperbolic equilibrium point. The local phase portrait of a finite semihyperbolic equilibrium point of a Hamiltonian system only can be a saddle, see again for example [9].
The local phase portrait of an infinite semi-hyperbolic equilibrium q can described using, for instance, Theorem 2.19 of [9].
When both eigenvalues are zero but the linear part of p(X ) at the equilibrium point q is not the zero matrix, we say that q is a nilpotent equilibrium point. The local phase portrait of a nilpotent equilibrium point can be studied using Theorem 3.5 of [9]. If q is nilpotent and finite, then it only can be a saddle, a cusp, or a center.
Finally, if the Jacobian matrix of p(X ) at the equilibrium point q is identically zero, and q is isolated inside the set of all equilibrium points, then we say that q is a linearly zero equilibrium point. The study of the local phase portraits of such equilibra needs special changes of variables called blow-ups, see for more details Chapter 3 of [9], or [1].
We want to classify in the Poincaré disk, modulo topological equivalence, the phase portraits of p(X ) being X the polynomial Hamiltonian vector fields given in (6). For doing that we must start by classifying the local phase portraits at all finite and infinite equilibria of p(X ) in the Poincaré disk. This will be made by using the techniques described in this subsection.

2.3.
Phase portraits in the Poincaré disk. In this subsection we shall see how to characterize the phase portraits of p(X ) in the Poincaré disk for the polynomial Hamiltonian vector fields X of (6).
A separatrix of p(X ) being X a Hamiltonian vector field (6) defined in the whole R 2 is an orbit which is either an equilibrium point, or a trajectory which lies in the boundary of a hyperbolic sector of a finite or infinite equilibrium point, or any orbit contained in the infinity S 1 . Neumann [16] proved that the set formed by all separatrices of p(X ), denoted by S(p(X )) is closed.
The open connected components of D 2 \ S(p(X )) are called canonical regions of X or of p(X ). A separatrix configuration is the union of S(p(X )) plus one solution chosen from each canonical region. Two separatrix configurations S(p(X )) and S(p(Y)) are topologically equivalent if there is an orientation preserving or reversing homeomorphism which maps the trajectories of S(p(X )) into the trajectories of S(p(Y)). The following result is due to Markus [15], Neumann [16] and Peixoto [17], that find it independently.
Theorem 2. The phase portraits in the Poincaré disk of two compactified polynomial differential systems p(X ) and p(Y) are topologically equivalent, if and only if, their separatrix configurations S(p(X )) and S(p(Y)) are topologically equivalent.
3. Phase portraits. We separate the study of the phase portraits of Hamiltonian system (2) in two cases: first for a = 0 where we subdivide in three subcases b = 0 and c = 0, c = 0 and b = 0, and bc = 0, and second for the case a = 0 which is subdivide in the subcases b = 0 and c = 0, c = 0 and b = 0, and bc = 0. Since c = 0 doing the c −2/3 -symplectic change of variables (x, y) → (5c) −1/3 (x, y) we can assume without loss of generality that c = 1 and the new Hamiltonian system isẋ = −y(1 + y 3 ),ẏ = x. (7) This system only has two finite real equilibria and two finite complex equilibria, but since the complex equilibria are not relevant for doing the phase portrait in the Poincaré disk in what follows we do not mention the finite and infinite complex equilibria. Thus these two finite equilibria are: the origin which as we know always is a center, and the saddle (0, −1), because the two eigenvalues of the linear part at this equilibrium are ± √ 3.
From (4) system (7) in the local chart U 1 becomeṡ At infinity, i.e., at z 2 = 0, this system has a unique equilibrium, the origin (z 1 , z 2 ) = (0, 0) which is a linear zero equilibrium. For studying its local phase portrait we need to do blow-ups. Since this is the first equilibrium that we must study using the changes of variables called blow-ups we will describe them with all details in this case. First we do the blow-up (z 1 , z 2 ) → (z 1 , w) with w = z 2 /z 1 , then we obtain the systemż We note that this change of variables blow up the origin of system (8) to the whole w-axis of system (9). Now doing a scaling of the independent variable we eliminate the common factor z 2 1 betweenż 1 andẇ of system (9), and we obtain the system Since the whole w-axis going back through the changes of variables goes to the origin of system (8) we must study the equilibria on such an axis. Thus, on z 1 = 0 the unique equilibrium of system (10) is the origin, and again it is linearly zero. So we do a second blow-up (z 1 , w) → (z 1 , u) with u = w/z 1 , and we arrive to the system Doing another scaling of the independent variable we eliminate the common factor z 2 1 between the two components of system (11), and we have Now, the unique equilibrium on z 1 = 0 is the origin. Since the eigenvalues of the linear part of this system at the origin are ±1, the origin is a saddle, with the local stable separatrices on the u-axis, and the unstable ones on the z 1 -axis, see Figure  2(a). Now we must go back through the changes of variables starting at system (12) and reaching system (8). Since the scaling for passing from system (11) to system (12) is z 2 1 , this change in the independent variable changes neither the orbits, nor their orientation, with the exception of the orbits on the u-axis of system (12) which become equilibria for system (11), see Figure 2(b). Going back from system (11) to system (10), and taking into account that for system (10) we have z 1 | w=0 = z 3 1 and w | z1=0 = −w 4 , we obtain the local phase portrait at the origin of system (10), which topologically is a saddle-node, see Figure  2(c).
