This is a preprint of: “Global Phase Portraits of Z2-Symmetric Planar Polynomial Hamiltonian Systems of Degree Three with a Nilpotent Saddle at the Origin”, Montserrat Corbera, Clàudia Valls, Internat. J. Bifur. Chaos Appl. Sci. Engrg., vol. 30(1), 2050006:1–30, 2020. DOI: [10.1142/S0218127420500066] GLOBAL PHASE PORTRAITS OF Z2 -SYMMETRIC PLANAR POLYNOMIAL HAMILTONIAN SYSTEMS OF DEGREE 3 WITH A NILPOTENT SADDLE AT THE ORIGIN MONTSERRAT CORBERA AND CLAUDIA VALLS Abstract. We characterize the phase portraits in the Poincaré disk of all planar polynomial Hamiltonian systems of degree 3 with a nilpotent saddle at the origin and Z2 -symmetric with (x, y) 7→ (−x, y). 1. Introduction and statement of the results In this paper we study the global phase portraits of a class of Z2 -symmetric planar polynomial Hamiltonian systems of degree three with a nilpotent sad- dle at the origin. We recall that a planar polynomial Hamiltonian system is a system of the form (1) x′ = Hy , y ′ = −Hx where H(x, y) is a real polynomial in the variables x and y. Here the prime denotes derivative with respect to the independent variable t. We say that system (1) has degree d if the maximum of the degrees of Hy and Hx is d. In this paper we will focus in the case in which d = 3. Let p ∈ R2 be a singular point of a polynomial differential system in R2 . Without loss of generality we can assume that p is at the origin of coordinates. We say that p is a nilpotent saddle if after a linear change of variables and a rescaling of the time (if necessary) the system can be written in the form x′ = y + P (x, y), y ′ = Q(x, y) where P (x, y) and Q(x, y) are real analytic functions without constant and linear terms, defined in a neighborhood of the origin. In this paper we will consider the case in which P and Q are polynomials of degree three. The global phase portraits in the Poincaré disk of all planar polynomial Hamiltonian vector fields with degree three having a nilpotent center at the origin have been provided in several studies (see for instance [2, 3]). However, 2010 Mathematics Subject Classification. 34C05, 34C07, 34C08. Key words and phrases. Polynomial Hamiltonian systems, nilpotent saddle, phase portrait, Poincaré compactification. 1 2 M. CORBERA AND C. VALLS no global phase portraits in the Poincaré disk are given for planar polynomial Hamiltonian vector fields having a nilpotent saddle at the origin. This is mainly due to the fact that there is a very important additional difficulty caused by the possible saddle connections which normally are very difficult to detect. In this paper we want to fill in this gap and provide the global phase por- traits in the Poincaré disk of all reversible and equivariant planar polynomial differential systems of degree three that are symmetric with respect to the y-axis and with a nilpotent saddle at the origin. Let X : U ⊂ R2 → R2 be the vector field associated to system (1). We say that system (1) is reversible with respect to the y-axis if it satisfies ( ) −1 0 (2) X(x, y) = −X(−x, y), 0 1 and we say that it is equivariant with respect to the y-axis if it satisfies ( ) −1 0 (3) X(x, y) = X(−x, y). 0 1 Other classes of polynomial vector fields in R2 with a Z2 -symmetry have been studied by several authors (see for instance [5, 6, 7, 8]). The main tool in this paper will be the Poincaré compactification of poly- nomial vector fields. The Poincaré compactification that we shall use for describing the global phase portraits of our Hamiltonian systems is stan- dard. For all the definitions and results on the Poincaré compactification see Chapter 5 of [4]. We say that two vector fields on the Poincaré disk are topologically equivalent if there exists a homeomorphism from one into the other which sends orbits to orbits preserving or reversing the direction of the flow. Our main result is the following one. Theorem 1. A Hamiltonian planar polynomial vector field of degree three with a nilpotent saddle at the origin and reversible with respect to the y-axis, after a linear change of variables and a rescaling of its independent variable can be written as one of the following classes: (I) x′ = y + by 3 , y ′ = x3 with b ∈ R; (II) x′ = y + sx2 y + by 3 , y ′ = x3 − sxy 2 with b ∈ R; (III) x′ = y + y 2 + by 3 , y ′ = x3 with b ∈ R; (IV) x′ = y + y 2 + sx2 y + by 3 , y ′ = ax3 − sxy 2 with a > 0 and b ∈ R; x2 √ (V) x′ =y+√ 2 , y ′ = ax3 − 2xy with a > −1; x2 √ (VI) x′ = y + √ 2 + sy 3 , y ′ = ax3 − 2xy with a > −1; x2 √ (VII) x′ = y + √ 2 + sy 2 + by 3 , y ′ = ax3 − 2xy with a > −1 and b ∈ R; x2 √ (VIII) x′ = y + √ 2 + sx 2 y + by 2 + cy 3 , y ′ = ax3 − 2xy − sxy 2 with a > −1 and b, c ∈ R, HAMILTONIAN SYSTEMS WITH A NILPOTENT SADDLE AT THE ORIGIN 3 with s ∈ {−1, 1}, a ̸= 0 in systems (V) and a2 + b2 ̸= 0 in systems (VII). Moreover, the global phase portraits of these families that have at most one infinite singular point, are topologically equivalent to the following ones of Figure 1: • 1.1–1.3 for systems (I); • 1.1–1.7 for systems (II); • 1.1, 1.3, and 1.8–1.12 for systems (III); • 1.1, 1.3, 1.5, and 1.7–1.33 for systems (IV ); • 1.2 and 1.34 for systems (V ); • 1.1–1.3, 1.13, 1.35, and 1.36 for systems (V I); • 1.1–1.3, 1.8–1.18, 1.25, and 1.35–1.45 for systems (V II); • 1.1–1.33, and 1.35–1.148 for systems (VIII). Note that the conditions a ̸= 0 in system (V) and a2 + b2 ̸= 0 in system (VII) are due to the fact that we are studying systems with degree three and otherwise in any of these cases, the corresponding system would be quadratic. The proof of Theorem 1 is given in sections 3–19. 2. Preliminary results A vector field is said to have the finite sectorial decomposition property at a singular point q if either q is a center, a focus or a node, or it has a neighborhood consisting of a finite union of parabolic, hyperbolic or ellip- tic sectors. We note that all the isolated singular points of a polynomial differential system satisfy the finite vectorial decomposition property. Theorem 2 (Poincaré Formula). Let q be an isolated singular point having the finite sectorial decomposition property. Let e, h an p denote the number of elliptic, hyperbolic and parabolic sectors of q, respectively. Then the index of q is (e − h)/2 + 1. The indices of a saddle, a center and a cusp are −1, 1 and 0, respectively. Theorem 3 (Poincaré–Hopf Theorem). For every vector field on the sphere S2 with a finite number of singular points, the sum of the indices of these singular points is 2. Nilpotent singular points of Hamiltonian planar polynomial vector fields are either saddles, centers, or cusps (for more details see Chapter Theorem 3.5 of [4] and take into account that Hamiltonian systems cannot have foci). 4 M. CORBERA AND C. VALLS 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 Figure 1. Phase portraits for the Hamiltonian planar poly- nomial vector fields of degree three with a nilpotent saddle at the origin and reversible with respect to the y-axis. HAMILTONIAN SYSTEMS WITH A NILPOTENT SADDLE AT THE ORIGIN 5 1.29 1.30 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.40 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49 1.50 1.51 1.52 1.53 1.54 1.55 1.56 Figure 1. Phase portraits for the Hamiltonian planar poly- nomial vector fields of degree three with a nilpotent saddle at the origin and reversible with respect to the y-axis. 6 M. CORBERA AND C. VALLS 1.57 1.58 1.59 1.60 1.61 1.62 1.63 1.64 1.65 1.66 1.67 1.68 1.69 1.70 1.71 1.72 1.73 1.74 1.75 1.76 1.77 1.78 1.79 1.80 1.81 1.82 1.83 1.84 Figure 1. Phase portraits for the Hamiltonian planar poly- nomial vector fields of degree three with a nilpotent saddle at the origin and reversible with respect to the y-axis. HAMILTONIAN SYSTEMS WITH A NILPOTENT SADDLE AT THE ORIGIN 7 1.85 1.86 1.87 1.88 1.89 1.90 1.91 1.92 1.93 1.94 1.95 1.96 1.97 1.98 1.99 1.100 1.101 1.102 1.103 1.104 1.105 1.106 1.107 1.108 1.109 1.110 1.111 1.112 Figure 1. Phase portraits for the Hamiltonian planar poly- nomial vector fields of degree three with a nilpotent saddle at the origin and reversible with respect to the y-axis. 8 M. CORBERA AND C. VALLS 1.113 1.114 1.115 1.116 1.117 1.118 1.119 1.120 1.121 1.122 1.123 1.124 1.125 1.126 1.127 1.128 1.129 1.130 1.131 1.132 1.133 1.134 1.135 1.136 1.137 1.138 1.139 1.140 Figure 1. Phase portraits for the Hamiltonian planar poly- nomial vector fields of degree three with a nilpotent saddle at the origin and reversible with respect to the y-axis. HAMILTONIAN SYSTEMS WITH A NILPOTENT SADDLE AT THE ORIGIN 9 1.141 1.142 1.143 1.144 1.145 1.146 1.147 1.148 Figure 1. Phase portraits for the Hamiltonian planar poly- nomial vector fields of degree three with a nilpotent saddle at the origin and reversible with respect to the y-axis. 3. Proof of the normal form in Theorem 1 Without loss of generality we can assume that a cubic planar Hamiltonian system with a nilpotent point at the origin is given by x′ = y + b2 x2 + a2 x3 + 2b3 xy + 2a3 x2 y + 3b4 y 2 + 3a4 xy 2 + 4a5 y 3 , (4) y ′ = −3b1 x2 − 4a1 x3 − 2b2 xy − 3a2 x2 y − b3 y 2 − 2a3 xy 2 − a4 y 3 . which corresponds to equation ∂H ∂H x′ = , y′ = − , ∂y ∂x where H(x, y) = y 2 /2 + a1 x4 + a2 x3 y + a3 x2 y 2 + a4 xy 3 + a5 y 4 + b1 x3 + b2 x2 y + b3 xy 2 + b4 y 3 . By hypothesis, systems (4) must have the symmetry (x, y, t) 7→ (−x, y, −t). Imposing this condition using (3) we get to the contradiction 2y = 0, so only condition (2) is possible. Imposing it, we must have that a2 = a4 = b1 = b3 = 0. Hence systems (4) become x′ = y + b2 x2 + 2a3 x2 y + 3b4 y 2 + 4a5 y 3 , (5) y ′ = −4a1 x3 − 2b2 xy − 2a3 xy 2 . Since systems (5) must have a saddle at the origin, by Theorem 3.5 of [4] we must have a1 < b22 /2. Therefore we obtain (5) with a1 < b22 /2. Now we provide the normal form to system (5) to transform it into other systems with less parameters. 10 M. CORBERA AND C. VALLS Case 1. Assume b2 ̸= 0 and a3 ̸= 0. By the change of coordinates and reparametrization of time of the form (6) X → αx, Y → βy, τ → γt, √ √ with γ = b2 / |a3 |, α = βγ and β = b2 2/γ 2 and we get system (VIII) with s = 1 when a3 > 0 and system (VIII) with s = −1 when a3 < 0. Case 2. Assume b2 ̸= 0, a3 = 0 and b2 b4 ̸= 0. By the change √ of coordinates √ and 1/4 √ reparametrization of time as in (6) with γ = 2 |b2 /b4 |/ 3, β = b2 2/γ 2 and α = βγ we get systems (VII) with s = 1 if b2 b4 > 0 and systems (VII) with s = −1 if b2 b4 < 0. Case 3. Assume b2 ̸= 0, a3 = b4 = 0 and a5 ̸= 0. By the change of coordinates √ and reparametrization of time as in (6) with γ = (b2 /(2|a5 |))1/4 , 2 β = b2 2/γ and α = βγ we get systems (VI) with s = 1 if a5 > 0 and systems (VI) with s = −1 if a5 < 0. Case 4. Assume b2 ̸= 0, a3 = b4 = a5 = 0. By the change √ of coordinates and reparametrization of time as in (6) with γ = 1, β = b2 2 and α = β we get systems (V). Case 5. Assume b2 = 0, b4 ̸= 0 and a3 ̸= 0. The √ change of coordinates and reparametrization of time as in (6) with γ = 2|a3 |/(3b4 ), β = 3b4 and α = βγ, we get systems (IV) with s = 1 if a3 > 0 and systems (IV) with s = −1 if a3 < 0. Case 6. Assume b2 = 0, b4 ̸= 0 and a3 = 0. The change of coordinates and reparametrization of time as in (6) with γ = (−4a1 /(9b24 ))1/4 , β = 3b4 and α = βγ, we get systems (III). Case 7. Assume b2 = 0, b4 = 0 and a3 ̸= 0. The change of coordinates √ √ and reparametrization of time as in (6) with β = |a3 |/ −a1 , γ = 2|a3 |/β and α = βγ, we get systems (II) with s = 1 if a3 > 0 and systems (II) with s = −1 if a3 < 0. Case 8. Finally, if b2 = 0, b4 = 0 and a3 = 0. The change of coordinates and reparametrization of time as in (6) with γ = ((−4a1 )/β 2 )1/4 , β = 1 and α = βγ, we get systems (I). In short, we have proved the first part (or the normal form part) of The- orem 1. 4. Global phase portrait of system (I) Consider system (I) x′ = y + by 3 , y ′ = x3 with b ∈ R. HAMILTONIAN SYSTEMS WITH A NILPOTENT SADDLE AT THE ORIGIN 11 In the local chart U1 system (I) u′ = 1 − bu4 − u2 v 2 , v ′ = −uv(bu2 + v 2 ). So, either there are no infinite singular points in the local chart U1 (when b ≤ 0) or there are two when b > 0, namely, (±b−1/4 , 0). Computing the eigenvalues of the Jacobian matrix at these points we get that (b−1/4 , 0) is an attracting node and (−b−1/4 , 0) is a repelling node. On the local chart U2 system (I) becomes u′ = b − u4 + v 2 , v ′ = −u3 v. The origin is a singular point if and only if b = 0. If b = 0 it is linearly zero, and using blow-up techniques we get that it is the union of two elliptic (one stable and one unstable) and four parabolic (two stable and two unstable) sectors. The finite singular points are on x = 0 and 1 + by 2 = 0. So, if b ≥ 0 there are no finite singular points among the origin √ and if b < 0 there are two more finite singular points which are (0, ± −1 b ). They are both nilpotent. Using Theorem 3.5 in [4] and that the system is Hamiltonian we conclude that they are both centers. Gluing all the information on the finite and infinite singular points we get that the global phase portraits are topologically equivalent to the following ones of Figure 1: 1.1 when b > 0, 1.2 when b = 0, and 1.3 when b < 0. 