Bifurcation Diagrams and Global Phase Portraits for Some Hamiltonian Systems with Rational Potentials

. In this paper we study the global dynamical behavior of the Hamil- tonian system ˙ x = H y ( x,y ), ˙ y = − H x ( x,y ) with the rational potential Hamil- tonian H ( x,y ) = y 2 / 2 + P ( x ) /Q ( y ), where P ( x ) and Q ( y ) are polynomials of degree 1 or 2. First we get the normal forms for these rational Hamiltonian sys- tems by some linear change of variables. Then we classify all the global phase portraits of these systems in the Poincar´e disk and provide their bifurcation diagrams.


Introduction and statement of the main results
A great deal of work has been done for studying the global dynamics of the planar polynomial differential systems, for example see [3,4,6,8,10,15]. Vulpe [16] studied the global phase portraits of the quadratic polynomial systems having a center. In [14] Schlomiuk gave the bifurcation diagrams for the global phase portraits of these quadratic systems. Artés and Llibre [2] provided the global phase portraits of all quadratic Hamiltonian systems. The authors of [12] presented the phase portraits of the quadratic polynomial vector fields having a rational first integral of degree 3. Guillamon et al. [11] gave an algorithm to obtain the phase portraits of the separable Hamiltonian system with the Hamiltonian function H(x, y) = F (x) + G(y). Colak et al. [5,7] presented the global phase portraits in the Poincaré disk of all Hamiltonian linear type centers of polynomial systems having linear plus cubic homogeneous terms, and gave their bifurcation diagrams.
In this paper we consider the Hamiltonian system (1)ẋ = H y (x, y),ẏ = H x (x, y) with a rational Hamiltonian function (2) H = H(x, y) = where P (x) and Q(y) are real polynomials of degree at most 2. We denote by the set L = {(x, y)|Q(y) = 0} the points where the Hamiltonian vector field are not defined. The system associated to the Hamiltonian function (2) has the form

TING CHEN AND JAUME LLIBRE
where P ′ (x) and Q ′ (y) indicate the derivatives of the polynomials P and Q with respect to x and y respectively, and the dot denotes derivative with respect to the time t. Under the time rescaling (4) dt dτ = Q 2 (y), the rational Hamiltonian system (3) becomes the polynomial system (5) x ′ = yQ 2 (y) − P (x)Q ′ (y), where x ′ and y ′ denote derivatives of x and y with respect to τ respectively. But the new system (5) is not Hamiltonian in general, we call it an integrable non-Hamiltonian system. Thus we can study the phase portraits of the Hamiltonian (3) analyzing the associated polynomial differential system (5). Martínez and Vidal [13] classified the phase portraits of the Hamiltonian system (1) with the potential V (x, y) = P (x)/Q(x). Our main result is the following one. where P (x) and Q(y) are polynomials of degree at most 2, after a linear change of variables and a rescaling of its independent variables, it can be written as one of the following classes: P (x) = ax + b and Q(y) = Ay 2 + By + C with aA = 0; (IV) H ± 22 = y 2 2 ± x 2 +∆ y 2 + By+ C , with ∆ = (c/a − b 2 /(4a 2 ))/|a|, B = B/ |a|, and C = C/|a|, if P (x) = ax 2 + bx + c and Q(y) = Ay 2 + By + C with aA = 0.
In each expression the sign " ± " corresponds to " + " for a > 0 and " − " for a < 0.
In Theorem 1.1 we get that the flow of the Hamiltonian system (3) associated to the rational Hamiltonians H − 11 and H − 21 are topologically equivalent to the one of the Hamiltonians H + 11 and H + 21 , respectively. For instance, the phase portraits of the rational Hamiltonian system of H − 11 can be obtained from H + 11 by simply doing the changes (x, b) → (−x, −b). We will classify the global phase portraits of the families of the systems of Theorem 1.1 in the Poincaré disk, and will provide the bifurcation diagrams of these phase portraits.
In order to classify the phase portraits of the polynomial systems (3), the main step is the characterization of the finite and infinite equilibrium points in the Poincaré compactification, for more details about the Poincaré compactification see [9]. The second step for determining the global flow of these polynomial vector fields is to characterize their separatrices. It is known that the separatrices of a polynomial differential system are all the infinite orbits, all the finite singular points, the separatrices of the hyperbolic sectors of the finite and infinite singular points, and the limit cycles. Since the existence of a first integral prevents the existence of limit cycles, we do not have to determine the limit cycles of system PHASE PORTRAITS AND BIFURCATION DIAGRAMS 3 (3). In the Poincaré disk D 2 , let Σ be the closed set formed by all the separatrices, the components of D 2 \ Σ are called the canonical regions (for more details see [9]). We denote by S and R the number of separatrices and canonical regions of a given phase portrait, respectively.
In the phase portraits of the following theorems the straight lines of the set L are indicated by the dash line " − − − ". A small circle in the phase portraits denotes an equilibrium point of the polynomial system (5), but no an equilibrium point of the rational Hamiltonian system (3), and we called it a virtual equilibrium point or a virtual singular point.    (1.13) S=2, R=3; (1.1) S=2, R=2; (1.12) S=4, R=1; (1.16) S=4, R=2; (1.14) S=2, R=2; (1.15) S=4, R=3; (    1.23 if B = 0, C > 0 and ∆ > C 2 /2, or C > B 2 /4 > 0, ∆ > F 4 and D 5 < 0,   We will give some preliminary definitions and theorems in the following section. We prove how to obtain the normal forms given in Theorem 1.1 in Section 3. We determine the phase portraits of the vector fields (I)-(IV) in Sections 4-7, respectively, in other words we prove Theorems 1.2-1.5. In a similar way to the proof of Theorem 1.5 we can obtain the phase portraits of the Hamiltonian H − 22 , but for the sake of simplicity we will not provide the proof of Theorem 1.6 here.    Figure 6. The bifurcation diagram of the phase portraits for  Figure 9. The bifurcation diagram of the phase portraits for  Figure 10. The bifurcation diagram of the phase portraits for  Figure 11. The bifurcation diagram of the phase portraits for 4 /8192 and f 9,10,11,12 (B, C) are convenient functions.

