This is a preprint of: “Z2 -symmetric planar polynomial Hamiltonian systems of degree 3 with nilpo- tent centers”, Fabio Scalco Dias, Jaume Llibre, Clàudia Valls, Electron. J. Differential Equations, vol. 2019(82), 1–29, 2019. Z2 -SYMMETRIC PLANAR POLYNOMIAL HAMILTONIAN SYSTEMS OF DEGREE 3 WITH NILPOTENT CENTERS FABIO SCALCO DIAS1 , JAUME LLIBRE2 AND CLAUDIA VALLS3 Abstract. We provide the normal forms and the global phase portraits in the Poincaré disk of all Z2 -symmetric planar polyno- mial Hamiltonian systems of degree 3 having a nilpotent center at the origin. 1. Introduction and statement of the results In this paper we study the global phase portrait of all Z2 -symmetric planar polynomial Hamiltonian systems of degree 3 having a nilpotent center at the origin. Let H(x, y) be a real polynomial in the variables x and y. Then a system of the form x′ = Hy y ′ = −Hx is called a polynomial Hamiltonian system. Here the prime denotes derivative with respect to the independent variable t. Poincaré in [21] defined a center for a vector field on the real plane as a singular point having a neighborhood filled with periodic orbits with the exception of the singular point. Let p ∈ R2 be a singular point of an analytic differential system in R2 , and assume that p is a center. Without loss of generality we can assume that p is at the origin of coordinates. Then after a linear change of variables and a rescaling of the time variable (if necessary), the system can be written in one of the following three forms (1) x′ = −y + P (x, y), y ′ = x + Q(x, y), (2) x′ = y + P (x, y), y ′ = Q(x, y), (3) x′ = P (x, y), y ′ = Q(x, y), 2010 Mathematics Subject Classification. 34C05, 34C07, 34C08. Key words and phrases. Polynomial Hamiltonian systems, nilpotent center, phase portrait, Poincaré compactification, Z2 -symmetric. 1 http://www.gsd.uab.cat 2 F.S. DIAS, J. LLIBRE AND C. VALLS 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 what follows a center of an analytic differential system in R2 is called linear type, nilpotent or degenerate if after an affine change of variables and a rescaling of the time it can be written as system (1), (2) or (3), respectively. Without loss of generality we can assume that a Hamiltonian system of degree three with a nilpotent center at the origin is given by (4) x′ = y + a2 x2 + 2a3 xy + 3a4 y 2 + a6 x3 + 2a7 x2 y + 3a8 xy 2 + 4a9 y 3, y ′ = −3a1 x2 − 2a2 xy − a3 y 2 − 4a5 x3 − 3a6 x2 y − 2a7 xy 2 − a8 y 3 . The classification of centers for real planar polynomial differential sys- tems started with the classification of centers for quadratic polynomial differential systems, and these results go back mainly to Dulac [10], Kapteyn [13, 14] and Bautin [2]. In [22] Vulpe provides all the global phase portraits of quadratic polynomial differential systems having a center. There are many partial results for the centers of planar poly- nomial differential systems of degree larger than two. For instance for polynomial differential systems of the form linear plus homogeneous nonlinearities of degree greater than three the centers at the origin are not characterized, but there are partial results for degree four and five for the linear type centers, see for instance Chavarriga and Giné [3, 4]. Some results for higher degree are known see for instance [12]. Recently Colak, Llibre and Valls [5, 6, 7, 8] provided the global phase portraits on the Poincaré disk of all Hamiltonian planar polynomial vector fields having only linear and cubic homogeneous terms which have a linear type center or a nilpotent center at the origin, together with their bi- furcation diagrams. More recently, Dias, Llibre and Valls [9] classified the global phase portraits of all Hamiltonian planar polynomial