5. Global phase portrait of system (II) Consider system (II) x′ = y + sx2 y + by 3 , y ′ = x3 − sxy 2 = x(x2 − sy 2 ) with b ∈ R and s ∈ {−1; 1}. We first study the infinite singular point. In the local chart U1 systems (II) become u′ = 1 − u2 (2s + bu2 + v 2 ), v ′ = −uv(s + bu2 + v 2 ). The infinite singular points satisfy v = 0 and 1 − 2su2 − bu4 = 0. If b = 0, there are two infinite singular points if s = 1 and zero if s = −1. On the other hand if b ̸= 0, either there are zero infinite singular points (when b < −1 and s = 1, or b < 0 and s = −1), or two (when b > 0, or when b = −1 and s = 1), or four (when b ∈ (−1, 0) and s = 1). When s = −1 and b > 0, or s = 1 and b ≥ 0, the two infinite singular points are nodes (one stable and one unstable). When s = 1 and b = −1 the two infinite singular points are linearly zero. Applying the blow up technique we conclude that they are two elliptic (one stable and one unstable) and four parabolic (two 12 M. CORBERA AND C. VALLS stable and two unstable) sectors. Finally, when s = 1 and b ∈ (−1, 0) the four infinite singular points are nodes (two stable and two unstable). In the local chart U2 systems (II) become u′ = b + 2su2 − u4 + v 2 , v ′ = vu(s − u2 ). The origin of the local chart U2 is a singular point if and only if b = 0. In this case it is linearly zero. Applying blow up techniques (see for instance [1] for details on this technique) we get that the origin of the local chart U2 is formed by two elliptic (one stable and one unstable) and four parabolic (two stable and two unstable) sectors if s = −1 and by two hyperbolic sectors (one stable and one unstable) if s = 1. Now we study the finite singular points. When b ≥ 0 there are no finite singular points among the origin. If b < 0 the finite singular √ points different from the origin and with x = 0 are of the form (0, ± −1/b) which are always real because b < 0. In this case computing the eigenvalues of the Jacobian matrix at these points and using that the system is Hamiltonian we conclude that they are both saddles if s = 1 and are both centers √ if s = −1. The finite singular points with x ̸= 0 are of the form x = ± sy 2 . So if s = −1 there are no finite singular points with x ̸= 0 and if s = 1 and b < −1 there are four singular points which are of the form ( 1 1 ) ( 1 1 ) ±√ ,√ , ±√ , −√ . −1 − b −1 − b −1 − b −1 − b Computing the eigenvalues of the Jacobian matrix at these points and taking into account that the system is Hamiltonian we conclude that they are all centers. √ The saddles (0, ± −1/b) (which exist when b < 0 and s = 1) are not connected with the saddle at the origin because the Hamiltonian evaluated on the saddles is never zero. On the other hand, they are connected among them due to the symmetry. In short gluing all the information on the finite and infinite singular points we get that the unique possible global phase portraits for system (II) are topologically equivalent to the following ones of Figure 1: 1.1 when b > 0 (either s = 1 or s = −1); 1.2 when b = 0 and s = −1; 1.3 when b < 0 and s = −1; 1.4 when b = 0 and s = 1; 1.5 when b ∈ (−1, 0) and s = 1; 1.6 when b = −1 and s = 1; and, finally, 1.7 when b < −1 and s = 1. 6. Global phase portrait of system (III) Consider system (III) x′ = y + y 2 + by 3 , y ′ = x3 with b ∈ R. HAMILTONIAN SYSTEMS WITH A NILPOTENT SADDLE AT THE ORIGIN 13 We first study the infinite singular points. In the local chart U1 system (III) is u′ = 1 − bu4 − u2 v(u + v), v ′ = −uv(bu2 + v(u + v)). The infinite singular points in the local chart U1 must satisfy v = 0 and 1 − bu4 = 0. So, either there are no infinite singular points in the local chart U1 (when b ≤ 0) or there are two when b > 0, namely, (±b−1/4 , 0). Computing the eigenvalues of the Jacobian matrix at these points we get that (b−1/4 , 0) is an attracting node and (−b−1/4 , 0) is a repelling node. On the local chart U2 system (III) becomes u′ = b − u4 + v + v 2 , v ′ = −u3 v. The origin is a singular point if and only if b = 0. If b = 0 it is nilpotent. Using Theorem 3.5 in [4] together with blow-up techniques we get that it is the union of one hyperbolic and one elliptic sectors. Now we study the finite singular points. The finite singular points are on x = 0 and y(1 + y + by 2 ) = 0. If b = 0 among the origin, there is a unique finite singular point (0, −1). Computing the eigenvalues of the Jacobian matrix at this point we get that it is nilpotent. Using Theorem 3.5 in [4] and that the system is Hamiltonian we conclude that it is a center. ( If b ̸= √ 0, the ) finite singular points, whenever they exist are (0, y± ) = −1± 1−4b 0, 2b . If b > 0, they exist whenever b ∈ (0, 1/4]. If b = 1/4, both points coalesce in the finite singular point (0, −2) which is linearly zero. Using blow up techniques we get that it is the union of two hyperbolic sectors (cusp). If b ∈ (0, 1/4) both singular points (0, y± ) are nilpotent. Using Theorem 3.5 in [4] and that the system is Hamiltonian we get that (0, y+ ) is a center and (0, y− ) is a saddle. We note that the possible saddle connection between the saddle (0, y− ) and the origin is when b = 2/9. Finally, if b < 0 the two finite singular points (0, y± ) exist and are again nilpotent. Using Theorem 3.5 in [4] and that the system is Hamiltonian we get that they are both centers. In short we have the following: If b > 1/4 the unique global phase portrait is topologically equivalent to 1.1 of Figure 1. If b = 1/4 the unique global phase portrait is topologically equivalent to 1.8 of Figure 1. If b ∈ (0, 1/4) we get three possible different global phase portraits: 1.9 attained for example when b = 11/90, 1.10 which is the saddle connection attained when b = 2/9, and 1.11 attained for example when b = 23/100. If b = 0 the unique global phase portrait is topologically equivalent to 1.12 of Figure 1 and if b < 0 the unique global phase portrait is topologically equivalent to 1.3 of Figure 1. 14 M. CORBERA AND C. VALLS 7. Global phase portrait of system (IV) Consider system (IV) x′ = y + y 2 + sx2 y + by 3 , y ′ = ax3 − sxy 2 = x(ax2 − y 2 ), with a > 0, b ∈ R and s ∈ {−1; 1}. We first study the infinite singular points. In the local chart U1 system (IV) becomes u′ = a − u2 (2s + bu2 + v(u + v)), v ′ = −uv(s + bu2 + v(u + v)). When v = 0 the infinite singular points on the local chart U1 satisfy a−2su2 − bu4 = 0. If b = 0, there are two infinite singular points √ if s = 1 and zero if s = −1. In this case the infinite singular points are (± a/2, 0). Computing √ the eigenvalues of the Jacobian √ matrix at these points we get that ( a/2, 0) is an attracting node and (− a/2, 0) is a repelling node. On the other hand if b ̸= 0, either there are zero (when b < 0 and s = −1, or b < −1/a and s = 1), or two (when b > 0, or b = −1/a and s = 1), or four (when b ∈√(−1/a, 0) and √ s = 1). In the case b > 0, the infinite singular points are (± −s+ b ab+1 , 0) and are a stable node and an unstable √ node, respectively. If b = −1/a and s = 1, the infinite singular points are (± a, 0) and they are nilpotent. Using Theorem 3.5 in [4] and blow up techniques we see that they are both formed by one hyperbolic and one elliptic sector. In case b √ ∈ (−1/a, 0) and s = 1 (1) √ there are four infinite singular points (u± , 0) = (± (−1 + 1 + ab)/b, 0) √ √ (2) and (u± , 0) = (± (−1 − 1 + ab)/b, 0). Computing the eigenvalues of (1) (2) the Jacobian matrix at these points we get that (u+ , 0) and (u− , 0) are (1) (2) attracting nodes and (u− , 0) and (u+ , 0) are repelling nodes. On the local chart U2 we get u′ = b − au4 + 2su2 + v + v 2 , v ′ = uv(s − au2 ). The origin (0, 0) is an infinite singular point if and only if b = 0. It is nilpotent. Taking into account Theorem 3.5 in [4] and applying blow-ups we get that the origin of U2 is formed by one elliptic (stable), one hyperbolic (unstable) and two parabolic (one stable and one unstable) sectors. Now we study the finite singular points. We consider separately the cases s = 1 and s = −1. Subcase s = 1. We first study the finite singular points at x = 0. If b = 0 the unique singular point is (0, −1). Computing the eigenvalues of the Jacobian matrix at this point we get that it is a saddle. If b = 1/4 then the singular point is (0, −2). It is nilpotent and using Theorem 3.5 in [4] we get that it is a cusp. If b > 1/4 there are no finite singular points HAMILTONIAN SYSTEMS WITH A NILPOTENT SADDLE AT THE ORIGIN 15 on x = 0. Finally, if b√∈ (−∞, 0) ∪ (0, 1/4) there are two finite singular points (0, y± ) = (0, −1±2b1−4b ). Computing the eigenvalues of the Jacobian matrix at these points we get that (0, y+ ) is a saddle and (0, y− ) is a center if b ∈ (0, 1/4) and a saddle if b < 0. The saddle (0, y+ ) cannot be connected with the origin. When b < 0, the saddle (0, y− ) cannot be connected with the origin and both saddles (0, y+ ) and (0, y− ) are also not connected. √ If x ̸= 0 the singular points are (±ȳ/ a, ȳ) with ȳ satisfying 1 + ȳ + 1+ab 2 √ a ȳ = 0. Hence, if b = −1/a the two finite singular points (∓1/ a, −1) are centers. If b = a−4 √2 4a the two singular points (± a , −2) are both nilpotent. Using Theorem 3.5 in [4] we get that they are both cusps. If b > a−4 4a there are no finite singular points with x ̸= 0 and if b < a−4 there are the four √ 4a finite singular points (±ȳ± / a, ȳ± ) with √ −a ± a2 (1 − 4b) − 4a ȳ± = . 2(1 + ab) Computing √ the eigenvalues of the Jacobian √ matrix at these points we get: (±ȳ+ / a, ȳ+ ) are centers and (±ȳ− / a, ȳ− ) are centers for b < −1/a and saddles for b > −1/a. ȳ− The saddles (± √ , ȳ ) can be connected with the origin if and only if a − b = 2a−9 9a and they are connected one with each other by symmetry. The ȳ− saddles (± √ , ȳ ) are never connected with the saddle (0, y− ) (whenever a − it exists) and may be connected with the saddle (0, y+ ) along the curve b = b(a) which is a real root of the polynomial (7) 4 + 12a − 3a2 − 36b − 102ab + 78a2 b − 9a3 b + 81b2 + 180ab2 − 534a2 b2 + 162a3 b2 − 9a4 b2 + 162ab3 + 1080a2 b3 − 864a3 b3 + 108a4 b3 + 81a2 b4 + 1440a3 b4 − 432a4 b4 + 576a4 b5 . Taking into account the information on the finite and infinite singular points together with the possible saddle connections and separating into different regions, we have the following possible global phase portraits of Figure 1 (whenever there is more than one possible phase portrait, in paren- thesis we give values of the parameters where they are realized): 1.1 if a > 0 and b > 1/4; 1.8 if a > 0 and b = 1/4; 1.13 if a ∈ (0, 4) and b ∈ (0, 1/4) or a ≥ 4 and a−4 1 a−4 4a < b < 4 ; and 1.5 if a ∈ (0, 4) and 4a < b < 0. If a > 4 and 0 < b < a−4 4a they are 1.14 (a = 6, b = 1/25); 1.15 (a = 6, b = 1/18. Here there is a saddle connection); 1.16 (a = 6, b = 3/50); 1.17 (a = 6, b ≈ 0.06681525. Here b = b(a) is a real solution of (7) and corresponds to a saddle connection); and 1.18 (a = 6, b = 7/100). 16 M. CORBERA AND C. VALLS If a ∈ (0, 4) and b ∈ (− a1 , a−44a ), they are 1.19 (a = 3, b = −1/10); 1.20 (a = 3, b ≈ −0.10277429. Here b = b(a) is a real solution of (7) and corresponds to a saddle connection); 1.21 (a = 3, b = −27/250); 1.22 (a = 3, b = −1/9. Here there is a saddle connection); and 1.23 (a = 3, b = −1/5). If a ≥ 4 and b ∈ (− a1 , 0), they are 1.23. If a > 0 and b < − a1 , they are 1.7. If a ∈ (0, 4) and b = a−4 4a , they are 1.24. If a > 4 and b = a−44a , they are 1.25. If a > 0 and b = − 1 a , they are 1.26. If a = 4 and b = 0 they are 1.27. If a ∈ (0, 4) and b = 0 they are 1.28. √ If a > 4 and b = 0, they are 1.29 (a = 21/5); 1.30 (a = 2(3+23 3) . Here a is a real solution of (7) with b = 0 and corresponds to a saddle connection); 1.31 (a = 22/5); 1.32 (a = 9/2. Here there is a saddle connection); and 1.33 (a = 5). Subcase s = −1. The unique finite singular points among the origin are located on x = 0. If b = 0 there is the singular point (0, −1) which is a center. If b = 1/4 then the singular point is (0, 2) which is nilpotent. Using Theorem 3.5 in [4] we get that it is a cusp. If b > 1/4 there are no finite singular points on√ x = 0. If b < 1/4 then there are two finite singular points (0, y± ) = (0, −1±2b1−4b ). Computing the eigenvalues of the Jacobian matrix at these points we have that (0, y+ ) is a center and (0, y− ) is a saddle if b > 0 and a center if b < 0. The saddle (0, y− ) can be connected with the origin if and only if b = 2/9. In short, again writing in parenthesis values of parameters where more than one possible phase portrait is realized in a region, we have the following global phase portraits of Figure 1: 1.3 if b < 0; 1.12 if b = 0; 1.1 if b > 1/4; 1.8 if b = 1/4; and if b ∈ (0, 1/4) they are: 1.9 (a = 1, b = 15/72), 1.10 (a = 1, b = 2/9. Here we have the saddle connection), and 1.11 (a = 1, b = 23/100). 