As in the pass from system (12) to system (11), the pass from system (10) to system (9) does not change the local phase portrait of system (10) except that the w-axis becomes filled with equilibria, see Figure 2(d).
Finally, undoing the first blow up and taking into account that for system (8) we have thatż 1 | z1=0 = z 3 2 , we get that the local phase portrait at the origin of system (8) is the unstable node of Figure 2(e).
In short, in the chart U 1 the unique infinite singular point is the origin which is an unstable node. We point out that for system (2) the equilibria in the local chart V 1 are the corresponding equilibria to the equilibria in U 1 but with the opposite stability due to the fact that the vector field p(X ) in of degree even. In order to (9) (e) System (8) Figure 2. The blow-ups of the origin of the chart U 1 for system (8). The dotted line represents a straight line of equilibria.
complete the study of the infinite singular points, as we explained in subsection 2.1, only it remains to see if the origin of the local chart U 2 is or not an equilibrium point. From (5) system (7) in the local chart U 2 writeṡ Hence, clearly the origin of U 2 is not an equilibrium.
Considering the previous study we have that the unstable separatrix of the saddle which enters into the quadrant x > 0 and y < 0 must cross the positive x-axis because it cannot intersect the stable separatrix of the saddle which also is in the quadrant x > 0 and y < 0, and such unstable separatrix must intersect the positive y-axis, at this moment it connects with the stable separatrix of the saddle that initially is in the quadrant x < 0 and y < 0 due to the fact that the phase portrait is symmetric with respect to the y-axis. This connection forces that the phase portrait of system (7) is topologically equivalent to the one described in Figure  1(1).
Another way to see the connection of the unstable separatrix of the saddle with the stable one described in the previous paragraph is drawing the level curves of the Hamiltonian of system (7) passing through the saddle.
3.1.2. Case c = 0 and b = 0. Under these conditions, system (2) iṡ System (13) has three finite equilibria: the origin which we know that is a center, and other two equilibriaẽ 3,4 = (±2 1/6 3 −1/2 , −2 −1/3 ), and these two symmetric equilibria are saddles because their eigenvalues are ± √ 6. Now, we do the study at the infinity, from (4) the system in U 1 iṡ At infinity, i.e., z 2 = 0, the origin is the unique equilibrium, which is degenerate, applying the blow-up technique in a similar way to subsection 3.1.1 we obtain the local phase portrait at the origin of system (14), this is shown in Figure 4(a).   (14), (b) system (15) In the local chart U 2 system (13) through (5) has the forṁ and the origin is the unique equilibrium point, which is degenerate. Doing blow-ups the local phase portrait at the origin of system (15) is described in Figure 4(b). From the previous study, we can give the local dynamics at all equilibria, finite and infinite, on the Poincaré disk, this is shown in Figure 5. Now we study the global phase portrait. Note that the local phase portrait obtained and considering that system (13) satisfiesẋ| y=0 = 0,ẏ| y=0 = x,ẋ| x=0 = −y andẏ| x=0 = 0, its force that one unstable separatrix of the saddle e 2 located at the region x > 0 and y < 0 must cross the positive x-axis because it cannot intersect the stable separatrices of itself, and such unstable separatrix must intersect the positive y-axis, at this moment it connects by symmetry with the stable separatrix of the saddle e 3 with is in the region x < 0 and y < 0. Analogously one stable separatrix of the saddle e 2 located at the region x > 0 and y < 0 must cross the negative y-axis because it cannot intersect the unstable separatrices of itself, at this moment it connects with one unstable separatrix of the saddle e 3 with is in the region x < 0 and y < 0 due to the fact that the phase portrait is symmetric with respect to the y-axis. These connections between these separatrices of the two finite saddles also can be studied using the energy level at these saddles, which for the symmetry of the Hamiltonian system, these energy levels at these two saddles are equal. These connections force that the global phase portrait on the Poincaré disk of system (13) is topologically equivalent to the one of Figure 1(2).