Preliminaries
Now we introduce some basic results that we will need for the analysis of the local phase portraits of the finite and infinite singular points of the polynomial vector fields.
2.1. Poincaré compactification. In this subsection we present some preliminaries about the Poincaré disk and the compactification of a polynomial vector field, for more details see Chapter 5 of [9].
We denote by P (R 2 ) the set of all polynomial vector fields on R 2 of the form where X(x 1 , x 2 ) and Y (x 1 , x 2 ) are real polynomials in the variables x 1 and x 2 of degree d. We consider R 2 as the plane in R 3 defined by s = (s 1 , s 2 , s 3 ) = (x 1 , x 2 , 1). The plane R 2 is tangent to the Poincaré sphere S 2 = {s ∈ R 3 : ||s|| = 1} at the north pole. We consider the central projection of R 2 onto the sphere S 2 , that is, for each point x ∈ R 2 = (x 1 , x 2 , 1) ∈ R 3 we take the straight line l through this point and the origin of R 3 and the central projection sends the point x to the two intersection points of the straight line l with the sphere S 2 . Then we obtain a vector field X ′ formed by two copies of X: one on the northern hemisphere {s ∈ S 2 : s 3 > 0}, and the other on the southern hemisphere {s ∈ S 2 : s 3 < 0}. Note that the equator S 1 = {s ∈ S 2 : s 3 = 0} corresponds to the infinity of R 2 . This vector field X ′ on S 2 \S 1 can be extended to a vector field p(X) on the whole sphere S 2 multiplying the vector field X ′ by x d 3 . The vector field p(X) is called the Poincaré compactification of the vector field X.
In order to compute the expression of the vector field p(X), we consider the six local charts given by for i = 1, 2, 3, and the diffeomorphisms defined as ϕ i (s) = −ψ i (s) = (s m /s i , s n /s i ) = (u, v) for m < n and m, n = i. We note that (u, v) will play different roles in every local chart. Then the expression of p(X) in the local chart U 1 is Analogously, the expression of p(X) in the local chart U 2 is The expression of p(X) in the local charts V i is the same as in U i multiplied by (−1) d−1 for i = 1, 2, 3. That is, if s is an equilibrium point, then −s is also an equilibrium point. Note that the local behavior near −s is the local behavior near s multiplied by (−1) d−1 . Hence it is enough to study the Poincaré compactification restricted to the northern hemisphere plus S 1 to study the vector field X. For drawing the phase portraits we will consider the orthogonal projection π(s) = (s 1 , s 2 ) of the northern hemisphere onto the closed unit disk D 2 centered at the origin of coordinates in the plane s 3 = 0. D 2 is called the Poincaré disk. Infinite equilibrium points of X are the equilibrium points of the vector field p(X) which are on S 1 . So for studying the infinite equilibrium points it suffices to look the ones at U 1 | v=0 and at the origin of U 2 . We compute the finite equilibrium points of X by the chart U 3 , which are the equilibrium points of p(X) in S 2 \ S 1 .
2.2. Some other preliminaries. For proving our main result we need to study different types of equilibria, where the linearly zero equilibria are studied using the changes of variables called blow-up's. For more details about the blow-up's see [1].
Sometime the explicit expressions of the finite equilibrium points and their eigenvalues in terms of the parameters are complicated, therefore it is hard to analyze their existence and their local phase portraits. For this reason we will provide an alternative way for studying the finite equilibria. Firstly we present one way to determine the real roots of the polynomial (12) f (x) = a 0 x n + a 1 x n−1 + · · · + a n .
We compute the discriminant sequence {D 1 , D 2 , · · · , D n } of the polynomial (12) (see more details in [17]) and determine the sign list of the discriminant sequence, where the sign function is Then we construct the associated revised sign list [r 1 , r 2 , · · · , r n ] which will give all the information about the number of real roots of the polynomial (12). For any sign list [s 1 , s 2 , · · · , s n ], the revised sign list [r 1 , r 2 , · · · , r n ] is obtained as follows: 1. If s k = 0 we write r k = s k . 2. If [s i , s i+1 , · · · , s i+j ] is a section of the given sign list such that s i+1 = · · · = s i+j−1 = 0 with s i s i+j = 0, we replace the subsection [s i+1 , s i+2 , · · · , s i+j keeping the number of terms.
As a result there are no zeros between nonzero elements of the revised sign list. Thus for instance the revision of the sign list [1, −1, 0, 0, 0, 0, 0, 1, −1, 0, 0] is From [17] we have the following theorem.
Theorem 2.1. For a polynomial (12) if the number of the sign changes of the revised sign list of [r 1 , r 2 , · · · , r n ] is v, and the number of nonzero members of the revised sign list is l, then the number of the distinct real roots of (12) equals l − 2v.
When we find the number of finite equilibrium points of a system, we can compute the topological indices of the equilibrium points, both finite and infinite. Here we will present two important theorems, the Poincaré Formula and the Poincaré-Hopf Theorem. The former allows to compute the index of an equilibrium point of a planar vector field. The latter is suitable for the systems in a 2-dimensional sphere. For more details about these theorems see Chapter 6 of [9].   We have the following remark.
Remark 2.5. Since the flow of Hamiltonian systems preserves the area, we have that any finite equilibrium of Hamiltonian systems must be either a center, or a union of an even number of hyperbolic sectors. In particular, the finite nilpotent equilibrium points of Hamiltonian planar polynomial vector fields are either saddles, centers, or cusps, for more details see Theorem 3.5 of [9].