8. Global phase portrait of system (V) Consider system (V) x2 √ √ x′ = y + √ , y ′ = ax3 − 2xy = x(ax2 − 2y) 2 with a > −1 and a ̸= 0. In the local chart U1 system (V) becomes 1 √ 1 √ u′ = a − uv(3 2 + 2uv), v ′ = − v 2 ( 2 + 2uv). 2 2 Since a ̸= 0 there are no infinite singular points. HAMILTONIAN SYSTEMS WITH A NILPOTENT SADDLE AT THE ORIGIN 17 In U2 system (V) becomes 3u2 v √ u′ = −au4 + √ + v 2 , v ′ = v(−au3 + 2uv). 2 The origin is a singular point which is linearly zero. Using blow-up tech- niques we get that: it is the union of two elliptic (one stable and one unsta- ble) and four parabolic (two stable and two unstable) sectors if a > 0; and it is the union of three elliptic (two stable and one unstable), one hyperbolic (unstable) and two parabolic (one stable and one unstable) sectors, if a < 0. The unique finite singular point is the origin. Taking into account the information on the infinite singular points we get that the global phase portraits of system (V) are topologically equivalent to 1.2 of Figure 1 if a > 0 and to 1.34 of Figure 1 if a < 0. 9. Global phase portrait of system (VI) Consider system (VI) x2 √ √ x′ = y + √ + sy 3 , y ′ = ax3 − 2xy = x(ax2 − 2y), 2 with a > −1 and s ∈ {−1; 1}. In the local chart U1 system (VI) is 1 √ v √ u′ = a − u(2su3 + v(3 2 + 2uv)), v ′ = − (2su3 + v( 2 + 2uv)). 2 2 The infinite singular points in the local chart U1 must satisfy v = 0 and a − su4 = 0. So, either there are no infinite singular points in the local chart U1 (when as < 0), or there is one (when a = 0), or there are two when as > 0, namely, (±(a/s)−1/4 , 0). In the case a = 0, the unique singular point is the origin which is linearly zero. Using blow-up techniques we get that it is the union of two elliptic (one stable and one unstable) and four parabolic (two stable and two unstable) sectors. When as > 0, computing the eigenvalues of the Jacobian matrix at (±(a/s)−1/4 , 0) we get that ((a/s)−1/4 , 0) is an attracting node if s = 1 and a repelling node if s = 1, and (−(a/s)−1/4 , 0) is a repelling node if s = 1 and an attracting node if s = −1. We now show that the origin of the local chart U2 is not a singular point. Indeed, on the local chart U2 system (VI) becomes 3 √ u′ = s + v 2 − au4 + √ u2 v, v ′ = uv( 2v − au2 ), 2 and the origin is not a singular point. Now we compute the finite singular points. If x = 0 the unique singular point is the origin if s = 1. On the other hand, if s = −1 then besides the origin on x = 0 we have the singular points (0, 1) and (0, −1) which 18 M. CORBERA AND C. VALLS are a saddle and a center, respectively. Moreover, the saddle (0, 1) is never connected with the saddle at origin. If a = 0 or as > 0 there are no finite singular points with x ̸= 0. If as < 0 we get the singular points ( ( 1 + a )1/4 √ 1 + a ) ± 21/4 , if a > 0, a3 a and √ ( ( ) 1/41 + a )1/4 1+a ±2 − ,− − if a ∈ (−1, 0), a3 a which are two centers. In short, gluing all the information on the finite and infinite singular points we get the following global phase portraits of Figure 1: 1.2 if a = 0, s = 1; 1.1 if a > 0, s = 1; 1.3 if a ∈ (−1, 0), s = 1; 1.35 if a = 0, s = −1; 1.36 if a > 0, s = −1; and 1.13 if a ∈ (−1, 0), s = −1. 10. Global phase portrait of system (VII) Consider system (VII) x2 √ √ x′ = y + √ + sy 2 + by 3 , y ′ = ax3 − 2xy = x(ax2 − 2y), 2 with a > −1, b ∈ R, s ∈ {−1; 1} and a2 + b2 ̸= 0. We first study the infinite singular points. In the local chart U1 if (a, b) ̸= (0, 0), system (VII) becomes 1 √ 1 √ u′ = a− u(2bu3 +v(3 2+2u(su+v))), v ′ = − v(2bu3 +v( 2+2u(su+v))). 2 2 The infinite singular points in the local chart U1 must satisfy v = 0 and a − bu4 = 0. So, either there are no infinite singular points in the local chart U1 (when either b = 0 and a ̸= 0, or ab < 0), or there is one (when a = 0 and b ̸= 0) which is the origin, or there are two when ab > 0, namely, (±(a/b)−1/4 , 0). When a = 0 and b ̸= 0, the origin is linearly zero. Applying blow up techniques we obtain that it is the union of two elliptic (one stable and one unstable) and four parabolic sectors (two stable and two unstable). When ab > 0, computing the eigenvalues of the Jacobian matrix at the points (±(a/b)−1/4 , 0) we get that ((a/b)−1/4 , 0) is an attracting node if b > 0 and a repelling node if b < 0, and (−(a/b)−1/4 , 0) is a repelling node if b > 0 and an attracting node if b < 0. In the local chart U2 for (a, b) ̸= (0, 0) we get 3u2 v √ u′ = b − au4 + sv + √ + v 2 , v ′ = v(−au3 + 2uv) 2 HAMILTONIAN SYSTEMS WITH A NILPOTENT SADDLE AT THE ORIGIN 19 The origin (0, 0) is not an infinite singular point if b ̸= 0. If b = 0 it is nilpotent and by Theorem 3.5 in [4] together with blow-up techniques we get that it is the union of one hyperbolic, one elliptic and two parabolic sectors. Now we study the finite singular points. We consider two different sub- cases. Subcase s = 1. If b = 0 the singular point is (0, −1). Computing the eigenvalues of the Jacobian matrix at this point we get that it is a center. If b = 1/4 the singular point is (0, −2). It is nilpotent and using Theorem 3.5 in [4] we get that it is a cusp. If b > 1/4 there are no finite singular points on x = 0. Finally,√ if b < 1/4 then there are the two finite singular points −1± 1−4b (0, y± ) = (0, 2b ). Computing the eigenvalues of the Jacobian matrix at these points we get that (0, y+ ) is a center and (0, y− ) is a saddle. The saddle (0, y− ) and the origin can be connected if and only if b = 2/9. Now we study the singular points with x ̸= 0. If a = 0 there are no finite singular points on x ̸= 0. If a ̸= 0 the singular points on x ̸= 0, whenever they exist, are of the form √√ √ 2y± −a ± a2 − 4ab(a + 1) (±x̄± , ȳ± ), x̄± = , ȳ± = . a 2ab Studying when these finite singular points are defined and in these cases computing the eigenvalues of the Jacobian matrix at these points we get: if a ∈ (−1, 0) and b > 0 we have the two centers (±x̄+ , ȳ+ ) and when a > 0 and b < 0 we have the two centers (±x̄− , ȳ− ). For any other value of the parameters (a, b) ∈ (−1, ∞) × R with a2 + b2 ̸= 0, there are no finite singular points on x ̸= 0. Taking into account the information on the finite and infinite singular points together with the possible saddle connections and separating into dif- ferent regions, we have the following possible global phase portraits of Figure 1 (whenever there is more than one possible phase portrait, in parenthesis we give values of the parameters where they are realized): 1.3 if b > 1/4 and a ∈ (−1, 0); 1.2 if b > 1/4 and a = 0; 1.1 if b > 1/4 and a > 0; 1.37 if b = 1/4 and a ∈ (−1, 0); 1.38 if b = 1/4 and a = 0; and 1.8 if b = 1/4 and a > 0. If b ∈ (0, 1/4) and a ∈ (−1, 0) they are 1.39 (a = −1/2, b = 1/10) and 1.40 (a = −1/2, b = 2/9. Here we have a saddle connection). If b ∈ (0, 1/4) and a = 0 they are 1.41 (b = 1/10); 1.42 (b = 2/9. Here we have a saddle connection); and 1.43 (b = 23/100). If b ∈ (0, 1/4) and a > 0 they are 1.9 (a = 1/2, b = 1/10); 1.10 (a = 1/2, b = 2/9. Here we have a saddle connection); and 1.11 (a = 1/2, b = 23/100). 20 M. CORBERA AND C. VALLS If b = 0 and a ∈ (−1, 0) they are 1.44. If b = 0 and a > 0 they are 1.12. If b < 0 and a ∈ (−1, 0) they are 1.13. If b < 0 and a = 0 they are 1.35 and if b < 0 and a > 0 they are 1.36. Subcase s = −1. First we study the finite singular points at x = 0. If b = 0 the singular point is (0, 1). Computing the eigenvalues of the Jacobian matrix at this point we get that it is a saddle. If b = 1/4 then the singular point is (0, 2) which is nilpotent. Using Theorem 3.5 in [4] we get that it is a cusp. If b > 1/4 there are no finite singular points on x = 0. Finally, if b <√1/4 and b ̸= 0 then there are the two finite singular points (0, y± ) = (0, 1± 2b1−4b ). Computing the eigenvalues of the Jacobian matrix at these points we get that (0, y+ ) is a center and (0, y− ) is a saddle. Now we study the singular points with x ̸= 0. If a = 0 there are no finite singular points on x ̸= 0. If a ̸= 0 and b = 0, there are the two singular points ( √ √ ) 2 1+a 1 ± ,1 + , a a where are centers. If a ̸= 0 and b ̸= 0 the singular points on x ̸= 0, whenever they exist, are of the form √√ √ 2y± a ± a2 − 4ab(a + 1) (±x̄± , ȳ± ), x̄± = , ȳ± = . a 2ab Studying when these finite singular points are defined and in these cases computing the eigenvalues of the Jacobian matrix at these points (using Theorem 3.5 in [4] whenever needed), we get: if a ∈ (−1, 0), b ∈ (a/(4(1 + a)), 0) we have that (±x̄+ , ȳ+ ) are centers and (±x̄− , ȳ− ) are saddles; if a ∈ (−1, 0) and b > 0 we have the two centers (±x̄+ , ȳ+ ); if a > 0 and b < 0 we have the two centers (±x̄− , ȳ− ); if if a > 0 and b ∈ (0, a/(4(1 + a)) we have the two saddles (±x̄+ , ȳ+ ) and the two centers (±x̄− , ȳ− ); if a > 0 and b = a/(4(1 + a)) we have the two cusps (±x̄+ , ȳ+ ); and for any other value of the parameters (a, b) ∈ (−1, ∞) × R with a2 + b2 ̸= 0, there are no finite singular points on x ̸= 0. Now we study the possible saddle connections. The saddle (0, y− ) and the saddle at the origin are never connected. The saddles (±x̄+ , ȳ+ ) (whenever they exist) are connected one with each other by symmetry and the saddles (±x̄− , ȳ− ) (whenever they exist) are also connected one with each other by symmetry. Moreover, the possible saddle connection between the saddles (±x̄+ , ȳ+ ) and the origin, or between the saddles (±x̄− , ȳ− ) and the origin is given on the curve b = 2a/(9(1 + a)). On the other hand, the possible saddle connection with the saddles (±x̄+ , ȳ+ ) and the saddle (0, y− ), or with the saddles (±x̄− , ȳ− ) and the saddle (0, y− ) is given on the curve b = b(a) HAMILTONIAN SYSTEMS WITH A NILPOTENT SADDLE AT THE ORIGIN 21 which is a real root of the polynomial − 12(3a2 + 3a + 1)a4 − 2(864a4 + 1728a3 + 1212a2 + 348a + 25)a2 b2 (8) + 36(2a + 1)3 (8a2 + 8a − 1)ab3 + 6(2a + 1)(36a2 + 36a + 11)a3 b + 81(2a + 1)4 b4 . Taking into account the information on the finite and infinite singular points together with the possible saddle connections and separating into dif- ferent regions, we have the following possible global phase portraits of Figure 1 (whenever there is more than one possible phase portrait, in parenthesis we give values of the parameters where they are realized): 1.3 if b > 1/4 and a ∈ (−1, 0); 1.2 if b > 1/4 and a = 0; 1.1 if b > 1/4 and a > 0; 1.36 if b ∈ (0, 1/4) and a ∈ (−1, 0), or b < 0 and a > 0; 1.35 if b ∈ (0, 1/4) and a = 0, or b < 0 and a = 0; 1.13 if b ∈ (a/(4(1 + a)), 1/4) and a > 0, or b < a/(4(1 + a)) and a ∈ (−1, 0); 1.25 if b = a/(4(1 + a)) and a ̸= 0; and 1.45 if b = 0 and a ̸= 0. If b ∈ (0, a/(4(1 + a))) and a > 0, or b ∈ (a/(4(1 + a)), 0) and a ∈ (−1, 0) they are 1.14 (a = 2, b = 7/50); 1.15 (a = 2, b = 2a/(9(1 + a)) = 4/27. Here there is a saddle connection); 1.16 (a = 2, b = 3/20); 1.17 (a = 2, b ≈ 0.15045562. Here b = b(a) is a real solution of equation (8) and so there is a saddle connection); and 1.18 (a = 2, b = 4/25). Finally, if b = 1/4 and a ∈ (−1, 0) they are 1.37; if b = 1/4 and a = 0 they are 1.38; and if b = 1/4 and a > 0 they are 1.8. 11. Global phase portrait of system (VIII): infinite singular points Consider system (VIII) x2 √ x′ = y + √ + sx2 y + by 2 + cy 3 , y ′ = ax3 − 2xy − sxy 2 2 with a > −1, b, c ∈ R and s ∈ {−1; 1}. We study in this section the infinite singular points. Lemma 4. On the local chart U1 we have: (a) no singular points if cs > 0 and as < 0, or c = 0 and as < 0, or ac < −1 and as > 0; (b) one singular point (the origin) if a = 0 and cs ≥ 0. It is the union of two elliptic (one stable and one unstable) and four parabolic sectors (two stable and two unstable); (c) two singular points which are an attracting node and a repelling node if c > 0 and a > 0, or c < 0 and a < 0, or c = 0 and as > 0; 22 M. CORBERA AND C. VALLS (d) two singular points which are the union of one elliptic and one √ hy- perbolic sector if cs < 0, as > 0 with c = −1/a and b ̸= −3s/( 2a), or the union of two elliptic sectors (one stable and one unstable) and four parabolic sectors (two stable √ and two unstable) if cs < 0, as > 0 with c = −1/a and b = −3s/( 2a); (e) three singular points which are an attracting node, a repelling node and the origin which is formed by two elliptic (one stable and one unstable) and four parabolic (two stable and two unstable) sectors if cs < 0 and a = 0; (f) four singular points which are two repelling and two attracting nodes if ac + 1 > 0, cs < 0 and as > 0. The origin of the local chart U2 is an infinite singular point if and only if c = 0. If b ̸= 0 it is the union of one hyperbolic and one elliptic sectors. If b = 0 it is the union of two hyperbolic sectors (one stable and one unstable) when s = 1 and the union of two elliptic (one stable and one unstable) and four parabolic sectors (two stable and two unstable) when s = −1. Proof. On the local chart U1 system (VIII) becomes 1 √ u′ = a − u(2cu3 + 3 2v + 2bu2 v + 4su + 2uv 2 ), 2 ′ 1 √ v = − v(2cu3 + 2v + 2bu2 v + 2su + 2uv 2 ). 