vector fields of degree three symmetric with respect to the x−axis having a nilpotent center at the origin. In this work we classify and provide the global phase portraits of all Z2 -symmetric planar polynomial Hamiltonian systems of degree 3 having a nilpotent center at the origin. Let X : U ⊂ R2 → R2 be the vector field associated to system (4). We define the matrix   −1 0 M= 0 1 3 Then it is Z2 –equivariant if (5) either MX (x, y) = X (−x, y), (6) or −MX (x, y) = X (x, −y), (7) or − X (x, y) = X (−x, −y), and X is Z2 –reversible if either (8) either MX (x, y) = −X (−x, y), (9) or −MX (x, y) = −X (x, −y), (10) or X (x, y) = X (−x, −y). Other classes of polynomial vector fields in R2 with a Z2 -symmetry have been studied by several authors, see for instance [15, 16, 17, 23]. Systems (4) satisfying equation (7) and (9) were studied in the article [5, 6] and [9], respectively. Systems (4) does not satisfy equations (5), (6) and (10). Hence it remains only to study the nilpotent centers of the Z2 -symmetric planar polynomial Hamiltonian systems of degree 3 satisfying (8). The classification will be done using the Poincaré compactification of polynomial vector fields, see section 2. 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 results are the following ones. Theorem 1. All planar polynomial Hamiltonian systems of degree 3 with a nilpotent center at the origin satisfying (8), after a linear change of variables and a rescaling of its independent variable t, can be written as one of the following seven systems: (I) x′ = y, y ′ = −x3 ; (II) x′ = y + x2 , y ′ = −2xy − x3 /a2 , with a 6= 0; (III) x′ = y + ax2 + y 3, y ′ = −2axy − x3 ; (IV) x′ = y + ax2 − y 3 , y ′ = −2axy − x3 ; (V) x′ = y + ax2 + y 2 + by 3 , y ′ = −2axy − x3 ; (VI) x′ = y + ax2 + by 2 − x2 y + cy 3, y ′ = −2axy − x3 + xy 2 ; (VII) x′ = y + ax2 + by 2 + x2 y + cy 3 , y ′ = −2axy − x3 − xy 2 , √ √ where a ∈ (−1/ 2, 1/ 2) and b, c ∈ R. 4 F.S. DIAS, J. LLIBRE AND C. VALLS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Figure 1. Global phase portraits of the vector fields in Theo- rem 2. 5 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 Figure 2. Continuation of Figure 1. 6 F.S. DIAS, J. LLIBRE AND C. VALLS 49 50 51 52 53 54 55 56 57 58 59 60 61 Figure 3. Continuation of Figure 1. Theorem 2. The global phase portraits of the seven families (I) − (V II) in Theorem 1 are topologically equivalent to the phase portraits of Figure 1: (a) 1 for systems (I) and (II); (b) 2 for systems (III); (c) 3 and 4 for systems (IV ); (d) 2, 3, 5 − 14 for systems (V ); (e) 2, 3, 7 − 11, 13 − 57 for systems (V I); (f) 1 − 14, 37, 38, 57 − 61 for systems (V II). 2. Preliminary results In this section we summarize the Poincaré compactification that we shall use for describing the global phase portrait of our Hamiltonian 7 systems. For more details on the Poincaré compactification see Chapter 5 of [11]. Let S2 be the sphere of points (s1 , s2 , s3 ) ∈ R3 such that s21 + s22 + s23 = 1, called the Poincaré sphere. Given a polynomial vector field X(x, y) = (x′ , y ′) = (P (x, y), Q(x, y)) in R2 of degree d (where d is the maximum of the degrees of the poly- nomials P and Q) it can be extended analytically to the Poincaré sphere by projecting each point x ∈ R2 identified with the point (x1 , x2 , 1) ∈ R3 in the Poincaré sphere using the straight line through x and the origin of R3 . The equator S 1 = {(s1 , s2 , s3 ) ∈ S2 : s3 = 0} corresponds to the infinity of R2 . In this way we obtain a vector field X̄ in S2 \ S1 . This vector field X̄ is formed by two copies of X: one on the northern hemisphere {(s1 , s2 , s3 ) ∈ S2 : s3 > 0} and another on the southern hemisphere {(s1 , s2 , s3 ) ∈ S2 : s3 < 0}. The local charts needed for doing the