3.1.3.
Case bc = 0. Here the associated system to Hamiltonian (3) iṡ Since b = 0, without loss of generality, we can do the b −2/3 -symplectic change of variables (x, y) → b −1/3 (x, y) and we obtain the systeṁ withc = 5c/b, for simplicity we just use again the letter c.
(iii) If c < 2/5, there are four finite equilibria, namely the origin, e 2 and e 3,4 = (±2 −1/3 3 −1/2 (2 − 5c) 1/2 , −2 −1/3 ). The equilibrium e 2 is a saddle if c < 0 and is a center if 0 < c < 2/5. The equilibrium points e 3 and e 4 are saddle because their eigenvalues are ± 3(2 − 5c). Thus we have concluded the local study of the finite equilibria of system (16). Now we are going to study the infinite equilibria using the local charts given by the Poincaré compactification. From (4) we have that the system in U 1 is given bẏ At infinity, i.e., z 2 = 0, when c > 0 there exists only the origin as equilibrium point. If c < 0 there are three equilibria, the origin and (±(−c) −1/2 , 0). In any case, the origin is degenerate, so we need to apply a blow-up for understanding its local phase portrait. Doing blow-ups we obtain the local phase portrait at the origin of system (18), which topologically is an unstable node, see Figure 6. The equilibria (±(−c) −1/2 , 0), have eigenvalues 10/c and 2/c so they are stable nodes, since c < 0. From (5) system (16) in the local chart U 2 iṡ , where clearly the origin of U 2 is not an equilibrium.
From the previous analysis we know the local dynamics at the all equilibria on the Poincaré disk, depending on c, the local dynamics of system (16) at its equilibria is topologically equivalent to the one shown in Figure 7.
Local phase portraits at the equilibria of system (16) when a = 0 and bc = 0. Now we will determine the global phase portrait according to this local information, using the symmetry and the connections between the equilibria. Note that the vector field (16) on the axes isẋ| x=0 = −y(1 + 5cy 3 ),ẏ| x=0 = 0,ẋ| y=0 = 0 anḋ y| y=0 = x. Thus in the cases c > 2/5 and c = 2/5 , following the same ideas as in the previous subsection we arrive that the global phase portrait of system (16) with c ≥ 2/5 is topologically equivalent to Figure 1(1).
Next for 0 < c < 2/5 the vector field over the axes and the local phase portrait force that one stable separatrix of the saddle e 3 located at the region x > 0 and y < 0 must cross the negative y-axis between the center because it cannot intersect neither the unstable separatrices of itself nor the x-axis, and by symmetry such stable separatrix must connect with one unstable separatrix of the saddle e 4 which is in the region x < 0 and y < 0. Using the same argument we arrive that the separatrices associated to one stable separatrix and one unstable separatrix of e 3 and e 4 form the boundary of the period annulus of the center e 2 . Now, we take the other unstable separatrix of e 3 , it must cross the positive x-axis because it cannot intersect the stable separatrices of itself, then by continuity of the flow it also must cross the positive y-axis, at this moment it connects with one stable separatrix of the saddle e 4 due to the fact that the phase portrait is symmetric with respect to the y-axis. These connections force that the global phase portrait on the Poincaré disk of system (16) is topology equivalent to the one of Figure 1(3). Again the connections of the separatrices of the saddles can be studied using the level curves of the Hamiltonian.