3. Proof of Theorem 1.1 (I) If P (x) = ax + b, Q(y) = Ay + B and aA = 0, we have V (x, y) = ax/A+b/A y+B/A . Without loss of generality we can take A = 1, and obtain the rational potential V (x, y) = ax+b y+B . By the linear change of variables the Hamiltonian function (2) will be the form (II) In the case V (x, y) = ax 2 +bx+c Ay+B with aA = 0, without loss of generality we can take A = 1, and have V (x, y) = a(x+b/(2a)) 2 +c−b 2 /(4a) y+B . Considering the linear change of variables and writing ∆ = c − b 2 /(4a), the Hamiltonian function (2) has the form H ± 21 = |a|( Y 2 2 ± X 2 ±∆ Y +B ). Doing the rescaling in time dt = dτ / |a| we get the normalized Ay 2 +By+C with aA = 0, without loss of generality we can take A = 1, and consequently V (x, y) = ay+b y 2 +By+C . Using a suitable change of variables we have the normalized Hamiltonian function (IV) Finally in the case V (x, y) = ax 2 +bx+c Ay 2 +By+C , without loss of generality we can take A = 1, and so V (x, y) = a(x+b/(2a)) 2 +c−b 2 /(4a) y 2 +By+C . Doing the change of variables the normalized Hamiltonian function (2) will be where ∆ = (c/a − b 2 /(4a 2 ))/|a|. This completes the proof of Theorem 1.1.

Proof of Theorem 1.2
The rational Hamiltonian function given in statement (I) of Theorem 1.1 is and the associated Hamiltonian system for H + 11 is Note that we can assume B ≥ 0 because system (14) is invariant under the trans- We apply the rescaling in time dt = (y + B) 2 dτ , and system (14) becomes We study the dynamics of the infinite equilibrium points of system (15) through the Poincaré compactification. In the local chart U 1 system (15) becomes On the infinity, i.e. on v = 0, the origin is the unique equilibrium point, and its the linear part is identically zero. In order to describe the local phase portrait at the origin of U 1 we do the blow-up (u, v) → (u, w) with w = v/u. Then we have the system Doing a rescaling of the time we eliminate the common factor u 2 between u ′ and w ′ . Then we obtain the system When u = 0 system (18) can have two equilibrium points E 1 = (0, 0) and If B = 0 system (18) has only the equilibrium point E 1 on u = 0, which is linearly zero. We do a second directional blow-up (u, w) → (u, w) with w = w/u and removing the common factor u, then we obtain the system Figure 12. Blow-up at the origin of system (16) in Figure 13. Blow-up at the origin of system (16) When u = 0 the origin is the unique equilibrium point of system (19), and it is a saddle, with the stable and unstable separatrices on the u-axis and w-axis, respectively (see Figure 12(a)). Going back through the change of variables, taking into account that both axes are invariant by system (16), we obtain . Hence we have that the phase portrait in a neighborhood of the origin of system (16) consists of two hyperbolic sectors and one parabolic sector (see Figure 12(d)).
If B > 0 the equilibrium E 1 is linearly zero and E 2 is a saddle. The local phase portrait of E 1 coincides with the one of the case B = 0, see Figure 12(b). We superpose the study of the equilibria E 1,2 and obtain the local behavior of system (18), see Figure 13(a). Going back through the change of variables and taking into account the behavior of the flow of system (16) on the axes, we have Then we obtain that the origin of system (16) has two hyperbolic and one parabolic sector, see Figure 13(b).
Next we check if the origin of the local chart U 2 is an equilibrium point, and we get Obviously, the origin of system (20) is not an equilibrium point.
System (15) has two equilibria in the infinite region, one in U 1 , and its diametrically opposite in V 1 . The flow in the chart V 1 has the same sense as in U 1 because the degree of system (15) is 3. Now we must analyze which of the equilibrium points of system (15) in the Poincaré disk are or not effectively equilibrium points of the rational system (14). We observe that the set L = {(x, y)|y + B = 0} corresponds to the straight line, which connects the origins in U 1 and V 1 . Therefore there exist no equilibria at infinity for the rational Hamiltonian system (14).
Having determined the infinite equilibria of system (15) and of the Hamiltonian system (14), we now analyze their finite equilibrium points. The linear part of system (15) is System (15) has the unique finite equilibrium point we get that p 1 is a hyperbolic attracting node of system (15). In addition, the equilibrium point p 1 is on the straight line y = −B, i.e. in the set L. Hence it is a virtual equilibrium point, and system (14) has no finite equilibrium points. Using the level of energy on the y-axis H + 11 | x=0 = y 2 /2 + b/(y + B) = h it follows that the integral curves cross the y-axis at most twice. Then the global phase portrait of the rational Hamiltonian system (14) is given by 1.1 of Figure 1. This concludes the proof of Theorem 1.2.

Proof of Theorem 1.3
Next we study the rational Hamiltonian system H + 21 = y 2 2 + x 2 +∆ y+B , here the Hamiltonian system of H + 21 is We apply the rescaling dt = (y + B) 2 dτ , and have the system which is a polynomial system of degree 3.
We analyze the infinity of system (23) through the Poincaré compactification. The associated system (23) in U 1 is Taking v = 0 the origin is the unique equilibrium point which is linearly zero. In order to understand the local phase portrait of this equilibrium point, we apply the directional blow-up (u, v) → (u, w) with w = v/u. And after eliminating the common factor u between u ′ and w ′ we have For u = 0 system (25) has two possible equilibrium points E 1 = (0, 0) and E 2 = (0, −1/B).
When B = 0 the origin of system (25) is the unique equilibrium point on u = 0 and it is linearly zero. Again we do another blow-up (u, w) → (u, w) with w = w/u, remove the common factor u, and obtain the system  For u = 0 the origin is the unique equilibrium, which is again linearly zero. We perform the directional blow-up (u, w) → (u, W ) with W = w/u and get the system after eliminating the common factor u. For u = 0 system (27) has the origin as the unique equilibrium point, which is a hyperbolic saddle. Going back through the change of variables we get that locally E 1 has two hyperbolic sectors and one parabolic sector, see Figure 14(c). Again going back through the change of variables until system (24) and taking into account the behavior of the flow of system (24) on the axes, we have Hence we obtain that the local phase portrait at the origin in U 1 has two hyperbolic sectors and two parabolic sectors see Figure 14(d).