calculations on the Poincaré sphere are Ui = {s ∈ S2 : si > 0}, Vi = {s ∈ S2 : si < 0}, where s = (s1 , s2 , s3 ), with the corresponding local maps ϕi (s) : Ui → R2 , ψi (s) : Vi → R2 , such that ϕi (s) = −ψi (s) = (sm /si , sn /si ) = (u, v) for m < n and m, n 6= i, for i = 1, 2, 3. We extend X̄ to a vector field p(X) at the whole sphere S2 by taking p(X) = v d X̄. The expression for the corresponding vector field on S2 in the local chart U1 is given by        ′ d 1 u 1 u ′ d+1 1 u (11) u = v − uP , +Q , , v = −v P , ; v v v v v v the expression for U2 is        ′ d u 1 u 1 ′ d+1 u 1 (12) u =v P , − uQ , , v = −v Q , ; v v v v v v and the expression for U3 is u′ = P (u, v), v ′ = −Q(u, v). The ex- pressions for the charts Vi are those for the charts Ui multiplied by (−1)d−1 , for i = 1, 2, 3. Hence for studying the vector field X it is enough to study its Poincaré compactification restricted to the north- ern hemisphere plus S1 . To draw the phase portraits we consider the projection, by π(s1 , s2 , s3 ) = (s1 , s2 ), of the closed northern hemisphere into the local disk D = {(s1 , s2 ) : s21 + s22 ≤ 1}, called the Poincaré disk. Finite singular points of X are the singular points Dπ ◦ p(X) in the interior of D, and they can be studied using U3 . Infinite singular points of X are the singular points of Dπ ◦ p(X) contained in S1 . Note that if 8 F.S. DIAS, J. LLIBRE AND C. VALLS s ∈ S1 is an infinite singular point, then −s is also an infinite singular point. Hence to study the infinite singular points it suffices to look for them only at U1|v=0 and at the origin of U2 . Now we see how to characterize the global phase portraits in the Poincaré disc of all Z2 -symmetric planar polynomial Hamiltonian sys- tems of degree 3 having a nilpotent center at the origin. We recall that a separatrix of p(X) is an orbit which is either a singu- lar point, or a limit cycle, or a trajectory which lies in the boundary of a hyperbolic sector at a singular point. It was proved by Neumann [19] that the set of all separatrices of p(X) is closed. We will denote it by S(p(X)). The canonical regions of p(X) are the open connected com- ponents of D\S(p(X)). The union of S(p(X)) with one solution chosen from each canonical region will be called a separatrix configuration. We say that two separatrix configurations S(p(X)) and S(p(Y )) are topo- logically equivalent if there is an orientation preserving (or reversing) homeomorphism which maps trajectories of S(p(X)) into trajectories of S(p(Y )). The following result is due to Markus [18], Neumann [19] and Peixoto [20]. Theorem 3. The phase portraits in the Poincaré disc of the two com- pactified polynomial differential systems p(X) and p(Y ) are topologi- cally equivalent if and only if their separatrix configurations S(p(X)) and S(p(Y )) are topologically equivalent. Finally we mention without getting into too much detail an impor- tant result that classifies the finite singular points of Hamiltonian pla- nar polynomial differential systems. For a detailed definition of the (topological) index of a singular point see for instance Chapter 6 of [11], it can be computed easily using the Poincaré formula which takes into account the parabolic sector, hyperbolic sector, and elliptic sectors at a singular point, for details see page 18 of [11]. 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 elliptic sectors. We note that all the isolated singular points of a polynomial differential system satisfy the finite vectorial decomposition property, see [11]. Theorem 4 (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. From Theorem 4 the following result follows easily. 