2 When v = 0 the infinite singular points on the local chart U1 satisfy a − 2su2 − cu4 = 0. If cs > 0 and as < 0, or c = 0 and as < 0, or ac < −1 and as > 0 there are no infinite singular points in the local chart U1 . This proves statement (a). If a = 0 and cs ≥ 0 the unique finite singular point is the origin, which is linearly zero. Applying blow-up techniques we get that it is the union of two elliptic (one stable and one unstable) and four parabolic sectors (two stable and two unstable). This proves statement (b). If c > 0 and a > 0 then there are two √ infinite singular points on the √ (2) −s ac+1 local chart U1 which are (u± , 0) = (± c + c , 0). Computing the (2) eigenvalues of the Jacobian matrix at these points we get that (u+ , 0) is an (2) attracting node and (u− , 0) is a repelling node. If c < 0 and a < 0 then there exits √ two infinite singular points in the √ (1) ac+1 local chart U1 which are (u± , 0) = (± −s c − c , 0). Computing the (1) eigenvalues of the Jacobian matrix at these points we get that (u+ , 0) is a (1) repelling node and (u− , 0) is an attracting node. HAMILTONIAN SYSTEMS WITH A NILPOTENT SADDLE AT THE ORIGIN 23 If c = 0 and as > 0 there are ( two infinite singular points on the local (0) √a ) chart U1 that are (±u , 0) = ± 2s , 0 . Computing the eigenvalues of the Jacobian matrix at these points we get that (u(0) , 0) is an attracting node and (−u(0) , 0) is a repelling node when s = 1, while (u(0) , 0) is a repelling node and (−u(0) , 0) is an attracting node when s = −1. Statement (c) is proved. If cs < 0, as > 0 with c = −1/a then √ there are two infinite singular points in the √ local chart U 1 , which are (± as, 0).√ They are nilpotent if b ̸= −3s/( 2a) and are linearly zero if b = −3s/( 2a). In the first case, applying Theorem 3.5 in [4] together with blow-up techniques we obtain that both are the union of one elliptic and one hyperbolic sector. In the second case, applying again blow-up techniques we conclude that they are the union of two elliptic sectors (one stable and one unstable) and four parabolic sectors (two stable and two unstable). This proves statement (d). If cs < 0 and a = 0 then there are three infinite singular points which are the origin (0, 0) and the points (±u(1) , 0) if s = 1, or (±u(2) , 0) if s = −1. The origin of the local chart U1 it is linearly zero. Computing the eigenvalues of the Jacobian matrix at this point we get that it is formed by two elliptic (one stable and one unstable) and four parabolic (two stable and two unstable) sectors. On the other hand, computing the eigenvalues of the Jacobian matrix at the points (±u(1) , 0) we get that: (u(1) , 0) is a repelling node and (−u(1) , 0) is an attracting node when s = 1, while (u(1) , 0) is an attracting node and (−u(1) , 0) is a repelling node when s = −1. Statement (e) is proved. Finally, when ac + 1 > 0, cs < 0 and as > 0 there are the four singular (1) (2) points (u± , 0) and (u± , 0). Computing the eigenvalues of the Jacobian (1) (2) matrix at these points we get that (u− , 0) and (u+ , 0) are attracting nodes (1) (2) while (u+ , 0) and (u− , 0) are repelling nodes. This concludes the proof of the lemma concerning the local chart U1 . On the local chart U2 system (VIII) becomes 3 √ u′ = c + bv + 2su2 + v 2 + √ u2 v − au4 , v ′ = −uv(−s − 2v + au2 ). 2 The origin (0, 0) is an infinite singular point if and only if c = 0. If b ̸= 0 it is nilpotent and by Theorem 3.5 in [4] together with blow-up techniques we get that it is the union of one hyperbolic and one elliptic sectors. If b = 0 it is linearly zero, using blow-up techniques we get that it is the union of two hyperbolic sectors (one stable and one unstable) when s = 1 and the union of two elliptic (one stable and one unstable) and four parabolic sectors (two stable and two unstable) when s = −1. The proof of the lemma is complete.  24 M. CORBERA AND C. VALLS 12. Global phase portraits of system (VIII) when a = 0 We compute the finite singular points. On x = 0 the finite singular points are (0, −1/b) if c = 0 (with b ̸= 0), and if c ̸= 0 they are ( √ ) ± −b ± b2 − 4c p = (0, y± ) = 0, 2c which exist for c < c2 = c2 (b) := b2 /4. On the other hand, if x ̸= 0 the finite singular points are √ ) ( √ √ √ ) ( ± 2 √ s( 2b − s) − 2c 2 q = ± x̄, − = ± 2 ,− s s s √ which exist for c < c1 = c1 (b) := (s( 2b − s))/2. For c = c1 (b) the singular points q ± become p± . The type of these finite singular points will be studied later. Now we investigate the possible saddle connections. The possible saddle connections between the saddles p± (whenever they exist) and the origin occurs when (9) c = c(b) = 2b2 /9. Due to the symmetry of the problem the saddles q ± are always connected. We study the possible saddle connections between the saddles q ± and the origin. Doing so, we get the curve √ (2 2b − 3s)s (10) c = c(b) = . 3 Furthermore, the possible saddle connection between the saddles p± and the saddles q ± (whenever they exist) is along the curve c = c(b) which is a real solution of the cubic √ √ √ (11) 36c3 − 12s 2bc2 + 36c2 − 10b2 c + 18s 2bc + 9c − 4s 2b3 − 2b2 = 0. Moreover, when c = 0 the point (0, −1/b) (whenever it is a saddle) cannot be connected with the saddle at the origin and can be connected with the √ points ± q (whenever they are saddles) if and only if s = −1 and b = 1/(2 2), but this will not happen. Finally, we observe that when p+ and p− are saddles, then y+ > 0 and y− < 0 and so they are never connected. We consider the cases s = 1 and s = −1. Subcase s = 1. In view of the infinite singular points, we consider the cases c = 0, c < 0 and c > 0 separately. HAMILTONIAN SYSTEMS WITH A NILPOTENT SADDLE AT THE ORIGIN 25 Subcase c = 0. If b = 0 there are no finite singular points among the origin. Now we study the finite singular points different from the√origin. If b ̸= 0 there are three finite singular points√(0, −1/b), q ± if b > 1/ 2 and one finite singular point (0, −1/b) if b ≤ 1/ 2. Computing the eigenvalues of the Jacobian matrix at the singular point √ (0, −1/b) and using Theorem √ 3.15 in [4] we get that it √ is a center if b > 1/ 2, a nilpotent √ saddle if b = 1/ 2 and a saddle if b < 1/ 2. Moreover, when b > 1/ 2, computing the eigenvalues of the Jacobian matrix at the singular points q ± we get that they are both saddles. Taking into account the information on the finite and infinite singular points together with the possible saddle connections and separating into dif- ferent regions, we have the following possible global phase portraits of Figure 1 (whenever there is more than one possible phase portrait, in parenthesis we give√values of the parameters where √ they are realized):√ 1.46 if b = 0. For b > 1/ 2 they are: 1.47 (b = 4/(3 2)); 1.48 (b = 3/(2√ 2) and √ corresponds to the saddle√connection in (10)); and 1.49 (b = (3 + 2 2)/(2 2)). Finally, when b ≤ 1/ 2 with b ̸= 0 they are 1.50. Subcase c < 0. We define the two regions with (b, c) ∈ R × R− : √ S1 = {b < 1/ 2, c ∈ [c1 , 0)}, √ √ S2 = {b < 1/ 2, c < c1 } ∪ {b ≥ 1/ 2, c < 0}. We study the finite singular points different from the origin. In the region S1 , there exist two finite singular points p± . Computing the eigenvalues of the Jacobian matrix at these points and using Theorem 3.5 in [4] (in the case c = c1 ) we get that they are both saddles. In the region S2 we have the four finite singular points p+ (which is a center) and p− , q ± which are saddles. Taking into account the information on the finite and infinite singular points together with the possible saddle connections and separating into different regions, we have the following possible global phase portraits of Figure 1 (whenever there is more than one possible phase portrait, in paren- thesis we give values of the parameters where they are realized): 1.51 in region √S1 . In region S2 they are: 1.52 (b = −4, c = −4); 1.53 (b = 1/2, c = (2 2b − 3)/3 and is the saddle connection in (10)); and 1.54 (b = 2, c = −2). 26 M. CORBERA AND C. VALLS Subcase c > 0. We define the regions with (b, c) ∈ R × R+ : √ √ √ R1 = {0 < b ≤ 1/ 2, c ∈ (0, c2 )} ∪ {1/ 2 < b < 2, c ∈ (c1 , c2 )} √ R2 = {b > 2, c ∈ (c1 , c2 )} ∪ {b < 0, c ∈ (0, c2 )}, √ R3 = {b > 1/ 2, c ∈ (0, c1 )}, R4 = {c > c2 } √ √ √ L1 = {1/ 2 < b < 2, c = c1 }, L2 = {b > 2, c = c1 }, √ √ √ L3 = {b > 0, b ̸= 2, c = c2 } ∪ {b < 0, c = c2 }, P = ( 2, c2 ) = ( 2, 1/2). We study the finite singular points different from the origin. In the region R1 , there exist two finite singular points p± . Computing the eigenvalues of the Jacobian matrix at these points we get that p+ is a saddle and p− is a center. In the region R2 , there exist again the two finite singular points p± . Computing the eigenvalues of the Jacobian matrix at these points we get that p+ is a center and p− is a saddle. In the region R3 we have four finite singular points: p± which are centers, and q ± which are saddles. In the region R4 there are no finite singular points. In the region L1 we have two finite singular points: p+ which is a nilpotent saddle and p− which is a center. In the region L2 we have two finite singular points: p+ which is a center and p− which is a nilpotent saddle. Finally, in the region L3 we have a unique finite singular point p− = p+ which is a cusp. √Finally, in the region P we get that the unique finite singular point (0, − 2) is linearly zero. Applying blow-up techniques we conclude that it is the union of two hyperbolic sectors (one stable and one unstable). Taking into account the information on the finite and infinite singular points together with the possible saddle connections and separating into dif- ferent regions, we have the following possible global phase portraits of Figure 1 (whenever there is more than one possible phase portrait, in parenthesis we give values of the parameters where they are realized): 1.35 in region R1 . In region R2 they are: 1.41 (b = 3, c = 19/10); 1.42 (b = 3, c = 2. Here there is the saddle connection given in (9)); 1.43 (b = 3, c = 43/20); and 1.35 (b = −3, c = 1/2). √ In region R3 they are: 1.55 (b = 1, c = 1/10); 1.56 (b = 3/2, c = −1 + 2. Here there is the saddle connection given in (10)); and 1.57 (b = 3/2, c = 1/10). In regions R4 , L1 , and L2 they are 1.2, 1.35, and 1.43, respectively. Finally in the regions L3 , or P they are 1.38. Subcase s = −1. Again, in view of the infinite singular points, we consider the cases c = 0, c < 0 and c > 0 separately. Subcase c = 0. We study√ the finite singular points different from the origin. If b > 0, or b ∈ (−1/ 2, 0), then there are three finite singular points: (0, −1/b) and q ± . Computing the eigenvalues of the Jacobian matrix at the singular point (0, −1/b) and using Theorem √ 3.15 in [4] we get that it is a center and q ± are two saddles. If b ≤ −1/ 2 then there is a unique finite HAMILTONIAN SYSTEMS WITH A NILPOTENT SADDLE AT THE ORIGIN 27 singular point (0, −1/b) which is a saddle. Finally, if b = 0 the unique finite singular points are q ± which are saddles. Taking into account the possible saddle connections we get that the unique global phase portraits are topologically equivalent to the following √ ones of Figure 1: 1.58 √ for b > 0; 1.59 for b = 0; 1.60 for b ∈ (−1/ 2, 0); and 1.50 for b ≤ −1/ 2. Subcase c < 0. We define the following regions with (b, c) ∈ R × R− √ √ S1 = {b ≤ −1/ 2} ∪ {b > −1/ 2, c ≤ c1 }, √ S2 = {b > −1/ 2, c ∈ (c1 , 0)}. We study the finite singular points different from the origin. In the region S1 , there exist the two finite singular points p± . Computing the eigenvalues of the Jacobian matrix at these points we get that p+ is a center and p− is a saddle (this saddle is a nilpotent saddle when c = c1 ). In the region S2 there are the four finite singular points: p± which are centers and q ± which are saddles. Taking into account the information in the finite and infinite singular points together with the possible saddle connections studied above, we conclude that the unique global phase portraits are topologically equivalent to the following ones of Figure 1: 1.35 in the region S1 and 1.61 in the region S2 . Subcase c > 0. If c > 0 we define the regions. We note that (b, c) ∈ R×R+ : √ R1 = {c = c2 , b ̸= 0, b ̸= − 2}, R2 = {c > c2 }, √ √ √ P = (− 2, c1 ) = (− 2, 1/2), R3 = {b < −1/ 2, c ∈ (0, c1 ]} \ P √ √ √ R4 = {b ∈ (− 2, −1/ 2), c ∈ (c1 , c2 )} ∪ {b ∈ [−1/ 2, 0), c ∈ (0, c2 )}, √ R5 = {b > 0, c ∈ (0, c2 )} ∪ {b < − 2, c ∈ (c1 , c2 )}. We study the finite singular points different from the origin. In the region R1 there are the three finite singular points p− = p+ and q ± . Computing the eigenvalues of the Jacobian matrix at this point and using Theorem 3.5 in [4] we conclude that p− is a cusp and q ± are saddles. In the region R2 there are the two saddles q ± . In the region R3 we have the two saddles p± (they are nilpotent when c = c1 ). In the region R4 we have the four finite singular points p− which is a center, p+ and q ± which are all saddles. In the region R5 we have the singular point p+ which is a center and the singular points p− and q ±√ which are all saddles. Note that the point P is the linearly zero point (0, 2) which is formed by six hyperbolic sectors (three stable and three unstable). Taking into account the information on the finite and infinite singular points together with the possible saddle connections and separating into different regions, we have the following possible global phase portraits of 28 M. CORBERA AND C. VALLS Figure 1 (whenever there is more than one possible phase portrait, in paren- thesis we give values of the parameters where they are realized): 1.62 in region R2 ; 1.63 in region P ; and 1.51 in region R3 . In the region R1 they are: 1.64 (b = −3, c = 9/4), 1.65 (b = −6/5, c = 9/25), and 1.66 (b = 1, c = 1/4). We recall the phase portraits in region R1 correspond to the cases in which the center and the saddle (0, y± ) coalesce. In the region R4 there are three possible global phase portraits: 1.67 (b = −6/5, c = 71/200); 1.68 (b = −6/5, c ≈ 0.35793523. Here c = c(b) is a real solution of the cubic equation given in (11) and so there is a saddle connection); and 1.69 (b = −6/5, c = 359/1000). Finally, in the region R5 there are six possible global phase portraits: 1.70 (b = 1, c = 1/10); 1.71 (b = 1, c = 2/9. This corresponds to the saddle connection given in (9)); 1.72 (b = 1 and c = 6/25); 1.52 (b = −3, c = 17/10); 1.73 (b = −3, c ≈ 1.85771681. Here c = c(b) is a real solution of the cubic equation given in (11) and so there is a saddle connection); and 1.74 (b = −3, c = 21/10). 