Assume that B = 0. The equilibrium point E 1 is linearly zero and E 2 is a saddle. Similarly to the above case the local phase portrait of E 1 is shown in Figure 14(c). We superpose the analysis of the equilibria E 1,2 , see the left of Figures 15(a) and 15(b) when B < 0 and B > 0, respectively. Going back through the change of variables and taking into account the phase portrait of the flow on the axes, we obtain that the local phase portrait at the origin of U 1 also has two hyperbolic and two parabolic sectors (see the right of Figures 15(a) and 15(b)). Now we check the origin of U 2 , in which system (23) becomes The origin of U 2 is not the equilibrium point.
Thus the associated polynomial system (23) has two equilibria at infinity, one in U 1 and its diametrically opposite in V 1 . Note that the degree of system (23) is 3 so the flow in the chart V 1 has the same sense as in U 1 . The set L = {(x, y)|y + B = 0} in the Poincaré disk is a straight line connecting the origin in U 1 and V 1 . We remark that in this case the origins in U 1 and V 1 are not equilibrium points for the rational Hamiltonian system (22) because they are in L.
This finishes the study of the infinite equilibrium points of system (23) and of the rational Hamiltonian system (22), we now focus on the finite region. Since the level of energy on the x-axis H + 21 | y=0 = x 2 + ∆/B = h has at most two solutions for any h ∈ R, the integral curves of systems (22) and (23) cut the x-axis at most once at each region x > 0 and x < 0 when B = 0. And by the level of energy on the y-axis H + 21 | x=0 = y 2 /2 + ∆/(y + B), the integral curves of systems (22) and (23) cut the y-axis at most twice. Next we consider the following cases: (i) ∆ = 0, (ii) ∆ > 0 and (iii) ∆ < 0.
(i.a) If ∆ = 0 and B = 0 system (23) has one finite equilibrium point p 1 = (0, 0), which is linearly zero. Performing the directional blow-up (x, y) → (x, w) with w = y/x and eliminating the common factor x, system (23) becomes When x = 0 the origin is the unique equilibrium point of system (29), and it is an attracting node. Going back through the change of variables and taking into account the behavior of the flow on both axes, we get that locally the origin of systems (23) consists of one hyperbolic sector, one elliptic sector and two parabolic sectors (see Figure 16(a)). In fact the equilibrium point p 1 is a virtual equilibrium point of the rational Hamiltonian system (22). Thus there exist no finite equilibria for system (22). We obtain the phase portrait 1.2 of Figure 1.
(i.b) If B = 0 system (23) has two finite equilibrium points p 1 = (0, 0) and p 2 = (0, −B). The eigenvalues of the equilibrium point p 1 are λ 1,2 = ± √ −2B 3 . If B > 0 the equilibrium p 1 is a center because system (23) is symmetric with the y-axis. The equilibrium point p 2 is linearly zero, doing the change of variables x → x and y → y − B, we translate this equilibrium point to the origin and we have the system Now we do the blow-up using (x, y) → (x, w) with w = y/x, and eliminate the common factor x between x ′ and w ′ . Then we have For x = 0 the origin of system (31) is the unique equilibrium and it is an attracting node. Going back through the change of variables, taking into account the behavior of the flow on the axes of system (30), we have Thus we get that the origin of system (30) has two hyperbolic and two parabolic sectors (see Figure 16(b)). Actually the equilibrium p 2 is a virtual equilibrium point of the associated system (22) because it is in the set L. Therefore there exists only one finite equilibrium for the rational Hamiltonian system (22). The phase portrait in this case is topologically equivalent to 1.3 of Figure 1.
If B < 0 we have that p 1 is a saddle. Similarly to the above case we obtain that the origin of system (30) consists of two elliptic sectors and two parabolic sectors (see Figure 16(c)). On the other hand the equilibrium p 2 is a virtual equilibrium point of the Hamiltonian system (22), hence we have the phase portrait 1.4 of Figure 1.
Next we consider the case ∆ = 0, the explicit expressions for the finite equilibrium points of system (23) and their eigenvalues in terms of the parameters b and B are complicated. Hence we need firstly to find the number of finite equilibrium points for system (23). Using Theorems 2.2 and 2.4 we count the indices of known equilibrium point, then we can deduce the type of the remaining finite equilibrium points.
The infinite equilibrium points in the Poincaré sphere are the origin of U 1 , and also the corresponding point in V 1 . The origins of U 1 and V 1 consist of two hyperbolic sectors respectively, by Theorem 2.2, they have index 0. Thus the finite equilibrium points must have total index 2 in the Poincaré sphere. On the other hand, it is easy to get that the cubic equation (32) have one, or two, or three real roots if 4B 3 ∆ + 27∆ > 0, or 4B 3 ∆ + 27∆ = 0, or 4B 3 ∆ + 27∆ < 0, respectively. Hence we have the following cases. (ii.c) If 4B 3 + 27∆ < 0 system (23) has three finite equilibrium points, there must be either just two centers and one saddle. Assume that the saddle is on the boundary of the period annulus of one center. We claim that this saddle cannot be on the boundary of the other center. If this were the case a straight line l through the center passing sufficiently close to the saddle would have four intersection points with the separatrices which are on the same energy level as the saddle (two with the boundary of the period annulus of one center and two with the boundary of the other center), see Figure 17 for an illustration. This means that on the straight line l, which could be defined by y = kx for some real number k, the equation We analyze the rational Hamiltonian system H 12 = y 2 2 + x+b y 2 +By+C given in statement (III) of Theorem 1.1, and the Hamiltonian system of H 12 is Note that we can assume B ≥ 0 because by the linear change (x, y, t, b, B, C) → (x, −y, −t, b, −B, C) system (34) is invariant. We will study the global phase portraits of system (34) when C = B 2 /4 and C = B 2 /4 separately.  System (35) in the local chart U 1 is For v = 0 the origin is the unique equilibrium, which is linearly zero. In order to understand the local behavior of this equilibrium point we apply the directional blow-up (u, v) → (u, w) with w = v/u. And after eliminating the common factor u 3 we have Assume that B = 0 system (37) has only one equilibrium point E 1 = (0, 0) on u = 0, and it is linearly zero. Again we do another blow-up (u, w) → (u, w) with w = w/u, remove the common factor u, and obtain the system For u = 0 the origin is the unique equilibrium, which is saddle. We recover the local behavior at the origin of system (36) going back through the blow-up, the equilibrium E 1 of system (37) has two hyperbolic sectors and two parabolic sectors (see Figure 19(b)), and the origin of U 1 consists of two hyperbolic and one attracting sectors (see Figure 19(c)).