9 Corollary 5. The indices of a saddle, a center and a cusp are −1, 1 and 0, respectively. To determine the possible number and local phase portraits of the finite singular points of the systems we will use the Poincaré-Hopf Theorem for vector fields in the 2−dimensional sphere. Theorem 6. For every vector field on the sphere S2 with a finite num- ber of singular points, the sum of the indices of these singular points is 2. We note that singular points with index 0 are more difficult to detect because they do not contribute to the total index of the singular points of the vector fields on the Poincaré sphere. To overcome this difficulty we present the following proposition, but first we make a remark and give some definitions. If a singular point p of an analytic vector field X has the two real parts of the eigenvalues of DX(p) non-zero then p is hyperbolic. If the eigenvalues of DX(p) are purely imaginary, then p is either a center or a focus. If only one eigenvalue of DX(p) is 0, then p is semi-hyperbolic. The hyperbolic and semi-hyperbolic singular points are called elemen- tary. If both eigenvalues of DX(p) are 0 but DX(p) is not identically zero, then p is nilpotent. Finally, if DX(p) is identically zero then p is linearly zero. The local phase portraits of hyperbolic, semi-hyperbolic and nilpotent singular points can be studied using, for instance, Theo- rems 2.15, 2.19 and 3.5 of [11], respectively. The linearly zero singular points must be studied using the changes of variables known as blow- ups, see for instance [1] and [11]. Remark 7. Nilpotent singular points of Hamiltonian planar polyno- mial vector fields are either saddles, centers, or cusps (for more details see Theorem 3.5 of [11] and taking into account that Hamiltonian sys- tems cannot have foci). 3. Proof of Theorem 1 Without loss of generality we can assume that a Hamiltonian system of degree three with a nilpotent center at the origin is given by x′ = Hy , y ′ = −Hx , where H(x, y) = y 2 /2 + a4 x3 + a5 x2 y + a6 xy 2 + a7 y 3 + a8 x4 + a9 x3 y + a10 x2 y 2 + a11 xy 3 + a12 y 4 . 10 F.S. DIAS, J. LLIBRE AND C. VALLS Therefore we have the following Hamiltonian system (13) x′ = y + a5 x2 + 2a6 xy + 3a7 y 2 + a9 x3 + 2a10 x2 y + 3a11 xy 2 + 4a12 y 3, y ′ = −(3a4 x2 + 2a5 xy + a6 y 2 + 4a8 x3 + 3a9 x2 y + 2a10 xy 2 + a11 y 3). In order that systems (13) be Z2 -reversible as in (8) it must be invariant under (x, y, t) → (−x, y, −t) and so we have that a4 = a6 = a9 = a11 = 0. Hence systems (13) become x′ = y + a5 x2 + 3a7 y 2 + 2a10 x2 y + 4a12 y 3 , (14) y ′ = −(2a5 xy + 4a8 x3 + 2a10 xy 2 ). Since systems (14) must have a center at the origin, by Theorem 3.5 of [11] we must have 2a8 > a25 . Case 1. Assume a10 > 0. By the change of coordinates and reparametriza- tion of the time of the form (15) x → αX, y → βY, t → γτ, √ √ p with α = −1/ 2a10 , β = − a8 /a10 and γ = a10 /2a8 , systems (14) can be written as √ ′ a5 2 3a7 a8 2 4a12 a8 3 X =Y − √ X − Y + X 2Y + Y , 2 a8 a10 a210 a5 Y ′ = √ XY − X 3 − XY 2 . a8 We obtain the normal form (V II). Note that, since 2a8 > a25 , we have −a5 1 a= √ <√ . 2 a8 2 Case 2. Assume a10 < 0. By the change of coordinates and √ reparametriza- p the form as in (15) with α = −1/ −2a10 , β = tion of the time of √ a8 /a10 and γ = −a10 /2a8 , systems (14) can be written as √ ′ a5 2 3a7 a8 2 4a12 a8 3 X =Y − √ X + Y − X 2Y + Y , 2 a8 a10 a210 a5 Y ′ = √ XY − X 3 + XY 2 . a8 We obtain the normal form (V I). Case 3. Assume a10 = 0 and a7 6= 0. By the change of coordinates √ 1/4 and reparametrization of the time as in (15) with α = −1/( 6a7 a8 ), β = √ √ 1/4 √ 1/4 1/3a7 , γ = − 3a7 /( 2a8 ) if a7 > 0, and α = 1/( −6a7 a8 ), β = 11 √ √ 1/4 1/3a7 , γ = − −3a7 /( 2 a8 ) if a7 < 0, systems (14) can be written as a5 4a12 3 X′ = Y ± √ X2 + Y 2 + Y , 2 a8 9a27 a5 Y ′ = ∓ √ XY − X 3 . a8 We obtain the normal