13. Global phase portraits of system (VIII) when c = 0, a ̸= 0 We compute the finite singular points different from the origin. If b = 0 there are no finite singular points with x = 0 different from the origin. If b ̸= 0 there is a unique finite singular point different with x = 0 which is (0, −1/b). Computing √ the eigenvalues of the Jacobian √ matrix at this point we get√that if b < s/ 2 it is a saddle, if b > s/ 2 it is a center and if b = s/ 2 it is nilpotent. Using Theorem 3.5 in [4] we get that if a > 1 it is a center, if a < 1 it is a saddle, and if a = 1 it is a saddle when s = 1 and a center when s = −1. Now we study the finite singular points with x ̸= 0. These are (whenever they exist) (x̄± , ȳ± ) and (−x̄± , ȳ± ) where √ √ √ √ √ ȳ± ( 2 + sȳ± ) −2ab − 3 2s ± (2ab + 3 2s)2 − 16(1 + a) x̄± = , ȳ± = . a 4 √ The total number of solutions (±x̄± , ȳ± ) can change when (2ab + 3 2s)2 − 16(1 + a) = 0 √ (in which case the solutions ȳ± coincide), or when ȳ± = 0, or when ȳ± = − 2s (in the last two cases x̄± = 0). The first condition leads to the values of b equal to b± where √ √ −3 2as ± 4 a2 (1 + a) b± = b± (a) = . 2a2 The second condition √ is never possible while the third one is possible if and only if b = 1/ 2s. Moreover x̄± evaluated at y = ȳ± with b = b+ is not defined for a ∈ (−1, 0) ∪ (0, +∞) when s = −1 and for a ∈ (−1, 0) ∪ (0, 1) when s = 1, while x̄± evaluated at y = ȳ± with b = b− is not defined for a > 1 when s = 1. So in these cases the number of real solutions (±x̄± , ȳ± ) HAMILTONIAN SYSTEMS WITH A NILPOTENT SADDLE AT THE ORIGIN 29 does not change on the curves b = b± . The type of the singular points (±x̄± , ȳ± ) will be studied later. The saddle (0, −1/b) (whenever it exists) and the origin are never con- nected while the saddles (x̄+ , ȳ+ ) (respectively (x̄− , ȳ− )) and the saddles (−x̄+ , ȳ+ ) (respectively (−x̄− , ȳ− )), whenever they exist, are connected by symmetry. The connection between the saddles (±x̄− , ȳ− ) and the saddle at the origin occurs on the curve √ −3s + 3 1 + a (12) b = b(a) = √ , 2a while the connection between the saddles (±x̄+ , ȳ+ ) and the saddle at the origin occurs on the curve √ −3s − 3 1 + a (13) b = b(a) = √ . 2a We will see that the points (±x̄+ , ȳ+ ) and (±x̄− , ȳ− ) are never simulta- neously four saddles and so we do not need to study the possible sad- dle connections among them. Finally, the connection between the saddles (±x̄± , ȳ± ) and the saddle (0, −1/b) occurs on the curve which is a real root with b = b1 (a) > 0 of the polynomial (14) √ √ −8as−16 2ab−3s(−1−4a+8a2 )b2 +2 2(3+10a+12a2 )b3 +2as(1+3a+3a2 )b4 . We study the cases s = 1 and s = −1 separately. Subcase s = 1. Taking into account the information on the finite and infinite singular points we distinguish between b = 0; a > 0 and b ̸= 0; and a < 0 and b ̸= 0. Subcase b = 0. If a > 0 there are no finite singular points among the origin. If a ∈ (−1, 0), besides the origin, there are the two finite singular points (±x̄+ , ȳ+ ) which are two centers. The unique possible global phase portraits are topologically equivalent to 1.4 in Figure 1 when a > 0 and 1.75 in Figure 1 when a ∈ (−1, 0). Subcase a > 0 and b ̸= 0. We consider the following regions for (a, b) ∈ R+ × (R \ {0}): √ √ S1 = {a > 1, b ≥ 1/ 2} ∪ {a ∈ (0, 1], b > 1/ 2}, √ S2 = {a > 1, b+ < b < 1/ 2}, S3 = {a > 1, b = b+ } ∪ {b = b− }, √ S4 = {a > 1, b− < b < b+ } ∪ {a ∈ (0, 1], b− < b ≤ 1/ 2}, S5 = {b < b− }. If (a, b) ∈ S1 then there are two unique finite singular points with x ̸= 0 which are (±x̄− , ȳ− ) that are both saddles. If (a, b) ∈ S2 the four finite singular points with x ̸= 0 exist, being (±x̄+ , ȳ+ ) two centers and (±x̄− , ȳ− ) 30 M. CORBERA AND C. VALLS two saddles. If (a, b) ∈ S3 we have ȳ− = ȳ+ , and the two finite singular points (with x ̸= 0) (±x̄− , ȳ− ) = (±x̄+ , ȳ+ ) become nilpotent. Using Theorem 3.5 in [4] we obtain that they are both cusps. If (a, b) ∈ S4 there are no finite singular points with x ̸= 0. If (a, b) ∈ S5 the four finite singular points with x ̸= 0 exist, being (±x̄+ , ȳ+ ) two saddles and (±x̄− , ȳ− ) two centers. Taking into account the information on the finite and infinite singular points together with the possible saddle connections and separating into different regions, we have the following possible global phase portraits of Figure 1 (whenever there is more than one possible phase portrait, in paren- thesis we give values of the parameters where they are realized): 1.27 in the region S3 and 1.28 in the region S4 . √ In the √ region S1 they are: 1.76 (a = 2, b = 1); 1.77 (a = 2, b = 3(−1 + 3)/(2 2). Here b = b(a) is as in (12) and so there is a saddle connection); and 1.78 (a = 2, b = 3/4). In the region √ √ S2 they are: 1.33 (a = 4, b = 69/100); 1.32 (a = 4, b = 3( 5 − 1)/(4 2). Here b = b(a) is as in (12) and so there is a saddle connection); 1.31 (a = 4, b = 63/100); 1.30 (a = 4, b ≈ 0.59902710. Here b = b1 (a) is a real root with b > 0 of the polynomial in (14) and so it corresponds to a saddle connection); and 1.29 (a = 4, b = 59/100). In the region S5 they are: 1.29 (a = 2, b = −71/25); 1.30 (a = 2, b ≈ −2.89367324. Here b = b1 (a) is a real root with b < 0 of the polynomial in (14) and so it corresponds to a √ saddle √connection); 1.31 (a = 2, b = −579/200); 1.32 (a = 2, b = −3(1 + 3)/(2 2). Here b = b(a) is as in (13) and so there is a saddle connection); and 1.33 (a = 2, b = −3). Subcase a ∈ (−1, 0) and b ̸= 0. We have the following regions for (a, b) ∈ (−1, 0) × (R \ {0}): √ R1 = {1/ 2 < b < b− }, L1 = {b = b− }, √ R2 = {b > b− }, R3 = {0 < b ≤ 1/ 2} ∪ {b < 0}. If (a, b) ∈ R1 the four finite singular points with x ̸= 0 exist, being (±x̄+ , ȳ+ ) two centers and (±x̄− , ȳ− ) two saddles. If (a, b) ∈ L1 we have ȳ+ = ȳ− , and the two finite singular points (with x ̸= 0) (±x̄− , ȳ− ) = (±x̄+ , ȳ+ ) become nilpotent. Using Theorem 3.5 in [4] we obtain that they are both cusps. If (a, b) ∈ R2 there are no finite singular points with x ̸= 0. If (a, b) ∈ R3 there are two unique finite singular points with x ̸= 0 which are (±x̄+ , ȳ+ ) and are both centers. Taking into account the information on the finite and infinite singular points together with the possible saddle connections and separating into different regions, we have the following possible global phase portraits of Figure 1 (whenever there is more than one possible phase portrait, in paren- thesis we give values of the parameters where they are realized): 1.44 in R2 HAMILTONIAN SYSTEMS WITH A NILPOTENT SADDLE AT THE ORIGIN 31 and 1.45 in R3 . In the √ region R√ 1 they are: 1.79 (a = −2/5, b = 9/10); 1.80 (a = −2/5, b = −3( 15 − 5)/(2 2). Here b = b(a) is as in (12) and so there is a saddle connection); and 1.81 (a = −2/5, b = 13/10). In the region L1 the cusps correspond to the case in which the saddles and the centers in the region R1 coalesce. So, the unique global phase portraits are topologically equivalent to 1.82. Subcase s = −1. Again, we distinguish between the cases b = 0; a > 0 and b ̸= 0; and a < 0 and b ̸= 0. Subcase b = 0. If a ∈ (−1, 0) there are two finite singular points besides the origin which are two saddles. If a ∈ (0, 1/8) then there are four singular points: (±x̄+ , ȳ+ ) which are saddles and (±x̄− , ȳ− ) which are centers. If a = 1/8 there are two finite singular points besides the origin which are two cusps (it corresponds to the fact that the saddles and the centers when a ∈ (0, 1/8) coalesce). Finally, if a > 1/8 there are no more finite singular points among the origin. Taking all this into account together with the possible saddle connections we conclude that the possible global phase portraits are: 1.83 when a ∈ (0, 1/8); 1.84 when a = 1/8; 1.2 when a > 1/8; and 1.85 when a ∈ (−1, 0). Subcase a > 0 and b ̸= 0. We have the following regions for (a, b) ∈ R+ × (R \ {0}): √ √ S1 = {b < −1/ 2} ∪ {a ∈ (0, 1), b = −1/ 2}, √ N1 = {a ∈ (0, 1), b = b− }, S2 = {a ∈ (0, 1), −1/ 2 < b < b− }, √ S3 = {a ∈ (0, 1), b > b− } ∪ {a ≥ 1, b ≥ −1/ 2}. If (a, b) ∈ S1 there are two unique finite singular points (centers) with x ̸= 0 which are (±x̄− , ȳ− ). If (a, b) ∈ N1 there are two finite singular points (cusps) with x ̸= 0 which are (±x̄− , ȳ− ) = (±x̄+ , ȳ+ ). If (a, b) ∈ S2 the four finite singular points with x ̸= 0 exist being (±x̄+ , ȳ+ ) two saddles and (±x̄− , ȳ− ) two centers. If (a, b) ∈ S3 there are no finite singular points with x ̸= 0. Taking into account the information on the finite and infinite singular points together with the information of the possible saddle connections we have the following possible global phase portraits of Figure 1 (whenever there is more than one possible phase portrait, in parenthesis we give values of the parameters when they are realized): 1.45 in the region S1 ; 1.86 (a = 1/2, b = −67/100) and 1.87 (a = 1/10, b = 1/10) in the region S2 ; 1.88 (a = 1/2, b = b− ) and 1.89 (a = 1/10, b = b− ) in the region N1 (they correspond to the cases in which the saddles and the centers in the region S2 coalesce); and 1.12 in the region S3 . 32 M. CORBERA AND C. VALLS Subcase a ∈ (−1, 0) and b ̸= 0. We consider the following regions for (a, b) ∈ (−1, 0) × (R \ {0}): R1 = {b < b− }, R2 = {b = b− }, √ √ R3 = {b− < b ≤ −1/ 2}, R4 = {−1/ 2 < b < 0} ∪ {b > 0}. If (a, b) ∈ R1 the four finite singular points with x ̸= 0 exist, being (±x̄+ , ȳ+ ) two centers and (±x̄− , ȳ− ) two saddles. If (a, b) ∈ R2 we have ȳ− = ȳ+ , and the two finite singular points (with x ̸= 0) (±x̄− , ȳ− ) = (±x̄+ , ȳ+ ) become nilpotent. Using Theorem 3.5 in [4] we obtain that they are both cusps. If (a, b) ∈ R3 there are no finite singular points with x ̸= 0. If (a, b) ∈ R4 there are two unique finite singular points with x ̸= 0 which are (±x̄+ , ȳ+ ) that are both saddles. Taking into account the information on the finite and infinite singular points together with the possible saddle connections and separating into different regions, we have the following possible global phase portraits of Figure 1 (whenever there is more than one possible phase portrait, in paren- thesis we give values of the parameters where they are realized): 1.27 in R2 and 1.28 in R3 . In the region R4 they are: 1.90 (a = −1/2, b = −1/2) and 1.91 of Figure 1 (a = −1/2, b = 1). In the region R1 they √ are: √1.29 (a = −9/10, b = −307/100); 1.30 (a = −9/10, b = −(10 + 10)/(3 2). Here b = b(a) is as in (12) and thus corresponds to a saddle connection); 1.31 (a = −9/10, b = −16/5); 1.32 (a = −9/10, b ≈ −3.31071911. Here b = b1 (a) is a real root with b < 0 of of the polynomial in (14) and thus corresponds to a saddle connection); and 1.33 (a = −9/10, b = −4). 