22
TING CHEN AND JAUME LLIBRE Figure 21. The phase portrait of system (35).
When B > 0 system (37) has two equilibria E 1 and E 2 = (0, −2/B) on u = 0. The equilibrium E 1 is linearly zero and E 2 is a saddle. Similarly to the above case, we obtain that E 1 consists of two hyperbolic sectors and two parabolic sectors. We superpose the study of the equilibrium points E 1,2 of system (37), see Figure 20(a). Going back to the origin of U 1 and taking into account the behavior of the flow on the axes, we have Thus the origin of U 1 consists of two hyperbolic sectors and one parabolic sector (see Figure 20(b)). In U 2 system (35) acquires the form (39) the origin is not the equilibrium point in U 2 .
For the polynomial system (35) we have found two equilibria in the infinite region, one in U 1 and its diametrically opposite in V 1 , which have the opposite sense. The set L = {(x, y)|y + B/2 = 0} is a straight line connecting the north and south poles. In this case the north and south poles are not equilibrium points for the rational Hamiltonian system (34) because they are in the set L.
To complete the global dynamics associated to H 21 , we are going to analyze more properties of the level of energy. Using the energy relations H 21 | y=0 = 4(x+b)/B 2 = h and H 21 | x=0 = y 2 /2 + b/(y + B/2) 2 = h, we have that the integral curves cross the x-axis at most once when B = 0 and cross the y-axis at most twice. Next we consider the finite equilibria for systems (35) and (34). System (35) has one finite equilibrium point p 1 = (−b, −B/2), which is an attracting node. We get the phase portrait of system (35), see Figure 21. Going back to the Hamiltonian system (34), p 1 is a virtual equilibrium point and the integral curves of system (34) have the opposite orientation with respect to system (35) in the region y < −B/2. Therefore in this case the phase portrait of the Hamiltonian system (34) is topologically equivalent to 1.11 of Figure 1.
(ii) Now we consider the case C = B 2 /4, and obtain the system (40) x ′ = −(x + b)(2y + B) + y(y 2 + By + C) 2 , y ′ = −(y 2 + By + C), doing the rescaling dt = (y 2 + By + C) 2 dτ . In the chart U 1 system (40) becomes For v = 0 the origin is the unique equilibrium point and it is linearly zero. We need to do the blow-up to understand the local phase portait in a neighborhood of the origin of system (41). We do the directional blow-up (u, v) → (u, w) with w = v/u and have after eliminating the common factor u 4 .
If B 2 − 4C < 0 the origin E 1 is the unique equilibrium point of system (42) on u = 0, which is linearly zero. We perform the second blow-up (u, w) → (u, w) with w = w/u and eliminate the common factor u 4 , then we have (43) For u = 0 system (43) has the origin as the equilibrium point, and it is a saddle. Going back through the blow-up until system (41) in U 1 and taking into account the behavior of the flow on the axes, we have that E 1 has two hyperbolic sectors and two parabolic sectors (see Figure 22(a)), and the origin of system (41) consists of one elliptic, one hyperbolic and two parabolic sectors (see Figure 22(b)). In this case the separatrix must coincide with the u-axis, because it is invariant for system (41).
If B > 0 and C = 0 system (42) has two equilibrium points E 1 and E 2 = (0, −1/B) on u = 0. Similarly to the above case E 1 is linearly zero which consists of two hyperbolic sectors and two parabolic sectors, and E 2 is a saddle (see Figure  22(c)). Going back through the blow-up and taking into account the behavior of the flow on the axes, we have u ′ | v=0 = −u 6 and v ′ | u=0 = Bv 5 . Then we see that the origin of U 1 in this case has four hyperbolic and two parabolic sectors (see Figure  22(d)).
If B 2 > 4C and C = 0 system (42) has three equilibrium points E 1 and E 2,3 = (0, (−B ± √ B 2 − 4C)/(2C)) on u = 0. The equilibria E 2,3 are saddles. Similarly we obtain that the local behavior of E 1,2,3 as it is shown in Figure 23(a). Going back through the blow-up and taking into account the behavior of the flow on the axes, we have u ′ | v=0 = −u 6 , u ′ | u=0 = −Cv 5 and v ′ | u=0 = Bv 5 . Therefore the origin of U 1 has also four hyperbolic and two parabolic sectors (see Figures 23(b) and 23(c)). In U 2 system (40) acquires the form the origin is not the equilibrium point in U 2 .
Hence the associated polynomial system (40) has two equilibria in the infinite region, one in U 1 and its diametrically opposite in V 1 , which have the same sense. When B 2 −4C > 0 we observe that the set L = {(x, y)|y 2 +By+C = 0} corresponds to a couple of straight lines y = (−B ± √ B 2 − 4C)/2, which connect the north and south poles in the Poincaré disk. Therefore there exist no equilibrium at infinity for the rational system (34). If B 2 − 4C < 0 the set L = {(x, y)|y 2 + By + C = 0} is empty and the rational system (34) has two equilibria in the infinite region.