form (V ). Case 4. Assume a7 = a10 = 0 and a12 6= 0. By the change of coordinates and reparametrization p of the time as in (15) with α = −1/(2(|a12 |a8 ) ), β = −1/(2 |a12 |), γ = (|a12 |/a8 )1/4 , systems (14) 1/4 can be written as a5 X ′ = Y − √ X 2 ± Y 3, 2 a8 a5 Y ′ = √ XY − X 3 . a8 We obtain the normal forms (III) and (IV ). Case 5. Assume a7 = a10 = a12 = 0 and a5 6= 0. By the change of coordinates and reparametrization of the time as in (15) with α = β = 1/a5 and γ = 1 systems (14) can be written as X ′ = Y + X 2, 4a8 3 Y ′ = −2XY − X . a25 We obtain the normal form (II). Case 6. Assume a7 = a10 = a12 = a5 = 0. By the change of coordinates and reparametrization of the time as in (15) with α = 1, √ √ β = −2 a8 and γ = −1/(2 a8 ) systems (14) can be written as obtain the normal form (I). In short we have proved Proposition 1. 4. Proof of Theorem 2 4.1. Global phase portrait of system (I). The phase portrait of this system is topologically equivalent to the phase portrait 1 of Figure 1, and this is proved in [9]. √ √ 4.2. Global phase portrait of system (II). Since a ∈ (−1/ 2, 1/ 2), a 6= 0, we have that (0, 0) is the unique finite singular point of systems (II). 12 F.S. DIAS, J. LLIBRE AND C. VALLS We now investigate the infinite singular points of systems (II). In the local chart U1 from (11) system (II) is 1 u′ = − − 3uv − u2 v 2 , v ′ = −v 2 (uv + 1). a2 When v = 0 there are no infinite singular points on U1 . In U2 from (12) systems (II) becomes u4 uv(u2 + 2a2 v) (16) u′ = v 2 + 3vu2 + , v′ = . a2 a2 The origin is an infinite singular point of the system, whose linear part is zero. So we need to do blow-ups to describe the local dynamics at this point. We perform the directional blow-up (u, v) 7→ (u, w) with w = v/u2 and we get 1  1  (17) u′ = u3 2 u + 3uw + uw 2 , w ′ = −u3 w 2 + 4w + 2w 2 . a a Now we eliminate the common factor u3 between u′ and w ′ and we get the vector field 1 1 (18) u′ = 2 u + 3uw + uw 2, w′ = − 2 w − 4w 2 − 2w 3. a a System (18) has the origin as its unique singular point. The eigenvalues of the linear part at the origin are 1/a2 and −1/a2 , so it is a hyperbolic saddle. Going back through the changes of variables until system (16) as shown in Figure 4, we have that the global phase portrait of system (II) is topologically equivalent to the phase portrait 1 of Figure 1. w w v u u u Systems (18) Systems (17) with Systems (16) the common factor u3 Figure 4. Blow-up of the origin of U2 of system (II). 13 4.3. Global phase portrait of systems (III). Again the origin (0, 0)√is the√unique finite singular point of these systems since a ∈ (−1/ 2, 1/ 2). We will now investigate the infinite singular points of systems (III). On the local chart U1 systems (III) become u′ = −1 − 3auv − u2 v 2 − u4 , v ′ = −v(av + uv 2 + u3 ). When v = 0 there are no infinite singular points on U1 . In U2 , systems (III) can be written as u′ = 1 + v 2 + 3au2 v + u4 , v ′ = uv(2av + u2 ). Again when v = 0 there are no infinite singular points on U2 . Therefore the global phase portrait of systems (III) are topologically equivalent to the phase portrait 2 of Figure 1. 4.4. Global phase portrait of systems (IV ). When a < 0 the singular points are the origin, √ √ (0, ±1) and (± −2a(1 − 2a2 )1/4 , 1 − 2a2 ). In this case we have that (0, 1) is a center and the other singular points are saddles. When a > 0 the singular points are the origin, √ √ (0, ±1) and (± 2a(1 − 2a2 )1/4 , − 1 − 2a2 ). Here, we have that (0, −1) is a center and the other singular points are saddles. Finally, when a = 0 the finite singular points are E0 = (0, 0) and E± = (0, ±1). We will study only the singular point E+ because the study of the other singular point is analogous. Since the singular point E+ is