14. Global phase portraits of system (VIII) when ac = −1, as > 0, cs < 0 The finite singular points (whenever they exist) are p± = (0, y± ) and (±x̄, ȳ) where ab 1 √ −2(1 + a)s y± = ± a(4 + ab2 ), ȳ = √ 2 2 3 2 + 2abs and √ √ −(1 + a)( 2ab + s(2 − a)) x̄ = 2 √ . a(3 2 + 2abs)2 The total number of solutions p± can change when a(4 + ab2 ) = 0 that is b = ± √2−a with a ∈ (−1, 0) and s = −1. The total number of solutions √ √ (±x, y) can change when either 3 2√+ 2abs = 0, that is,√b = −3/( 2as) (in which case x and y disappear), √ or 3 2 + 2abs ̸= 0 and 2ab + s(2 −√ a) = 0, that is, √ b = (a−2)s/( 2a) (in which case x = 0). Note that (a−2)s/( 2a) = −3/( 2as) if and only if a = −1, and so it is never possible. HAMILTONIAN SYSTEMS WITH A NILPOTENT SADDLE AT THE ORIGIN 33 The saddles p− and p+ whenever they exist cannot be connected one with each other. The saddles p± and the origin can√be connected if and only if s = −1 and along the curve b = b2 (a) = ±3/( −2a). On the other hand the points (±x̄, ȳ) are never saddles. We distinguish between the cases: s = 1 and s = −1. Subcase s = 1. In this case, taking into account that a > 0 we have the following regions for (a, b) ∈ R+ × R: √ √ √ S1 = {b < −3/( 2a)} ∪ {−3/( 2a) < b < (a − 2)/( 2a)}, √ √ L1 = {b = −3/( 2a)}, S2 = {b ≥ (a − 2)/( 2a)}. We study the finite singular points different from the origin. If (a, b) ∈ S1 then there exist the four finite singular points p± which are two saddles and (±x̄, ȳ) which are two centers. If (a, b) ∈ L1 , then there exist the two finite singular points p± which are two saddles. In the region S2 there are the two finite singular points p− (which is a center) and p+ (which is a saddle). Taking into account the information on the finite and infinite singular points and the fact that there are no saddle connections in this case we conclude that the global phase portraits are topologically equivalent to the following ones of Figure 1: 1.26 in the region S1 ; 1.6 in the region L1 ; and 1.92 in the region S2 . Subcase s = −1. In this case a < 0, i.e. a ∈ (−1, 0). We consider the following regions for (a, b) ∈ (−1, 0) × R: √ √ √ R1 = {b > 2/ −a} ∪ {(2 − a)/( 2a) < b < −2/ −a} √ √ M1 = {b = 2/ −a} ∪ {b = −2/ −a} √ √ √ R2 = {−2/ −a < b < 2/ −a}, M2 = {b = 3/( 2a)}, √ √ R3 = {b < (2 − a)/( 2a), b ̸= 3/( 2a)}. We study the finite singular points different from the origin. If (a, b) ∈ R1 then there exist the two singular points p+ , which is a center and p− , which is a saddle. If (a, b) ∈ M1 , there exist the unique finite singular p− = p+ which is a cusp. In the region R2 there are no finite singular points. In the region M2 there exist the two singular points p± which are two saddles and in the region R3 there are the four singular points p± which are saddles and (±x̄, ȳ) which are centers. Taking into account the information on the finite and infinite singular points together with the possible saddle connections and separating into different regions, we have the following possible global phase portraits of Figure 1 (whenever there is more than one possible phase portrait, in paren- thesis we give values of the parameters where they are realized): 1.93 in R2 , 1.6 in M2 , and 1.26 in R3 . In the region R1 they are: 1.94 (a = −1/2, 34 M. CORBERA AND C. VALLS b = 29/10); 1.95 (a = −1/2, b = 3. Here we have a saddle connection); 1.96 (a = −1/2, b = 4); √ and 1.92 (a = −1/2, b = −3). √ In the region M1 they are 1.97 when b = 2/ −a) and 1.98 when b = −2/ −a. Note that these global phase portraits correspond to the ones in the region R1 which the saddle and the center coalesce. 15. Global phase portraits of system (VIII) when ac ̸= 0 and ac ̸= −1: finite singular points on x = 0 We study the case ac ̸= 0 with either ac < −1 and as > 0 or cs > 0 and as < 0. In this section we focus on the finite singular points on x = 0. If b2 − 4c < 0 there are no finite singular points on x = 0. If b2 = 4c there is a unique finite singular point on x = 0 which is (0, −2/b). Computing √ the eigenvalues of the Jacobian √ matrix at this point we get that if b ̸= 2s it is nilpotent and if b = 2s it is linearly zero. In the√ first case using Theorem 3.5 in [4] we get that it is a cusp and if b = 2s then using blow-up techniques we conclude that it is the union of two hyperbolic sectors (one stable and one unstable) if s = 1 and is the union of six hyperbolic sectors (3 stable and 3 unstable) if s = −1. If b2 > 4c we get that(there are √ two) finite singular points with x = 0 ± −b± b2 −4c which are p = (0, y± ) = 0, 2c . We introduce the notation 1 √ b2 c1 = c1 (b) = (−1 + 2bs), c2 = c2 (b) = 2 4 We distinguish between the cases s = 1 and s = 1. Subcase s = 1. In this case computing the eigenvalues of the Jacobian matrix at the points p± we get the following regions √ R1 = {(b, c) ∈ R2 : c1 < c < c2 , b > 2} ∪ {(b, c) ∈ R2 : 0 < c < c2 , b < 0}, √ √ ∪ {(b, c) ∈ R2 : c < c1 , b < 1/ 2} ∪ {(b, c) ∈ R2 c < 0, b ≥ 1/ 2}, √ R2 = {(b, c) ∈ R2 : 0 < c < c1 , b > 1/ 2}, √ R3 = {(b, c) ∈ R2 : 0 < c < c2 , 0 < b ≤ 1/ 2} √ √ ∪ {(b, c) ∈ R2 : c1 < c < c2 , 1/ 2 < b < 2}, √ R4 = {(b, c) ∈ R2 : c1 < c < 0, b < 1/ 2}, √ √ L1 = {c = c1 , b < 2}, L2 = {c = c1 , b > 2}. If (b, c) ∈ R1 then p+ is a center and p− is a saddle. If (b, c) ∈ R2 , both p± are centers. If (b, c) ∈ R3 then p+ is a saddle and p− is a center, if (b, c) ∈ R4 both p± are saddles. If (b, c) ∈ L2 then p+ is a center and√p− is nilpotent. √ Using Theorem 3.15 in [4] we get that it is a saddle if 2 < b ≤ 3/ 2 HAMILTONIAN SYSTEMS WITH A NILPOTENT SADDLE AT THE ORIGIN 35 √ √ √ or b > 3/ √ 2 and a > 1/(2 − 2b), and it is a center if b > 3/ 2 and − a≤ √ 1/(2 − √2b). On the other hand,√if (b, c) ∈ L1 then p√ is a center if 1/ 2 < b < 2 and a saddle if b < 1/ 2 (note that b = 1/ 2 corresponds to c = 0). Moreover p+ is nilpotent. √ Using Theorem 3.5 in√[4] we get that it is a saddle if a ≤ 1/(2 − 2b) and a center if a > 1/(2 − 2b). Subcase s = −1. In this case computing the eigenvalues of the Jacobian matrix at the points p± we get the following regions √ √ R1 = {(b, c) ∈ R2 : c1 < c < c2 , b < − 2} ∪ {(b, c) ∈ R2 : c < 0, b ≤ −1/ 2}, √ ∪ {(b, c) ∈ R2 : c < c1 , b > −1/ 2} ∪ {(b, c) ∈ R2 : 0 < c < c2 , b > 0}, √ R2 = {(b, c) ∈ R2 : 0 < c < c1 , b < −1/ 2}, √ √ R3 = {(b, c) ∈ R2 : c1 < c < c2 , − 2 < b ≤ −1/ 2} √ ∪ {(b, c) ∈ R2 : 0 < c < c2 , −1/ 2 < b < 0}, √ R4 = {(b, c) ∈ R2 : c1 < c < 0, b > −1/ 2}, √ √ √ L1 = {c = c1 , b < − 2}, L2 = {c = c1 , − 2 < b < − 2}, √ L3 = {c = c1 , b > −1/ 2}. If (b, c) ∈ R1 then p− is a saddle and p+ is a center. If (b, c) ∈ R2 , both p± are both saddles. If (b, c) ∈ R3 then p− is a center and p+ is a saddle, if (b, c) ∈ R4 both p± are both centers. If (b, c) ∈ L1 then p− is a saddle and p+√is nilpotent.√Using Theorem√ 3.15 in [4] we get that √ it is a saddle if −3/ 2 ≤ b <√− 2 or b < −3/ √2 and a ≥ 1/(2 + 2b), and it is a center if b < −3/ 2 and a < 1/(2 + 2b). On the other hand, if (b, c) ∈ L2 then p+ is is a saddle and p− is nilpotent. √ Using Theorem 3.5 in [4] √ we get that it is a saddle if a < 1/(2 + 2b) and a center if a ≥ 1/(2 + 2b). Finally, if (a, b) ∈ L3 then p+ is a center and p− is nilpotent. √ Using again Theorem 3.5 in √ [4] we get that it is a saddle if a < 1/(2 + 2b) and a center if a ≥ 1/(2 + 2b). 16. Global phase portraits of system (VIII) when ac ̸= 0 and ac ̸= −1: preliminaries on finite singular points with x ̸= 0 We continue determining the finite singular points of systems (VIII) with x ̸= 0. We first introduce four auxiliary lemmas. Auxiliary lemmas. Lemma 5. There exist at most four finite singular points for system (VIII) with x ̸= 0 which are, whenever they exist, √ √ √ √ ( ) ( ) ȳ+ ( 2 + sȳ+ ) ȳ− ( 2 + sȳ− ) (15) ± , ȳ+ and ± , ȳ− a a 36 M. CORBERA AND C. VALLS with √ √ √ 2ab + 3 2s ± (2ab + 3 2s)2 − 16(1 + a)(ac + 1) (16) ȳ± = − . 4(1 + ac) Proof. We compute the Gröebner basis for the polynomials in system (VIII) and we obtain six polynomials {p1 , . . . , p6 } where 1 ( √ ) ( √ ) p2 = xy 2(1+a)+(3 2+2ab)y+2(1+ac)y 2 , p6 = −x ax2 − 2y−sy 2 . 2 From p6 = 0 we get √ 2 y( 2 + sy) (17) x = . a Recall that a ̸= 0. Moreover since x ̸= 0 we must have, in particular, y ̸= 0. Now solving p2 = 0 with respect to y we get y = ȳ± as in (16). Substituting these values of y and the corresponding x given in (17) into the Gröebner basis we get pi = 0 for i = 1, . . . , 6 and so we have at most two solutions in y and two solutions in x for each solution in y. In short, there are at most four solutions of system (VIII) with x ̸= 0 and they are, whenever they exist, the ones in the statement of the lemma. This concludes the proof of the lemma.  Lemma 6. The finite singular points with x ̸= 0 are either elementary or nilpotent. They are nilpotent if and only if √ 2a2 b2 + 6 2abs + 1 − 8a c = c0 := 8a(1 + a) and in this case, if they exist, they are two cusps. Proof. We compute the Gröebner basis for the polynomials ẋ, ẏ and the determinant of the linear part of system (VIII) and we get a set of 22 poly- nomials. Recall that xy ̸= 0 (because if y = 0 it follows easily from system (VIII) that x = 0). One of the polynomials in the Gröebner basis can be written as −16a(1 + a)2 xy(c − c0 )(c − c1 ). Hence we must study the cases c = c1 and c = c0 . If c = c1 , substituting √ √ it into the Gröebner basis we get a solution of the form y = −s 2 = − 2/s. This is not possible because then x = 0 (see (17)). On the other hand, if c = c0 , substituting it into the Gröebner basis we get the following cases: √ (i) b = −s(1 − 2a)/(√2a); (ii) b = −s(5 +√2a)/( 2a); (iii) b = −3s/( 2a); √ √ (iv) y = −4s(1 + a)/(3 2 + 2asb) with b ̸= −3s/( 2a). We consider each case (i)–(iv) separately. Note that in statement (iv) we have that y = ȳ+ = ȳ− given in (16) (because c = c0 ). HAMILTONIAN SYSTEMS WITH A NILPOTENT SADDLE AT THE ORIGIN 37 Substituting the √ value of b in case (i) into the Gröebner basis we get the solution y = −s 2 which is not possible because then x = 0 (see Lemma 5). On the other hand, substituting the √ value of b in case (ii) into the Gröebner √ basis we get the solutions y = s 2 and x = ±2/ as. Both solutions are included in case (iv) (whenever b is as in (ii)) so we will study them later when we study the general case (iv). Substituting the value of b in (iii) into the Gröebner basis we get the solution x = y = 0 which is not possible. Finally, introducing the value of y in (iv) into the Gröebner basis we get a polynomial quadratic in x. Solving it we obtain x = ±x with √ √ 8(1 + a)(−1 + 2a − 2abs) (18) x= s √ . a(3 2 + 2sab)2 Note that x is precisely the value of the component x in the solutions given in (15) with ȳ+ = ȳ− because c = c0 and so the four solutions become two (whenever x exists). Substituting these two values of x into the Gröebner basis we get that it is identically zero. Now taking c = c0 , and computing the eigenvalues of the Jacobian matrix at the points (±x, y) with y as in (iv) and x as in (18) we get that it is not the 2 × 2-zero matrix. Hence both points (±x, y) are nilpotent. Applying Theorem 3.5 in [4] we conclude that both of them are cusps. To conclude the proof of the lemma we just note that taking c = c0 and solving the polynomials ẋ = ẏ = 0 we get exactly two solutions with x ̸= 0 (which are (±x, y) and two solutions with x = 0. This concludes the proof of the lemma.  Lemma 7. In the case ac + 1 > 0, cs < 0 and as > 0 there cannot be two centers with x = 0. Proof. Note that when s = 1 the √ regions in which we can √ have two centers √ with x = 0 are: √ 0 < c <√c 1 , b > 1/ 2, or c = √ c1 , b > 3/ 2, a ≤ 1/(2 − 2b), or c = c1 , 1/ 2 < b < 2 and a > 1/(2 − 2b). In the three cases we have that c > 0 which is not possible because then cs > 0. When s = −1, the regions √ in which we can have √ two centers with x√= 0 are: c1 < c < 0, b > −1/ 2, or c = c1 , b > −1/ 2 and a ≥ 1/(2 + 2b). In both cases we have that c < 0 which is again not possible because then cs > 0. This concludes the proof of the lemma.  Now we investigate the possible saddle connections. Saddle connections. The saddles p+ and p− (whenever they exist) cannot be connected. The saddles p± (whenever they exist) and the origin can be connected along the surface c = c(a, b) in the parameter space with 2b2 (19) c= . 9 38 M. CORBERA AND C. VALLS The saddles that are not on x = 0, that is, the saddles (±x̄± , ȳ± ) with x̄± ̸= 0 (whenever they exist) are connected by symmetry and they are connected with the saddle at the origin along the surface c = c(a, b) in the parameter space with √ 2ab2 + 6 2bs − 9 (20) c= . 