By the level of energy on the x-axis H 12 | y=0 = h the integral curves, which have a point at the x-axis, cross the x-axis exactly once. And by the energy on the y-axis H 12 | x=0 = y 2 /2 + b/(y 2 + By + C), we have that the integral curves cross the y-axis at most twice. Now we study the finite equilibrium points of systems (34) and (40). If B 2 − 4C < 0 they have no equilibria. Therefore we obtain that the phase portraits of systems (34) and (40) are topologically equivalent to 1.12 of Figure 1.
If B 2 − 4C > 0 system (40) has two equilibria points p 1,2 = (−b, (−B ± √ B 2 − 4C)/2), which are an attracting node and a repelling node. The equilibrium points p 1,2 of system (40) are two virtual equilibrium points in the rational Hamiltonian system (34). Thus there exist no finite equilibria for system (34). The phase portrait of system (34) is topologically equivalent to 1.13 of Figure 1. Thus we conclude the proof of Theorem 1.4.

7.
Proof of Theorem 1.5 Next we study the Hamiltonian system H + 22 = y 2 2 + x 2 +∆ y 2 +By+C , and the associated vector field of H + 22 is System (45)  Hence we only need to analyze the global phase portraits of system (45) with B ≥ 0. It is easy to get that system (45) is symmetry with respect to the y-axis, in particular, it is symmetry with respect to the x-axis when B = 0. We are going to study the dynamics for system (45) in two cases C = B 2 /4 and C = B 2 /4.
Through the local chart we study the infinity of system (46) using the Poincaré compactification. The associated system (46) in the local chart U 1 is For v = 0 the origin is the unique equilibrium, which is linearly zero. In order to understand the local behavior of this equilibrium point, we apply the directional blow-up (u, v) → (u, w) with w = v/u. And after eliminating the common factor u 2 we have System (48) can have two possible equilibrium points E 1 = (0, 0) and E 2 = (0, −2/B) on the u = 0.
When B = 0 the origin E 1 of system (48) is the unique equilibrium point for u = 0 and it is a linearly zero. Again we do the blow-up (u, w) → (u, w) with w = w/u, and after removing the common factor u 3 we obtain the system For u = 0 system (49) has the origin as the unique equilibrium point, which is a saddle. We recover the local phase portrait at the origin of system (47) going back through the blow-up. The equilibrium E 1 of system (48) consists of two hyperbolic sectors and one repelling parabolic sector (see Figure 24(b)), the origin of U 1 has four hyperbolic sectors (see Figure 24(c)).
When B > 0 system (48) has two equilibria E 1 and E 2 on the u = 0. Similarly to the above case E 1 is a linearly zero equilibrium, which has two hyperbolic and one parabolic sectors. And the equilibrium E 2 is a saddle with eigenvalues λ 1,2 = ±8/B 2 . Going back through the blow-up and taking into account the behavior of the flow on the axes, we have u ′ | u=0 = Bv 3 , u ′ | v=0 = −u 3 and v ′ | u=0 = 2v 3 +O(v 4 ). Then we obtain that the origin of system (47) also consists of four hyperbolic sectors see Figure 25.
We analyze the local chart U 2 , system (46) in this chart is The origin is not the equilibrium point of system (50).
For the polynomial system (46) we have found two equilibria in the infinite region, one in U 1 and its diametrically opposite in V 1 , which have the opposite sense. We observe that the set L = {(x, y)|y + B/2 = 0} corresponds to y = −B/2, which is a straight line connecting the north and south poles. Therefore there exist no equilibria at infinity for the rational Hamiltonian system (45).   For x = 0 this system does not have any equilibrium points, therefore we analyze the vector field in a neighborhood of the origin. Note that x ′ | w=0 = −2 and w ′ | x=0 = 0, then the local phase portrait at the origin of system (51) is given in Figure 26(a). We reconstruct the flow through the blow-up to get the behavior of the flow close to the origin of system (46), it has two elliptic sectors and two parabolic sectors, one attracting and one repelling, Figure 26(b) shows this local phase portrait.
Hence we obtain the phase portrait of system (46), see Figure 27(a). Going back to the Hamiltonian system (45), p 1 is a virtual equilibrium point. By the rescaling time dt = x 3 dτ the integral curves of system (45) has the opposite sense with respect to the ones of system (46) in the region y < 0. Thus we have the phase portrait 1.14 of Figure 1.
If ∆ = 0 and B > 0 system (46) has two finite equilibria p 1 = (0, 0) and p 2 = (0, −B/2), where p 2 is in the set L and it is linearly zero. The eigenvalues of p 1 are ±B 2 √ 2i/2 and system (46) is symmetry with respect to the y-axis, hence p 1 is a center. We translate the equilibrium point p 2 to the origin though the change of variables (x, y) → (u, v − B/2), and we have the system Applying the directional blow-up (u, v) → (u, w) with w = v/u and eliminating the common factor u 2 we have System (53) does not have any equilibrium points on u = 0, hence we analyze the vector fields in a neighborhood of the origin. Note that u ′ | w=0 = −2 and w ′ | u=0 = Bw 4 /2 > 0, then the local phase portrait at the origin of system (53) is given in Figure 28(a). We reconstruct the flow through the blow-up to get the local phase portrait of the flow at the origin of system (52). It has one elliptic, one hyperbolic and two parabolic sectors, see Figure 28(b).
In this subcase we have the phase portrait of system (46), as it is shown in Figure  27 Observe that we have D 1 > 0, and D 4 < 0 because ∆ > 0, hence the sign list of this discriminant sequence is determined only by the signs of B. It is given in Theorem  (54) has two real distinct roots, accordingly system (46) has two finite equilibrium points, which are not in the L.