nilpotent, using Theorem 3.5 of [11] we obtain that (0, 1) is a saddle. We will now investigate the infinite singular points of systems (IV ). On the local chart U1 systems (IV ) become u′ = −1 − 3auv + u4 − u2 v 2 , v ′ = −v(av − u3 + uv 2). The infinite singular points are P± = (±1, 0). The eigenvalues of the linear part at P+ are (4, 1). Hence it is an repelling hyperbolic node. On the other hand, the eigenvalues of the linear part of the systems at P− are (−4, −1) Hence it is an attracting hyperbolic node. In U2 systems (IV ) are given by u′ = −1 + v 2 + 3au2 v + u4 , v ′ = uv(2av + u2 ). Hence the origin is not a singular point in U2 . 14 F.S. DIAS, J. LLIBRE AND C. VALLS Taking into account the local information on the finite and infinite singular points together √ with the fact that √ the system is Hamiltonian 2 1/4 (and so the saddles (± 2a(1 − 2a ) , − 1 − 2a2 ) are connected but are not connected with the saddle at (0, −1)) we get that when a 6= 0 the global phase portrait of systems (IV ) are topologically equivalent to the phase portrait 3 of Figure 1. On the other hand, when a = 0 the global phase portrait of systems (IV ) are topologically equivalent to the phase portrait 4 of Figure 1. 4.5. Global phase portrait of systems (V ). We study the infinite singular points of systems (V ). 4.5.1. Infinite singular points. In U1 systems (V ) become u′ = −1 − 3auv − bu4 − u3 v − u2 v 2 , v ′ = −v(av + bu3 + u2 v + uv 2 ). When v = 0 the candidates for singular points of systems (V ) are the roots of the polynomial 1 + bu4 . Therefore, if b ≥ 0 there are no infinite singular points on the local chart U1 . If b < 0 the points (|b|−1/4 , 0) are repelling hyperbolic nodes and (−|b|−1/4 , 0) are attracting hyperbolic nodes. Now we study the origin of U2 of systems (V ) which in U2 write as (19) u′ = b + v + v 2 + 3au2 v + u4 , v ′ = uv(2av + u2 ). If b 6= 0 the origin is not singular. If b = 0 the origin is singular. Since the origin is nilpotent, using Theorem 3.5 of [11] together with blow-up techniques we obtain that the phase portrait of the origin consists of one hyperbolic, one elliptic and two parabolic sectors, see Figure 5. u Figure 5. Local phase portrait at the origin of system (19) for b = 0. In short we have Table 1 15 Parameters Infinite singular points in chart U1 and U2 b>0 There are no infinite singular points b<0 In the chart U1 there are one attracting node and one repelling node. In the chart U2 there are no infinite singular points b=0 In the chart U1 there are no infinite singular points. In the chart U2 the origin has a one hyperbolic, one elliptic and two parabolic sectors Table 1. Infinite singular points in the local charts U1 and U2 . 4.5.2. Finite singular points. If b = 0 the finite singular points are p p0 = (0, −1) and p± = (± 2a(1 − 2a2 ), 2a2 − 1) (whenever a > 0). Computing the eigenvalues of the Jacobian matrix at these points we get that (0, −1) is a center if a > 0 and a saddle if a < 0. If a = 0 it is nilpotent, using Theorem 3.5 in [11] we get that it is a saddle. On the other hand, the points p± exist if and only if a > 0. In this case, computing the eigenvalues of the Jacobian matrix at these points we get that they are both saddles. Assume now that b 6= 0. In this case the candidates for finite singular points of systems (V ) other than the origin are  √  −1 ± 1 − 4b E1,2 = 0, , 2b s √ √   a(1 − A) 1 − A E3,4 = ± ,− , b 2b and s  √ √  a(1 + A) 1 + A E5,6 = ± ,− b 2b where A = 4b(2a − 1) + 1. Note that E1,2 exist whenever b2 −√4c ≥ 0. 