9(1 + a) The saddles (±x̄+ , ȳ+ ) and the saddles (±x̄− , ȳ− ) can be connected (when- ever they exist ) if and only if c = c0 (in which case they become two cusps). Finally, the saddles (±x̄± , ȳ± ) with x̄± ̸= 0 and the saddles p± (whenever they exist) can be connected on a surface c = c(a, b) in the parameter space which is any real solution (whenever it exists) of the equation √ ( √ ) (21) (2b − 2s(1 + 2c)) C1 (a, b, c) + 2 2bsC2 (a, b, c) = 0, where C1 = C1 (a, b, c) is C1 = −64a2 b4 + 24ab6 + 96a2 b6 − 192a3 b6 + 16a2 b8 + 48a3 b8 + 48a4 b8 + 576a2 b2 c − 252ab4 c − 1328a2 b4 c + 1632a3 b4 c + 120ab6 c + 264a2 b6 c + 144a3 b6 c − 1248a4 b6 c + 48a3 b8 c + 144a4 b8 c + 144a5 b8 c − 1296a2 c2 + 792ab2 c2 + 5760a2 b2 c2 − 2880a3 b2 c2 − 1152ab4 c2 − 3684a2 b4 c2 − 6816a3 b4 c2 + 8544a4 b4 c2 + 456a2 b6 c2 + 936a3 b6 c2 + 144a4 b6 c2 − 2592a5 b6 c2 + 48a4 b8 c2 + 144a5 b8 c2 + 144a6 b8 c2 − 648ac3 − 7776a2 c3 − 2592a3 c3 + 18b2 c3 + 3456ab2 c3 + 13608a2 b2 c3 + 26208a3 b2 c3 − 17280a4 b2 c3 − 516ab4 c3 − 4772a2 b4 c3 − 10272a3 b4 c3 − 2448a4 b4 c3 + 13824a5 b4 c3 − 264a3 b6 c3 − 1392a4 b6 c3 − 2592a5 b6 c3 − 1728a6 b6 c3 − 81c4 − 3240ac4 − 17496a2 c4 − 12960a3 c4 − 1296a4 c4 + 90b2 c4 + 3672ab2 c4 + 16560a2 b2 c4 + 27360a3 b2 c4 − 4320a4 b2 c4 − 23040a5 b2 c4 + 200a2 b4 c4 + 2784a3 b4 c4 + 9696a4 b4 c4 + 13824a5 b4 c4 + 6912a6 b4 c4 − 324c5 − 5184ac5 − 18144a2 c5 − 20736a3 c5 − 5184a4 c5 + 144ab2 c5 − 288a2 b2 c5 − 6336a3 b2 c5 − 19584a4 b2 c5 − 23040a5 b2 c5 − 9216a6 b2 c5 − 324c6 − 2592ac6 − 7776a2 c6 − 10368a3 c6 − 5184a4 c6 and C2 = −64a2 b4 + 24ab6 + 80a2 b6 + 96a3 b6 + 576a2 b2 c − 252ab4 c − 840a2 b4 c − 1392a3 b4 c + 88a2 b6 c + 288a3 b6 c + 336a4 b6 c − 1296a2 c2 + 792ab2 c2 + 2448a2 b2 c2 + 6624a3 b2 c2 + 72ab4 c2 − 708a2 b4 c2 − 2808a3 b4 c2 − 4416a4 b4 c2 + 120a3 b6 c2 + 384a4 b6 c2 + 432a5 b6 c2 − 648ac3 − 1296a2 c3 − 10368a3 c3 + 18b2 c3 − 312ab2 c3 + 1944a2 b2 c3 + 10512a3 b2 c3 + 16608a4 b2 c3 − 296a2 b4 c3 − 2100a3 b4 c3 − 4800a4 b4 c3 − 3888a5 b4 c3 − 81c4 − 108ac4 − 2592a2 c4 − 16848a3 c4 − 15984a4 c4 + 24ab2 c4 + 1944a2 b2 c4 + 9504a3 b2 c4 + 17376a4 b2 c4 + 10368a5 b2 c4 + 54c5 + 216ac5 − 1728a2 c5 − 9504a3 c5 − 14688a4 c5 − 6912a5 c5 . HAMILTONIAN SYSTEMS WITH A NILPOTENT SADDLE AT THE ORIGIN 39 17. Global phase portraits of system (VIII) when ac < −1 and as > 0, or cs > 0 and as < 0 By Theorems 2 and 3, since there are no singular points at infinity and the saddle at the origin have total index −2 on the Poincaré sphere, the remaining finite singular points on the Poincaré sphere have to have total index 4. Moreover, studying the finite singular points with x = 0, we have the following possibilities: (i) there are no singular points in x = 0, (ii) a center and a saddle, (iii) two centers, (iv) two saddles, (v) a cusp. In all this chapter whenever there is more than one possible phase portrait, in parenthesis we give values of the parameters where they are realized. In case (i) the total index of the singular points with x = 0 in the Poincaré sphere is −2. Hence, the remaining singular points with x ̸= 0 have to have total index 2. Hence, in view of Lemmas 5 and 6 and taking into account the symmetry of the system, they can only be two centers. The global phase portraits of the systems in case (i) are topologically equivalent to 1.3 in Figure 1 and are realized whenever b2 < 4c. In case (ii) the total index of the singular points with x = 0 in the Poincaré sphere is −2. Hence, the remaining singular points with x ̸= 0 have to have total index 4. Hence, in view of Lemma 6 and taking into account the symmetry of the system, they can only be two centers. Taking into account the possible saddle connections we have that the possible global phase portraits are topologically equivalent to the following ones of Figure 1: 1.39 (s = 1, a = −4/5, b = 3, c = 17/10); 1.40 (s = 1, a = −4/5, b = 3, c = 2. Here c is as in (19) and so there is a saddle connection); and 1.36 (s = 1, a = −4/5, b = 1/2, c = 1/20). In case (iii) the total index of the singular points with x = 0 in the Poincaré sphere is 2. Hence, the remaining singular points with x ̸= 0 have to have total index 0. Hence, in view of Lemmas 5 and 6 and taking into account the symmetry of the system, they can only be: (iii.1) no points, (iii.2) two cusps (that in view of Lemma 6 it only happens when c = c0 ), (iii.3) two centers and two saddles. Taking into account the possible saddle connections we conclude that the global phase portraits are topologically equivalent to the following ones of Figure 1: for case (iii.1) to 1.99 (s = 1, a = −4/5, b = 3, c = 3/2) and 1.3 (s = −1, a = 1, b = 3, c = −1/2); for case (iii.2) to 1.100 (s = 1, a = −2/5, b = 2, c = c0 ) and to 1.101 (s = −1, a = 1/2, b = −1/2, c = c0 ); for case (iii.3) to 1.102√(s = 1, a = −2/5, b = 2, c = 9/10), 1.103 (s = 1, a = −2/5, b = 2, c = (60 2 − 61)/27. Here c is as in (20) and so we have a saddle connection), 1.104 (s = 1, a = −2/5, b = 2, c = 4/5), and 1.105 (s = −1, a = 1/4, b = 18/25, c = −1). In case (iv) the total index of the singular points with x = 0 in the Poincaré sphere is 6. Hence, the remaining singular points with x ̸= 0 have 40 M. CORBERA AND C. VALLS to have total index 8. Hence, in view of Lemmas 5 and 6 and taking into account the symmetry of the system, they can only be four centers. Taking into account that there are no saddle connections, we conclude that the global phase portraits are topologically equivalent to 1.7 in Figure 1 which is realized for example when s = 1, a = 1, b = −3 and c = −2. In case (v) the total index of the singular points with x = 0 in the Poincaré sphere is −2. Hence the remaining singular points with x ̸= 0 have to have total index 4. Hence, in view of Lemmas 5 and 6 and taking into account the symmetry of the system, they can only be two centers. The global phase portraits of the systems in case (v) are topologically equivalent to 1.37 in Figure 1 and it is realized whenever b2 = 4c. 18. Global phase portraits of system (VIII) when ac > 0 By Theorem 2, the two nodes at infinity and the saddle at the origin have total index 1 on the Poincaré sphere. Then by Theorem 3 the remaining finite singular points on the Poincaré sphere have to have total index 0. Moreover, studying the finite singular points with x = 0 in the case ac > 0, we have the following possibilities: √ (i) there are no points, (ii) a cusp (whenever c = b2 /4 and b ̸= − 2 when s = −1), (iii) a center and a saddle p, (iv) two saddles p1 and p2 , (v) two centers; and √ (vi) a point which is the union of six hyperbolic sectors (whenever b = − 2, c = b2 /4 and s = −1). In all the chapter whenever there is more than one possible phase portrait, we give in parenthesis values of the parameters where they are realized. In case (i) the total index of the infinite singular points and the finite singular points with x = 0 in the Poincaré sphere is 1. Hence, the remaining finite singular points with x ̸= 0 have to have total index 0. Hence, in view of Lemma 6 and taking into account the symmetry of the system, they can only be: (i.1) do not exist, (i.2) two cusps, (i.3) two centers and two saddles p3 , p4 . Taking into account the fact that there are no saddle connections in this region, we conclude that the global phase portraits are topologically equivalent to the following ones of Figure 1: 1.1 in case (i.1) (s = 1, a = 1/10, b = 1/2, c = 4), to 1.106 in case (i.2) (s = −1, a = 1/10, b = 0, c = c0 ), and to 1.107 in case (i.3) (s = −1, a = 1/10, b = −1, c = 1). In case (ii) the total index of the infinite singular points and the finite singular points with x = 0 in the Poincaré sphere is 1. Hence, the remaining finite singular points with x ̸= 0 have to have total index 0. So, in view of Lemma 6 and taking into account the symmetry of the system, they can only be: (ii.1) do not exist, (ii.2) two cusps, (ii.3) two centers and two saddles √ p3 , p4 . We recall that here c = b2 /4 and that we exclude the case b = − 2 with s = −1. Taking into account that there are no saddle connections, the global phase portraits are topologically equivalent to the following ones of Figure 1: for case (ii.1) to 1.8 (s = 1, a = 1/2, b = 1); for case (ii.2) (we must HAMILTONIAN SYSTEMS WITH A NILPOTENT SADDLE AT THE ORIGIN 41 √ have s = −1 and a = 1/(8 + 6 2b + 2b2 )) to 1.108 (b = 1), 1.109 (b = −3), and 1.110 (b = −1/2); and for case (ii.3) to 1.111 (s = −1, a = 1/2, b = −2), 1.112 (s = −1, a = 1/2, b = −1), and 1.113 (s = −1, a = 3/100, b = 1). In case (iii) the total index of the infinite singular points and the finite singular points with x = 0 in the Poincaré sphere is 1. Hence, the remaining finite singular points with x ̸= 0 have to have total index 0. Hence, in view of Lemma 6 and taking into account the symmetry of the system, they can only be: (iii.1) do not exist, (iii.2) two cusps, (iii.3) two centers and two saddles p3 and p4 . In case (iii.1) we need to investigate the possible saddle connection be- tween the saddle p and the origin. Doing so, we get that they can be connected along the surface (19). Taking all the above into account and recalling that we know the region in the parameters space where condition (iii.1) holds (because we know the region in the parameter space where con- dition (iii) holds and we also know that none of the solutions with x ̸= 0 exist), we conclude that the global phase portraits are topologically equiv- alent to the following ones of Figure 1: 1.9 (s = 1, a = 1, b = 3, c = 17/10); 1.10 (s = 1, a = 1, b = 3, c = 2. Here c is the value in (19)); 1.11 (s = 1, a = 1, b = 3, c = 21/10); and 1.13 (s = 1, a = 1/4, b = 1/2, c = 1/20). In the case (iii.2) we need to investigate the possible saddle connections between the saddle p and the saddle at the origin, between the cusps and the origin, and between the cusps and the saddle p. The possible saddle connection between the saddle p and the saddle at the origin is along the surface in (19) and between the cusps p1 , p2 and the saddle p is, whenever it exist, along the surface which is a solution of the equation in (21). Finally, the possible saddle connection between the cusps and the origin is along the surface in (20). Taking into account that we know the region in the parameters space where condition (iii.2) holds (because we know the region in the parameter space where condition (iii) holds and we also know that c = c0 ), we conclude that the global phase portraits are topologically equivalent to the following ones of Figure 1: 1.25 (s = 1, a = 1, b = −6, c = c0 ); 1.114 √ (s = −1, a = √ 1/10, b = 449/2000, c = c0 ); 1.115 (s = −1, a = 1/10, b = 2(−135 + 66 5)/79, c = c0 . Here b is the solution of (19)); 1.116 (s = −1, a = 1/10, b = 23/100, c = c0 ); 1.117 (s √ = −1, a = 1/2, b = −87/100, c = c0 ); 1.118 (s = −1, a = 1/2, b = −3 2/5, c = c0 . Here b is the solution of (19)); 1.119 (s = −1, a = 5/2, b = −1.2733525026, c = c0 . Here b is the real solution of the surface in (21)); and 1.120 (s = −1, a = 5/2, b = −319/250, c = c0 ). In case (iii.3) we need to investigate the possible saddle connections. The connections between the saddles p3 , p4 and the saddle at the origin is given along the surface c = c(a, b) given in (20) and the connection between the saddle p and the origin is along the surface c = c(a, b) given in (19). Finally, 42 M. CORBERA AND C. VALLS the possible connection between the saddle p and the saddles p3 , p4 is along the surface c = c(a, b) which is any real solution (whenever it exists) of the equation (21). Taking all the above into account and recalling that we know the region in the parameters space where condition (iii.3) holds (because we know the region in the parameter space where condition (iii) holds and we also know that all the four solutions with x ̸= 0 exist), we conclude that the global phase portraits are topologically equivalent to the following ones of Figure 1: 1.14 (s = 1, √a = 1, b = −11/2, c = 1/10); 1.15 (s = 1, a = 1, b = −11/2, c = (103 − 66 2)/36. Here c is a value in (20)); 1.16 (s = 1, a = 1, b = −11/2, c = 269/1000); 1.17 (s = 1, a = 1, b = −11/2, c = 0.27064859. Here c is the value of the real solution of (21)); 1.18 (s = 1, a = 1, b = −11/2, c = 3/10); 1.121 (s = −1, a = 1/10, b = 1/10, c = 1/1000); 1.122 (s = −1, a = 1/10, b = 1/10, c = 2b2 /9. Here c is the value in (19)); 1.123 (s = −1, a = 1/10, b = 1/10, c = 23/10000); 1.124 (s = −1, a = 1/4, b = −6/5, c = 44/125); 1.125 (s = −1, a = 1/4, b = −6/5, c = 0.35787810. Here c is the value of the real solution of (21)); 1.126 (s = −1, a = 1/4, b = −6/5, c = 359/1000); and 1.127 (s = −1, a = 1/4, b = −1, c = 2/9. Here c is the value of the real solution of (19)). In case (iv) the total index of the infinite singular points and the finite singular points with x = 0 in the Poincaré sphere is −1. Hence, the remain- ing finite singular points with x ̸= 0 have to have total index 2. Taking into account Lemma 6 and the symmetry of the system they must be two centers. The saddles p1 and p2 cannot be connected one with each other and they can be connected with the saddle at the origin along the surface in (20). Taking all the above into account and recalling that we know the region in the parameters space where condition (iv) holds, we conclude that the global phase portraits are topologically equivalent to the following ones of Figure 1: 1.128 (s = 1, a = −1/2, b = −1, c = −1) and 1.129 (s = −1, a = 1, b = −3/2, c = 1/2. Here c is the value in (20) and so there is a saddle connection). In case (v) the total index of the infinite singular points and the finite singular points with x = 0 in the Poincaré sphere is 3. Hence, the remaining finite singular points with x ̸= 0 have to have total index −2. Taking into account Lemma 6 and the symmetry of the system they must be two saddles p3 and p4 . The saddles p3 and p4 are connected by symmetry. We recall that the possible saddle connection between the saddles p3 , p4 and the saddle at the origin is along the surface in (20). Taking all the above into account and recalling that we know the region in the parameters space where condition (v) holds, we conclude that the global phase portraits are topologically equivalent to the following ones of Figure 1: 1.130√ (s = 1, a = 1, b = 4/5, c = 1/25); 1.131 (s = 1, a = 1, b = 1, c = (6 2 − 7)/18. HAMILTONIAN SYSTEMS WITH A NILPOTENT SADDLE AT THE ORIGIN 43 Here c is the value in (20)); 1.132 (s = 1, a = 1, b = 1, c = 1/20); and 1.133 (s = −1, a = −1/2, b = −1/2, c = −1/10) √ In case (vi) we have s = −1, b = − 2 and c = 1/2. In this case we must have√ a > 0 (because c > 0). Among the origin and the finite singular point (0, 2) which is formed by the union of six hyperbolic sectors (3 stable and 3 unstable) we have two finite singular points with x = 0 that are two centers. The possible global phase portraits are topologically equivalent to 1.134 of Figure 1. 