Finally in the case c < 0 the vector field and the local phase portrait ( Figure  7(a)) force that one stable separatrix of the saddle e 3 must cross the negative y-axis because it neither intersects the unstable separatrices of itself nor the positive x-axis, it connects by symmetry with the unstable separatrix of the saddle e 4 . The other stable separatrix of e 3 must connect with an unstable orbit of the infinite equilibrium point located at the end of the positive x-axis. By symmetry the analogous happens with the other unstable separatrix of e 4 . Now, we are going to study the possible connections of the unstable separatrices of the saddle e 3 using the energy levels of the saddles. Defining the energy level associated to the equilibria e 2 , e 3 (or e 4 ) as h 2 = 3 · 5 −5/3 c −2/3 /2 and h 3 = (1 − c)/2 5/3 , respectively, and letĉ = 11 − 4 · 10 1/3 − 10 2/3 /15 ≈ −0.1506 the value of c when h 2 coincide with h 3 . We note that h 2 < h 3 if c <ĉ, h 2 = h 3 if c =ĉ, and h 2 > h 3 ifĉ < c < 0. On the other hand, the equation H| y=0 = x 2 /2 = h > 0 has two solutions for any h > 0, and H| x=0 = cy 5 + y 2 /2 = h, then applying the Descartes's rule when h > 0 we obtain that H| x=0 has exactly one negative root and has two or zero positive roots.
In the case c <ĉ, for h = h 3 there are no crossings of the level curves H = h 3 with the positive y-axis, and for h = h 2 there is exactly one point of the level curve H = h 2 on the positive y-axis which is e 2 . Thus, there exists a homoclinic orbit of e 2 surrounding the origin. With these informations we conclude that the global phase portrait of system (16) when c <ĉ is topologically equivalent to the one of Figure 1(4).
In the case c =ĉ (bifurcation value), for h = h 2 = h 3 there is one crossing point of the level curves H = h 2 = h 3 on the positive y-axis and it is e 2 . Thus, there exist heteroclinic orbits connecting e 2 , e 3 and e 4 . Using the previous analysis we are able to complete the global phase portrait of system (16) when c =ĉ, which is topologically equivalent to the one of Figure 1(5).
Ifĉ < c < 0, then for h = h 3 there are two crossing points of the level curves H = h 3 with the positive y-axis, one of them is located between the center and the saddle e 2 , and the other is located between e 2 and the end of the positive y-axis, and for h = h 2 , as in the case c <ĉ, there is exactly one point (with multiplicity 2) of the level curve H = h 2 on the positive y-axis which is e 2 . Thus, there exists two heteroclinic orbits connecting e 3 and e 4 crossing the positive yaxis (both heteroclinic orbits connecting e 3 and e 4 are forming the boundary of the period annulus of the center). So the global phase portrait of system (16) in this case is topologically equivalent to the one shown in Figure 1(6).

3.2.
Phase portraits when a = 0. For the case when a = 0 we do the a −2/3symplectic change of variables (x, y) → a −1/3 (x, y) and we can suppose without loss of generality that a = 1 and the new Hamiltonian system iṡ whereb = b/a andc = c/a, for simplicity we just use b and c instead ofb andc respectively.
In order to give a complete analysis of all the global phase portraits is necessary study separately the cases when the coefficient c is null or not.
(iii) If b < 0 there are five finite equilibria for system (20), namely e 1 = (0, 0), e 2 , e 3 and The points e 4 and e 5 are of the saddle type. Now we are going to study the infinite equilibria. From (4) system (20) in the local chart U 1 iṡ At infinite, i.e., at z 2 = 0, we have two cases: if b ≥ 0 the origin is the unique equilibrium which is an unstable node, because its eigenvalues are 1 and 5, if b < 0 we have three equilibria, the origin with the same local phase portrait as before, and other two equilibria given by (±(−b) −1/2 , 0) whose eigenvalues are −10 and −2, so they are stable nodes. Therefore, in the local chart V 1 also the origin is a infinite singular point which is an stable node and if b < 0 there exist other two infinite singular points that are unstable nodes.
In the chart U 2 , from (5) the system (20) is given bẏ Here, the origin if it is an equilibrium for system (21), but it is degenerate. We apply the blow-up technique and obtain that the local phase portrait of the origin of system (21) is shown in Figure 8 and consists in one hyperbolic sector, one elliptic sector, one attracting and one unstable parabolic sectors and their distribution depends on sign of b. More specifically if b < 0 the local local phase portrait at the origin of system (21) is shown in Figure 8(a), and if b ≥ 0 is shown in Figure 8(b).
(a) (b) Figure 8. Local phase portrait at the origin of system (21).
All the possible local phase portraits for system (3) when c = 0 and a = 0 are topologically equivalent to the one shown in Figure 9, and depends on the sign of b. Sinceẋ| x=0 = −y,ẏ| x=0 = 0,ẋ| y=0 = 0 andẏ| y=0 = x, the study of the global dynamics for the case b ≥ 0 follows in the same way that for the analysis made for in section 3.1.2, and the global phase portrait on the Poincaré disk is topologically equivalent to the one of Figure 1(2).