We continue determining the finite equilibria of systems (46) and (45). The infinite equilibrium points in the Poincaré sphere are the origin in U 1 , and also the corresponding point in V 1 . The origins of U 1 and V 1 consist of four hyperbolic sectors, hence they have index −2 by Theorem 2.2. Thus, in the Poincaré disk, the finite equilibrium points must have total index 2. Note that the two remaining finite equilibrium points are p 1,2 , in particularly p 1,2 = (0, ± 4 √ 2∆) when B = 0. By Corollary 2.3 and Remark 2.5 the equilibria p 1,2 must be centers. In this subcase we have the phase portrait of system (46), see Figure 29(a). Going back to the Hamiltonian system (45), the integral curves of system (45) have the opposite sense with respect to the ones of system (46) in the region y < −B/2. Hence in this subcase we have the phase portrait 1.16 of Figure 1.
(i.c) Assume that ∆ < 0 system (46) has two additional finite equilibria p 1,2 = (± √ −∆, −B/2) in the set L, which are distinct to the points from (54). The equilibrium points p 1,2 are an attracting node and a repelling node, hence each has index 1. So the known equilibrium points have total index 2. By Theorem 2.4 the remaining finite equilibrium points must have total index 0. We will determine the cases in which (54) has two, one or zero distinct real roots. These roots correspond the equilibria of systems (45) and (46), which are not in the set L. (ii) Next we consider the case C = B 2 /4, then system (45) becomes (56) x ′ = −(x 2 + ∆)(2y + B) + y(y 2 + By + C) 2 , y ′ = −2x(y 2 + By + C), after the rescaling time dt = (y 2 + By + C) 2 dτ . And system (56) in the chart U 1 is For v = 0 the origin is the unique equilibrium and it is linearly zero. Doing the directional blow-up (u, v) → (u, w) with w = v/u and eliminating the common factor u 3 , system (57) becomes When C > B 2 /4 the equilibrium point E 1 = (0, 0) of system (58) is the unique equilibrium point on u = 0 and it is linearly zero. We do another blow-up (u, w) → (u, w) with w = w/u, removing the common factor u 2 , we have For u = 0 the origin is the unique equilibrium point of system (59), which is a saddle. We recover the local phase portrait at the origin of system (58) going back through the blow-up, the equilibrium E 1 of system (58) consists of four hyperbolic sectors, as it is shown in Figure 30(b). Again going back through the blow-up until system (57) and taking into account the flow on the axes, we obtain that the origin of U 1 has two hyperbolic sectors see Figure 30(c), and the u-axis contains two separatrices because it is invariant for system (57).
If C = 0 and B > 0 system (58) has two equilibria E 1 and E 2 = (0, −1/B) on u = 0. From system (58) it is easy to get that E 2 is a saddle. Similarly to the above case we obtain that in a neighborhood of the equilibrium E 1 has four hyperbolic sectors doing a blow-up. In Figure 31(a) we superpose the study of the two equilibria E 1,2 obtained by the blow-up at the origin of system (57). Going back through this blow-up and taking into the behavior of the flow on both axes, we obtain that the local phase portrait at the origin of system (58) consists of six hyperbolic sectors, see Figure 31(b).
In the local chart U 2 system (56) becomes the origin is not the equilibrium point of system (60). For the polynomial system (56) we have found two equilibria at infinity, one in U 1 and its diametrically opposite in V 1 , which have the same sense. When B 2 − 4C > 0 we observe that the set L = {(x, y)|y 2 + By + C = 0} corresponds to two straight lines y 1,2 = (−B ± √ B 2 − 4C)/2, which connect the north and south poles. Therefore there exist no equilibria at infinity for the rational Hamiltonian system (45). If B 2 − 4C < 0 the set L = {(x, y)|y 2 + By + C = 0} is empty, system (45) has two equilibria in the infinite region.
This finishes the study of the infinite equilibrium points of system (56) and of the rational Hamiltonian system (45), we now focus on the finite region in three subcases: ∆ > 0, ∆ = 0 and ∆ < 0.
Let (0, y 0 ) be an equilibrium point of system (56), then the linear part of (56) at (0, y 0 ) is where F 1 = y 2 0 + By 0 + C and Since ∆ > 0 and C = B 2 /4 we have F 1 = 0 from (56). Hence the equilibrium (0, y 0 ) is either elementary or nilpotent, that is, it is either a saddle or a center or a cusp, and it is not in the set L.
Now we analyze the location of the five distinct real roots. Let g(x) be a polynomial of degree d(g(x)) = n, and we define the following polynomials (65) where λ i > 0, d(g i (x)) < d(g i−1 (x)) and d(g m (x)) = 0.
Theorem 7.1 (Sturm Theorem). Assume that g 0 (x), g 1 (x),· · · , g m (x) are as the polynomials (65) where g(a)g(b) = 0. For each x ∈ R, V (x) denotes the number of the sign changes of the sequence {g 0 (x), g 1 (x), · · · , g m (x)}. We have that the number of real roots of g( Using the above Sturm criterion we take the polynomial g 0 (y) = g(y) from (61), and from (65) we obtain We have that under conditions C < B 2 /4, 0 < ∆ < F 4 and D 5 > 0 the polynomial Table 1 y − − + + sign(g 3 (y)) + + + + sign(g 4 (y)) − − + + sign(g 5 (y)) + + + + V (y) 5 4 1 0 g 0 (y) has five real roots, so V (−∞) and V (+∞) are determined as in Table 1. We also compute V ((−B + √ Hence polynomial (61) has one, three and one real roots in the intervals ((−B + √ Next we will analyze the local phase portraits of the finite equilibria. For this we first study the infinite equilibrium points. The infinite equilibrium points in the Poincaré sphere are the origin of U 1 , and also the corresponding origin of V 1 . The origins of U 1 and V 1 consist of six hyperbolic sectors when C < B 2 /4, hence, by Theorem 2.2, they have index −4. In the Poincaré disk the finite equilibrium points must have total index 3. Thus the finite equilibrium points must be four centers and one saddle. The only possible phase portrait of system (45) in this case is topologically equivalent to 1.20 of Figure 1.