19. Global phase portraits of system (VIII) when ac + 1 > 0, cs < 0 and as > 0 By Theorems 2 and 3 the four nodes at infinity and the saddle at the origin have total index 6 on the Poincaré sphere, the remaining finite sin- gular points on the Poincaré sphere have to have total index −4. Moreover, studying the finite singular points with x = 0 in the case ac + 1 > 0, cs < 0 and as > 0, and taking into account Lemma 7, we have the following possi- bilities: (i) there are no singular points in x = 0 (since b2 < 4c we must have c > 0 and so s = −1), (ii) 2 √ a cusp (since b = 4c we must have c > 0 and so s = −1 and also b ̸= − 2), (iii) a center and a saddle p, (iv) two saddles p1 , p2 ; and (v) a point which is the union of six hyperbolic √ sectors (three stable and three unstable) whenever s = −1, b = − 2 and c = b2 /4. Again, in all the chapter whenever there is more than one possible phase portrait, we give in parenthesis values of the parameters where they are realized. In case (i) the total index of the infinite singular points and the finite singular points with x = 0 in the Poincaré sphere is −6. Hence, the re- maining finite singular points with x ̸= 0 have to have total index −4. In view of Lemma 6 and taking into account the symmetry of the system, they can only be two saddles. The two saddles must be connected by symmetry. We have s = −1 and we see that there are no possible saddle connections between these saddles and the saddle at the origin. Taking all the above into account we conclude that the global phase portraits are topologically equivalent to 1.135 in Figure 1. The phase portrait is realized for example when a = −1/2, b = −2, c = 3/2 and s = −1. In case (ii) the total index of the infinite singular points and the finite singular points with x = 0 in the Poincaré sphere is again 6. By Lemma 6 and taking into account the symmetry of the system the remaining finite singular points with x ̸= 0 can only be two saddles. The two saddles must be connected by symmetry. We have s = −1 and we see that there are no possible saddle connections between these saddles and the saddle at the origin (neither with the cusp and the saddle at the origin). Taking all the above into account we conclude that the global phase portraits are topolog- ically equivalent to the following ones of Figure 1: 1.136 (a = −1/2, b = 2, 44 M. CORBERA AND C. VALLS c = 1); 1.137 (a = −1/2, b = −2, c = 1); and 1.138 (a = −1/2, b = −1, c = 1/4). In case (iii) the total index of the infinite singular points and the finite singular points with x = 0 in the Poincaré sphere is again 6. Hence, the remaining finite singular points with x ̸= 0 have to have total index −4. So, in view of Lemma 6 taking also into account the symmetry of the system, they can only be two saddles p3 , p4 . The two saddles p3 , p4 must be connected by symmetry. The possible saddle connection between these saddles and the saddle at the origin is along the surface in the parameter space given in (20), and between the saddle p and the origin is along the surface c = c(a, b) given in (19). Finally, the possible connections between the saddle p and the saddles p3 , p4 is along the surface c = c(a, b) in the parameter space which is any real solution (whenever it exists) of the equation (21). Taking all the above into account and recalling that we know the region in the parameters space where condition (iii) holds, we conclude that the global phase portraits are topologically equivalent to the following ones of Figure (1): 1.139 (s = 1, a = 1/2, b√= −3/2, c = −17/10); 1.140 (s = 1, a = 1/2, b = −1/2, c = −(35 + 12 2)/54. Here c is the value in (20)); 1.141 (s = 1, a = 1/2, b = −1/2, c = −9/10); 1.142 (s = −1, a = −1/2, b = 1, c = 1/10); 1.143 (s = −1, a = −1/2, b = 1, c = 2/9. Here c is the value in (19)); 1.144 (s = −1, a = −1/2, b = 1, c = 6/25); 1.145 (s = −1, a = −1/2, b = −1/2, c = 3/100); 1.146 (s = −1, a = −1/2, b = −1/2, c = 0.05793018. Here c is the value in (21)); and 1.147 (s = −1, a = −1/2, b = −1/2, c = 3/50). In case (iv) the total index of the infinite singular points and the finite singular points with x = 0 in the Poincaré sphere is 2. Hence, the remaining finite singular points with x ̸= 0 have to have total index 0. Hence, in view of Lemma 6 and taking into account the symmetry of the system, they can only be: (iv.1) none; (iv.2) two cusps, (iv.3) two saddles (p3 , p4 ) and two centers. In case (iv.1) we see that there are no saddle connections between the saddles p1 (or p2 ) and the origin. Hence, taking also into account that we know the region in the parameters space where condition (iv.1) holds (since we know the region in the parameter space where condition (iv) holds and we also know that in this case there are no solutions with x ̸= 0), we conclude that the global phase portraits are topologically equivalent to 1.5 of Figure 1 which can be realized for instance when s = 1, a = 1/4, b = −1 and c = −1. In case (iv.2) we can prove that there are again no saddle connections between the saddles p1 (or p2 ) and the saddle at the origin. Taking also into account that we know the region in the parameters space where condition (iv.2) holds (since we know the region in the parameter space where con- dition (iv) holds and c = c0 ), we conclude that the global phase portraits HAMILTONIAN SYSTEMS WITH A NILPOTENT SADDLE AT THE ORIGIN 45 are topologically equivalent to 1.24 of Figure 1 which can be realized for example when s = 1, a = 1, b = −1 and c = c0 . In case (iv.3) the possible saddle connection between the saddles p1 (or p2 ) and the saddle at the origin is along the surface in (20), between the saddles p3 , p4 and the saddle at the origin is along the surface in (19), and between the saddles p1 , p2 and the saddles p3 , p4 is along the surface which is any real solution of the equation in (21). Taking also into account that we know the region in the parameters space where condition (iv.3) holds (because we know the region in the parameter space where condition (iv) holds and we also know that all the four solutions with x ̸= 0 exist), we conclude that the global phase portraits are topologically equivalent to the following ones of Figure 1: 1.23 (s = 1, a = 1,√b = −1/5, c = −63/100); 1.22 (s = 1 a = 1, b = −1/5, c = −(223 + 30 2)/450. Here c = c(a, b) is the real solution of the equation (19)); 1.21 (s = 1, a = 1, b = −1/5, c = −117/200); 1.20 (s = 1, a = 1, b = −1/5, c = −0.54752598. Here c = c(a, b) is the real solution of the equation (21)); and 1.19 (s = 1, a = 1, b = −1/5, c = −27/50). √ In case (v) we have s = −1, b = − 2, c = 1/2. In this √ case, since a < 0, the unique finite singular point among the origin is (0, 2) which is the formed by the union of six hyperbolic sectors (3 stable and 3 unstable). So, the global phase portraits are topologically equivalent to 1.148 of Figure (1) which is achieved for instance when a = −1/2. 20. Global phase portraits of system (VIII): final remarks We note that system (VIII) provides all phase portraits in Figure 1 except 1.34, which is not possible because from Lemma 4 the origin of U2 cannot be the union of one hyperbolic and three elliptic sectors. In the rest of this section we explain the procedure that we have used in Sections 17, 18 and 19 to give all possible phase portraits. We do it for the case (iv.3) with s = 1 in Section 19 to illustrate the procedure and the other cases have been treated in a similar way. In case (iv.3) with s = 1 in Section 19 there are four singular points at infinity so ac + 1 > 0, c < 0 and a > 0 (see Lemma 4), and there are two singular points with x = 0 (two saddles). So, from the results in Section 15 we get that √ (a, b, c) ∈ {c1 < c < 0, b < 1/ 2, a > −1}, √ where c1 is defined in Section 15. Let b∗1 = (a − 2)/( 2a). Joining the above two conditions we get that (a, b, c) ∈ S1 with { } { ∗1 } ∗1 1 S1 = a > 0, b ≤ b , c2 < c < 0 ∪ a > 0, b < b < √ , c1 < c < 0 , 2 46 M. CORBERA AND C. VALLS where again c1 , and c2 are defined in Section √ 15. The four singular points with x ̸= 0 are defined when (2ab + 3 2s)2 − 16(1 + a)(ac + 1) > 0 and x̄2± evaluated at ȳ± is positive. On the other hand, the determinant of the Jacobian matrix evaluated at x = x̄± is ( √ ) 2 2 d± = −2x̄± 2abȳ± + 3acȳ± + a + 3 2sȳ± + 3ȳ± +1 . So (±x̄± , ȳ± ) are two saddles and two centers when either √ d+ > 0 and d− < 0, or d+ < 0 and d− > 0. Let b∗2 = (2a − 1)/( 2a). Computing the set √ where (±x̄± , ȳ± ) are two saddles and two centers for a > 0 and b < 1/ 2 we get that (±x̄+ , y+ ) are saddles and (±x̄− , y− ) are centers when (a, b, c) ∈ S2 with { } 3 S2 = a > 0, b < − √ , c2 < c < c0 , 2a and (±x̄+ , y+ ) are centers and (±x̄− , y− ) are saddles when (a, b, c) ∈ S3 with { } −3 ∗2 S3 = 0 < a ≤ 1, √ < b < b , c2 < c < c0 ∪ 2a { } 3 1 a > 1, − √ < b < √ , c2 < c < c0 . 2a 2 In short, the conditions of case (iv.3) in Section 19 with s = 1 are satisfied for (a, b, c) ∈ S0 = S1 ∩ (S2 ∪ S3 ). The projection onto the (a, b)-plane of the region S0 is the set √ A = {0 < a ≤ 1, b < b∗2 } ∪ {a > 1, b < 1/ 2}. The values of c on the region S0 are delimited by one of the surfaces c = c2 , c = c1 , c = c0 , and c = 0. Moreover the projection on the (a, b)-plane of the intersection of the surfaces c = c0 with c = 0, and of c = c1 with c = 0 are the curves √ √ √ b = b∗3 = −(−3 2 ± 4 1 + a)/(2a) and b = 1/ 3, respectively. The surface c = c2 does not intersect c = 0. The projection on the (a, b)-plane of the intersection of the surfaces c = c0 with c = c1 and of c = c2 with c = c1 are the curves b = b∗2 and b = b∗1 , respectively. Finally the projection on the (a, b)-plane √ of the intersection of the surfaces c = c0 and c = c3 is the curve b = −3/( 2a). In a similar way, we compute the projection on the (a, b)-plane of the intersections between the surfaces c = c0 , c = c1 and c = c2 with the surfaces providing all possible saddle connections (given in (19), (20) and (21)); and of the intersections between these last surfaces. All the above mentioned intersections give a set of curves that divide A into several regions. For each one of these regions we pick a point (a, b), and for this value of (a, b) we find the set c ∈ I for which (a, b, c) ∈ S0 . HAMILTONIAN SYSTEMS WITH A NILPOTENT SADDLE AT THE ORIGIN 47 Moreover, we also compute the values of c on the surfaces c = 0, c = c0 , c = c1 , c = c2 and the ones given in (19), (20) and (21). These last values divide I in several regions. Finally we pick a value of c in each one of these regions and draw the phase portrait for the selected triple (a, b, c). Acknowledgements The first author is partially supported by a FEDER-MINECO Grant MTM2016-77278-P. The second author is partially supported by FCT/Portugal through UID/MAT/04459/2013. References [1] M.J. Álvarez, A. Ferragut and X. Jarque A survey on the blow up technique, Int. J. Bifurcation and Chaos 21 (2011), 3103–3118. [2] I.E. Colak, J. Llibre, and C. Valls, Hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields, Advances in Mathematics 259 (2014), 655 – 687. [3] F. S. Dias, J. Llibre, and C. Valls, Polynomial Hamiltonian systems of degree 3 with symmetric nilpotent centers, Mathematics and Computers in Simulation 144 (2018), 60–77. [4] F. Dumortier, J. Llibre and J.C. Artés, Qualitative Theory of Planar Differential Systems, Springer Verlag, New York, (2006). [5] J. Llibre and C. Valls, Global phase portraits for the Abel quadratic polynomial differential equations of second kind with Z2 -symmetries, to appear in the Bull. Cana- dian Math. Soc. [6] J. Li and Y. Liu, Global bifurcation in a perturbed cubic system with Z2 -symmetry, Acta Math. Appl. Sinica 8 (1992), 131–143. [7] J. Li and Y. Liu, Z2 -equivariant cubic system which yields 13 limit cycles, Acta Math. Appl. Sinica 30 (2014), 781–800. [8] P. Yu and M. Han, On limit cycles of the Liénard equation with Z2 symmetry, Chaos, Solitons & Fractals 31 (2007), 617–630. Facultat de Ciències i Tecnologia. Universitat de Vic - Universitat Central de Catalunya (UVic-UCC), C. de la Laura,13, 08500 Vic, Spain E-mail address: montserrat.corbera@uvic.cat Departamento de Matemàtica, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal E-mail address: cvalls@math.ist.utl.pt