If b < 0, we note that the energy level of the saddle are Here there exists a bifurcation valueb = −1/(2(1 + √ 5)) where the four saddles are in the same level of energy. We obtain that one stable separatrix of the saddle e 2 located at the quadrant x > 0 and y < 0 must cross the negative y-axis because it cannot intersect neither the stable separatrices of itself, nor the positive x-axis, at this moment it connects with the unstable separatrix of the saddle e 3 , that is in the quadrant x < 0 and y < 0 due to the fact that the phase portrait is the symmetric with respect to the y-axis. The same behavior happens with one unstable separatrix of e 4 and one stable separatrix of e 5 .
By the previous analysis we must divide the study of the dynamics in three cases: (i) If b <b it holds 0 < h 4 < h 2 so the saddles e 4 and e 5 are in the boundary of the period annulus of the origin and we can force that the phase portrait of system (20) in this case is topologically equivalent to the one Figure 1(11).
(ii) If b =b the four saddles are connected through their separatrices forming the boundary of the period annulus of the center at the origin, thus the global dynamics is as the one of Figure 1(12).
(iii) If b >b it holds 0 < h 2 < h 4 , which implies that the saddles e 2 and e 3 form the boundary of the period annulus of the center at the origin and we can complete the global dynamics as it is shown Figure 1(13). We can conclude that in this case c = 0 and ab = 0 the global phase portrait is topologically equivalent to the ones of Figure 1(2), (11), (12) or (13).

3.2.2.
Case c = 0. The equilibrium points of system (19), according to second equation are characterized by x = 0 or 1 + 4x 2 y + 2by 3 = 0. In the first case, by the first equation implies that there are two equilibria, namely, e 1 = (0, 0) that we know is always a center, and (0, (5c) −1/3 ), which is a saddle if c < 0 and b > 5c/2, or c > 0 and b < 5c/2, and is a center for c < 0 and b < 5c/2, or c > 0 and b > 5c/2; and it is nilpotent for b = 5c/2. In order to study the local behavior at e 2 when b = 5c/2 we apply Theorem 3.5 of [9] for nilpotent equilibria in a similar way that to studyẽ 3 in subsection 3.1.3. Using the notations and results of Theorem 3.5 of [9] we obtain y = f (x) = ((5c) 1 [9] if c > 8/15, or c < 0 we have that a > 0 and then e 2 is a saddle. If 0 < c < 8/15 then a < 0, thus e 2 is a center or a focus, but since our system is Hamiltonian it is a center. If c = 8/15 the equilibrium is a saddle.
, with eigenvalues both zero, but with linear matrix not null, so it is nilpotent. After doing the computations for studding this case using Theorem 3.5 in [9] we arrive that e 3,4 are cusp (under the notation of Theorem 3.5 we have that F (x) =âx 2 withâ = ( Next, we analyze when these points are well defined. More precisely Direct computations provide the following summary: (II.3.1) If b < 4/3 and 2b/5 < c < c * then there exist four more equilibria, namely, e 3 , e 4 , e 5 and e 6 . The equilibria e 5,6 are always saddles, because their characteristic polynomial is in the form α 1 λ 2 + β 1 where α 1 > 0 and β 1 < 0. The equilibria e 3,4 have characteristic polynomial of the same form α 2 λ 2 + β 2 where α 2 > 0. If b < 0 and 2b/5 < c < b 2 /4, then β < 0 and these equilibria are saddles. If b < 0 and b 2 /4 < c or 0 < b < 4/3 and 2b/5 < c, then β is positive and the equilibria e 3,4 are centers. Explicitly, for (II.3) the new equilibria are given by , From the analysis of the linear part of the system at these equilibria we have the following results: If c < c * the equilibria e 3 and e 4 are always saddles because their characteristic polynomials have the form αλ 2 + β, where αβ < 0.