From the above analyzing we have B = 0, C < 0 and ∆ ≥ C 2 /2, or B = 0, C > 0 and ∆ > C 2 /2, or B > 0, C < B 2 /4, ∆ > 0 and D 5 < 0, or C > B 2 /4 > 0, ∆ > F 4 and D 5 < 0 when systems (45) and (56) have three equilibria which are not in the set L. If C < B 2 /4 the origins of U 1 and V 1 consist of six hyperbolic sectors, which have index −4. In the Poincaré disk the finite equilibrium points must have total index 3. Therefore the three finite equilibrium points must be centers. In this subcase we have the phase portrait 1.22 of Figure 1.
In short systems (45) and (56) have two equilibria, which are not in the set L, if and only if C > B 2 /4 > 0, ∆ > F 4 and D 5 = 0. In this subcase the origins of U 1 and V 1 consist of two hyperbolic sectors which have index 0. In the Poincaré sphere the finite equilibrium points must have total index 2. Hence the finite equilibrium points must be one center and one cusp. The phase portrait is topologically equivalent to 1.24 of Figure 1.
(ii.a.5) Now we study the case when the polynomial (61) has one distinct real root. For the sake of simplicity, we obtain the possible revised sign lists and their associated discriminant sequences as it is shown in the Table 2. We denote by RSL and DS the revised sign lists and the discriminant sequences in the following tables. In summary systems (45) and (56) have one equilibria, which is not in the set L, if and only if B = 0, C > 0 and 0 < ∆ ≤ C 2 /2, or C > B 2 /4 > 0, ∆ > 0 and D 5 > 0.  The origins of U 1 and V 1 have two hyperbolic sectors, by Theorem 2.2, have index 0. In the Poincaré sphere, the finite equilibrium points must have total index 2. Thus the finite equilibrium point must be a center. In this subcase we have the phase portrait 1.25 of Figure 1.  (ii.b.1) If C > B 2 /4 system (56) has one equilibrium point p 1 = (0, 0), which is a center. In this subcase we have that the phase portrait is topologically equivalent to 1.25 of Figure 2.
(ii.b.2) If C = 0 and B = 0 system (56) has two equilibrium points p 1 = (0, 0) and p 2 = (0, −B), which are in the set L and are linearly zero. To understand the local phase portrait of the origin we need to do the blow-up (x, y) → (x, w) with w = y/x, and we get the system after eliminating the common factor x between x ′ and w ′ . When x = 0 system (66) has one equilibrium located at the origin, which is an attracting node. Going back through the change of variables to the origin of system (56) and taking into account the behavior of the flow on the axes, it has one hyperbolic sector, one elliptic sector and two parabolic sectors, see Figure 33(a). To determine the local behavior of p 2 , we translate this equilibrium point to the origin doing the change of variables x → x and y → y − B, we have the system Now we do the blow-up (x, y) → (x, w) with w = y/x, and eliminating the common factor x between x ′ and w ′ we have The origin of system (68) is a repelling node. Going back through the change of variables we get that the origin of system (67) consists of two elliptic and two parabolic sectors (see Figure 33(b)). Going back to the associated Hamiltonian system (45) the equilibrium points p 1,2 are two virtual equilibrium points. Therefore the phase portrait in this case is topologically equivalent to 1.26 of Figure 2.
(ii.b.3) If C < B 2 /4 and C = 0 system (56) has three equilibrium points p 1 = (0, 0) and p 2,3 = (0, (−B ± √ B 2 − 4C)/2). The equilibrium points p 2,3 are in the set L and they are linearly zero. The local phase portrait of p 2 consists of two elliptic sectors and two parabolic sectors, or two hyperbolic sectors and two parabolic sectors when C < 0 or C > 0 respectively (see Figures 33(b) and 34). The local phase portrait of p 3 consists of two elliptic and two parabolic sectors, see Figure 33(b). On the other hand, the equilibrium point p 1 is a saddle and a center when C < 0 and C > 0, respectively. Going back to the Hamiltonian system (45) the equilibrium points p 2,3 are two virtual equilibrium points. Hence we have the phase portraits 1.27 and 1.28 of Figure 2 when C < 0 and 0 < C < B 2 /4.
Similarly to the case (ii.a) the polynomial (61) has at least one distinct root.  Table 3 we have that systems (45) and (56) have three equilibria, which are not in the set L, if and only if B > 0, 0 < C, C = B 2 /4, F 6 < ∆ < 0 and D 5 < 0.  Assume that B 2 /4 < C then the origins of U 1 and V 1 consist of two hyperbolic sectors which have index 0 by Theorem 2.2. Thus, in the Poincaré disk, the finite equilibrium points must have total index 1. Hence the finite equilibrium points must be two centers and one saddle. The phase portrait is topologically equivalent to 1.23 of Figure 1.
The origins of U 1 and V 1 have six hyperbolic sectors when C < B 2 /4, hence, by Theorem 2.2, have index −4. And the known equilibrium points p 1,2,3,4 have total index 4. Thus the remaining finite equilibrium points must have total index -1 in Poincaré disk. Then the remaining finite equilibrium points must be one center and two saddles. Going back to the Hamiltonian system (45), the equilibria p 1,2,3,4 are four virtual equilibrium points, therefore we obtain that the phase portrait of system (45) is topologically equivalent to 1.29 of Figure 2. When B 2 /4 < C the phase portrait of system (45) is topologically equivalent to 1.24 of Figure 1. If C < B 2 /4 system (56) has more four equilibrium points p 1,2,3,4 , which have total index 4. Hence the remaining finite equilibrium points must be one cusp and one saddle so that their index is -1. Going back to system (45), the equilibrium points p 1,2,3,4 are four virtual equilibria. Hence the phase portrait is topologically equivalent to 1.30 of Figure 2.
When B 2 /4 < C the phase portrait of system (45) is topologically equivalent to 1.25 of Figure 1. If C < B 2 /4 the known infinite and finite equilibrium points have total index 4. Hence the remaining finite equilibrium points must be one saddle so that it has index -1. Going back to the Hamiltonian system (45) we have the phase portrait 1.31 of Figure 2.