Next, we will study the infinite equilibria. From (4) system (19) in the local chart U 1 becomeṡ At infinity (z 2 = 0) the equilibrium points of system (23) are characterized by z 1 = 0 or real roots of the polynomial 1 + bz 2 1 + cz 4 1 . In the first case the origin is the unique equilibrium and it is an unstable node because their eigenvalues are 1 and 5. In the second case taking z = z 2 1 and let p(z) = 1 + bz + cz 2 , we have the following sub-cases: (II.2) If b < 0, p(z) has a unique root given by z = −2/b, so system (23) has two more equilibria, namely, (± −2/b, 0) and they are degenerate, so we need to shift the equilibria to the origin and then apply a blow-up for understanding the local phase portrait at these equilibrium points. For p 2 = ( −2/b, 0) the technique of blow-up give that the local phase portrait atẽ 3 after translating to the origin is as in Figure 10(a) and consists in one hyperbolic sector, one elliptic sector, one attracting and one repeller parabolic sectors. For p 3 = ( −2/b, 0) the study of the local phase portrait through the blow-up technique give that it is as in Figure  10 These equilibria at infinity are stable nodes because their eigenvalues 5  Figure 11. Figure 11. Local phase portraits at the equilibria of system associated to Hamiltonian (3) when ac = 0.
In summary we have eight possible local dynamics on Poincaré disk, according to Figure 11 the cases are: (i) Figure 11(a): Figure 11 Using similar arguments as in the previous phase portraits in the Poincaré disk, we obtain that the global flow associated to the parameters in the case: Figure 11(a), i.e., in the case (i), the global phase portrait is topologically equivalent to the one of Figure 1(1). Figure 11(b), i.e., in the case (ii), the global phase portrait is topologically equivalent to the one of Figure 1(3). Figure 11(d), i.e., in the case (iv), first we note thatẋ| x=0 = −y(1 + 4y 3 ), y| x=0 = 0,ẋ| y=0 = −x 4 andẏ| =0 = x, this information with the local phase portrait forces that the unstable separatrix of e 2 which is in the region x > 0 and y < 0 must cross the positive x-axis and then must cross the positive y-axis connecting, by the symmetry, with the stable separatrix of e 2 located in the region x < 0 and y < 0. Furthermore, the unstable separatrix of the cuspê 3 located in the region x > 0 and y < 0 must also cross the positive x-axis and then the positive y-axis connecting with the stable separatrix of the other cuspê 4 . These connections force that the global phase portrait on the Poincaré disk is as in Figure 1(7). Figure 11(e), i.e., in the case (v), the global phase portrait is topologically equivalent to the ones of Figure 1(8), (9) or (10). In this case we have a bifurcation curve that depends of b and c where the energy level h 2 of the saddle e 2 coincides with the energy level h 5 = h 6 of the other two saddles e 5 and e 6 (resp.).
Where h 2 = 3 10 5 2/3 c 2/3 and Consider the vector field on the axes, given in the previous analysis of the case (iv). In order to complete the global dynamics we need to know the connections of the unstable separatrix of the saddles e 5 and e 2 . One unstable separatrix of e 5 must cross transversally the positive x-axis because it cannot intersect neither a stable separatrix of itself nor the unstable separatrix of e 2 , and then by continuity must cross orthogonally the positive y-axis, at this moment it connects with one stable separatrix of e 6 . Now we are going to study the possible connections of the remaining separatrices of e 5 and e 6 and the separatrices of e 2 .
The energy levels h 5 = h 6 and h 2 of the saddles coincide for the bifurcation value that depends of b and c. (ii) If h 2 = h 5 the separatrices of e 2 located in the region x > 0 and y < 0 connect with the ones of e 5 enclosing the center at e 3 in the same region, for symmetry the same happens with e 6 and e 4 . In view of this information that we obtain that the global dynamics in this case is topologically equivalent to the one of Figure 1 (9).
(iii) If h 2 > h 5 one stable separatrix of e 5 is a homoclinic orbit, forming the boundary of the periodic annulus of the center e 3 , connecting with one unstable separatrix of itself. Thus, we conclude that the global dynamics in this case is topologically equivalent to the one of Figure 1(10). More precisely for case b 2 − 4c < 0, ∆ > 0, 0 ≤ b < 4/3 and c > 2b/5 we have that in the bifurcation curve h 2 = h 5 , see Figure 12(a), the global phase portrait is topologically equivalent to Figure 1(9), in the region when h 2 < h 5 the global phase portrait is topologically equivalent to the one of Figure 1(8), and in the region when h 2 > h 5 the global phase portrait is topologically equivalent to the one of Figure  1 (10).
Second, in the case b 2 − 4c < 0, ∆ > 0, b ≤ 0 and c > b 2 /4 we have that in the bifurcation curve, as before, the global phase portrait is topologically equivalent to the one of Figure 1(9), for values in the region h 2 > h 5 the global phase portrait is topologically equivalent to the one of Figure 1 (8) and for values in the region h 2 < h 5 (Figure 12(b)) the global phase portrait is topologically equivalent to the one of Figure 1(10).   (6). In this case we have a bifurcation curve where the energy level of the three saddles coincide, in the region that satisfy the existence conditions, see Figure 13(a). On this curve the global phase portrait is topologically equivalent to the one of Figure 1 (5). For the parameters located in the region h 2 < h 5 we have that global phase portrait is topologically equivalent to the one of Figure 1(4). And for the parameters in the region h 2 < h 5 we have that global phase portrait is topologically equivalent to the one of Figure 1 Proceeding in a similar way as for example in the case a = 0 and bc = 0, we verify that for b < b 1 there exists a homoclinic orbit of e 2 forming the boundary of the period annulus of the center at the origin; there is a separatrix connectingẽ 3 withẽ 4 cutting the positive y-axis. These informations force that the global phase portrait in this case is topologically equivalent to the one of Figure 1 (14). For b = b 1 we have that the three saddles are in the boundary of the period annulus of the center, thus the global phase portrait in this case is topologically equivalent to the one of Figure 1 (15). If b 1 < b < 0 the two saddlesẽ 3 andẽ 4 are in the boundary of the period annulus of the center; the separatrices of e 2 located in the region x > 0 and y < 0 connect with the infinite equilibrium point p 5 surrounding its elliptic sector. By symmetry we obtain that the global phase portrait in this case is topologically equivalent to the one of Figure 1(16).
For the parameters associated to the case of Figure 11(f), when we are in the case (vi), it is verified that the energy level h 3 of the saddle e 3 and e 4 is the same, and the level of energy of the saddle e 5 and e 6 , called h 5 , is the same. Here, a bifurcation curve appears when h 3 = h 5 , i.e., the four saddle is in the same energy level, in this case two separatrices of e 3 and two separatrices of e 4 are forming the boundary of the period annulus of e 2 ; the four saddles are in the boundary of the period annulus of the center at the origin. This forces that the global flow is topologically equivalent to the one of Figure 1(17). In the region h 3 > h 5 (see Figure 13(b)) it holds that 0 < h 5 < h 3 < h 2 (where h 2 is the level of energy of the center e 2 , and h = 0 is the level of energy of the center at the origin), so the origin is enclosed by a boundary of the period annulus formed by two separatrices of each saddle e 5 and e 6 located in the region y < 0, on the other hand the separatrices of e 3 and e 4 are forming the boundary of the period annulus of the center e 2 , this implies that the global phase portrait in this case is topologically equivalent to the one of Figure 1 (18). Finally in the region h 3 < h 5 it holds 0 < h 3 < h 2 < h 5 , so in this case the period annulus of the origin has the boundary formed by separatrices of e 3 and e 4 . Furthermore, the boundary of the period annulus of e 2 is formed by separatrices of e 3 and e 4 too. From previous arguments it is clear that one separatrix of e 5 connects with one separatrix of e 6 crossing the y-axis. Thus the global phase portrait is topologically equivalent to the one of Figure 1 (19). Finally, for the parameters associated to the case of Figure 11(h), i.e., when (viii) holds, we note that the energy level of e 2 is h 2 = 3/(10(5c) 2/3 ), for the saddle e 3 and e 4 is b(9b − 16) − 20c + 16 + 3b − 6 (2 √ 9b 2 − 16b − 20c + 16 + 4b − 8) 2/3 and for the saddle e 5 and e 6 is For studying all the possible global dynamics we consider first the relation between h 3 and h 5 , in Figure 14 Thus we have five regions in relation to the energy level of the saddle. In the first region it holds that h 2 < h 5 < h 3 , so the boundary of the period annulus of the center at the origin is formed by separatrices of e 2 , one unstable separatrix of e 3 must cross the positive x-axis, and then must cross the positive y-axis connecting with a stable separatrix of e 4 by the symmetry. Furthermore we know form previous analysis that one unstable separatrix of e 5 connects with one stable separatrix of e 6 crossing the positive y-axis. These informations show that the global phase portrait under these conditions is topologically equivalent to the one of Figure 1 (20).
In the second region we have that the energy level of h 2 coincide with h 5 , so the three saddle on this energy level e 2 , e 5 and e 6 form the boundary of the period annulus of the center at the origin. As before one unstable separatrix of e 5 connects with one stable separatrix of e 6 crossing the positive y-axis. Thus the global phase portrait under these conditions is topologically equivalent to the one of Figure 1 (21).
In the other three regions (h 5 < h 2 < h 3 , h 5 < h 2 = h 3 and h 5 < h 3 < h 2 ) the energy level h 5 is less than the energy levels h 2 and h 3 . It is checked that when h 2 = h 3 the level curve is formed by disconnected components and the saddle e 2 is not connected with the saddle e 5 or e 6 . In Figure 15 it is shown the evolution of level curve H = h 2 through the regions h 5 < h 2 < h 3 , h 5 < h 2 = h 3 and h 5 < h 3 < h 2 . In these last three cases some separatrices of the saddle e 5 and e 6 are forming the For the region h 2 < h 3 = h 5 we verify that there exists a homoclinic orbit of e 2 forming the boundary of the period annulus of the center at the origin; the saddles e 3 , e 4 , e 5 and e 6 are connected by some of their separatrices, the separatrix connecting e 3 with e 4 is cutting the positive y-axis. These informations force that the global phase portrait in this case is topologically equivalent to the one of Figure  1 (23). For the parameters (b, c) such that the energy level coincide h 2 = h 3 = h 5 , some separatrices of all the saddles e 2 , e 3 , e 4 , e 5 and e 6 form the boundary of the period annulus of the center as it is shown in Figure 1 (24). Thus the global phase portrait in this case is topologically equivalent to the one of Figure 1 (24). In the region where the parameters (b, c) verify 0 < h 3 = h 5 < h 2 some separatrices of the four saddles e 3 e 5 and e 6 are in the boundary of the period annulus of the center; the separatrices of e 2 located in the region x > 0 and y < 0 connect with the infinite equilibrium points p 3 and p 5 . By symmetry we obtain that the global phase portrait in this case is topologically equivalent to the one of Figure 1(25).
Region h 3 < h 5 : In this region there are two bifurcation curves given by the functions f 23 (b, c) = h 2 − h 3 = 0 and f 25 (b, c) = h 2 − h 5 = 0, as it is shown in Figure 16 (b).
So we have five subregion to analyze. First, if h 2 < h 3 < h 5 , there exist a homoclinic orbit formed by two separatrices of e 2 that form the boundary of the period annulus of the origin, next the saddle e 3 and e 4 are connected one of their separatrices crosses the positive y-axis, and other separatrices of each one are surrounding the period annulus of the center, passing next to e 2 and continue to an infinite equilibria. Thus the separatrices of e 5 and e 6 cannot cross the y-axis and they connect with infinite equilibria. These information force that the global phase portrait is topologically equivalent to the one of Figure 1(26).
In the other regions always the energy levels of e 3 and e 4 is less or equal than the energy levels of the others saddles, so e 3 and e 4 are always in the boundary of the period annulus of the center. In the region where the parameters b and c satisfy that h 2 = h 3 < h 5 and the three saddles e 2 , e 3 and e 4 form the period annulus of the center at the origin, this implies that the separatrices of e 5 and e 6 must connect at the infinite with infinite equilibria. Thus we obtain that the global phase portrait is topologically equivalent to the one of Figure 1(27).
In the region where b and c satisfy that h 3 < h 2 < h 5 we have that the boundary of the period annulus is formed by separatrices of e 3 and e 4 , the level curve passing through e 2 surround this annulus, two separatrices of this saddle border the annulus, approaching to the saddle e 3 and e 4 and then continue to an infinite equilibrium point. As before the separatrices of e 5 and e 6 must connect at infinite with an infinite equilibrium point, one of them first approaches to the saddle e 2 . From the previous information we conclude that the global phase portrait is topologically equivalent to the one of Figure 1(28).
In the region where b and c satisfy that h 3 < h 2 = h 5 we have, as before that some of the separatrices of the saddles e 3 and e 4 form the boundary of the period annulus of the center, and the saddles e 2 , e 5 and e 6 are connected by some of their separatrices, and their other separatrices must connect at infinite with an equilibrium point, from this the global phase portrait is topologically equivalent to Figure 1(29).
Finally, in the region where h 3 < h 5 < h 2 we have that the boundary of the period annulus is formed by separatrices of e 3 and e 4 surrounded for outside by two separatrices of e 5 and e 6 . Here one unstable separatrix of e 6 cross the negative x-axis connecting with one stable separatrix of e 5 . The separatrices of e 2 must connect at infinite by continuity. Thus we conclude that the global phase portrait is topologically equivalent to the one of Figure 1(30).
This completes the proof of Theorem 1.