2 On the other hand, E3,4 exist whenever √ A = 0 and 0 < a < 1/ 2 with b 6= 0, or A > 0 and 0 ≤√a < 1/ 2 with b 6= 0 and E5,6 exist whenever √ A = 0 and 0 < a < 1/ 2 with √ b 6 = 0, or A > 0 and 0 ≤ a < 1/ 2 with b > 0, or A > 0, −1/ 2 < a < 0 and b < 0. Computing the eigenvalues of the Jacobian matrix at these points we get that all the points are either hyperbolic or nilpotent except the case A > 0, a = 0 and b = 1/4 in which case the point is linearly zero. In this last case, using blow-up techniques we get that the point is the union of two 16 F.S. DIAS, J. LLIBRE AND C. VALLS hyperbolic sectors. Finally, using Theorem 3.5 in [11] for the nilpotent points we get Table 2 for the finite singular √ points. We recall that when we write a√> 0 means that 0 < a < 1/ 2 and when we write a < 0 we mean −1/ 2 < a < 0. Conditions Equilibria different from (0, 0) A<0 There are no singular points b=0 a>0 p0 center, p± saddles a≤0 p0 saddle A ≥ 0, a < 0, b > 1/4 There are no singular points A = 0, a > 0 E3 , E4 cusps A > 0, a > 0 b > 1/4 E3 , E4 saddles, E5 , E6 centers A > 0, a > 0 b = 1/4 E1 cusp, E3 , E4 saddles, E5 , E6 centers A > 0, a > 0 0 < b < 1/4 E1 , E5 , E6 centers, E2 , E3 , E4 saddles A > 0, a > 0 b<0 E1 center, E2 , E3 , E4 saddles A > 0, a = 0, 0 < b < 1/4 E1 saddle, E2 center A > 0, a = 0, b < 0 E1 , E2 saddles A = 0, a = 0, b = 1/4 E1 union of 2 hyperbolic sectors A > 0, a < 0 b = 1/4 E1 cusp A > 0, a < 0 0 < b < 1/4 E1 saddle, E2 center A > 0, a < 0 b<0 E1 , E5 , E6 saddles, E2 center Table 2. Finite singular points of systems V . Now taking into account the local information on the finite and infi- nite singular points together with the fact that our system is Hamilton- ian (used whenever convenient to obtain possible saddle connections), we have Table 3 where we have listed the phase portraits that systems (V) can be topologically equivalent with, taking into account the above mentioned regions. 4.6. Global phase portrait of systems (V I). Without loss of gen- erality we can assume that b ≥ 0 (otherwise changing (x, y, a, b, c) → (−x, −y, −a, −b, c) we get the same system with b ≥ 0). We recall that the Hamiltonian of system (V I) is 1 b 1 1 c H = y 2 + ax2 y + y 3 + x4 − x2 y 2 + y 4 . 2 3 4 2 4 We now study the infinite singular points of these systems. We distinguish between the cases c ≥ 1, 0 ≤ c < 1 and c < 0. 17 Conditions Phase portraits A<0 Figure 2 b=0 a>0 Figure 5 b=0 a≤0 Figure 6 A ≥ 0, a < 0, b > 1/4 Figure 2 A = 0, a > 0 Figure 7 A > 0, a > 0 b > 1/4 Figure 8 A > 0, a > 0 b = 1/4 Figure 9 A > 0, a > 0 0 < b < 1/4 Figure 10 A > 0, a > 0 b<0 Figure 3 A > 0, a = 0, 0 < b < 1/4 Figure 11 A > 0, a = 0, b < 0 Figure 12 A = 0, a = 0, b = 1/4 Figure 13 A > 0, a < 0 b = 1/4 Figure 13 A > 0, a < 0 0 < b < 1/4 Figure 11 A > 0, a < 0 b<0 Figure 14 Table 3. Topologically equivalent phase portraits of systems V . 4.6.1. Infinite singular points. In the local chart U1 systems (V I) can be written as u′ = −1 + 2u2 − cu4 − v(3au + u2v + bu3 ), v ′ = −v(av − u + uv 2 + bu2 v + cu3 ). When v = 0, the candidates for singular points are the roots of the polynomial −1 + 2u2 − cu4, i.e. q √ q √ c(1 + 1 − c) c(1 − 1 − c) ± and ± , c 6= 0. c c In the local chart U2 systems (V I) can be written as (20) u′ = v 2 + 3au2 v + bv − 2u2 + c + u4 , v ′ = uv(2av + u2 − 1). 4.6.2. Case c ≥ 1. When c > 1 there are no infinite singular points on the local chart U1 . If c = 1 the points (±1, 0) are infinite singular points on U1 . We will study only the singular point (1, 0) because the study of the other singular point is analogous. When b 6= −3a the point (1, 0) is nilpotent. First we translate (1, 0) the origin. Applying Theorem 3.5 of [11] we obtain that the phase portrait of the singular point (1, 0) consists of one hyperbolic, one elliptic and two parabolic sectors, see Figure 5. 18 F.S. DIAS, J. LLIBRE AND C. VALLS When b = −3a, on the other hand, the point (1, 0) is an infinite singular point, whose linear part is zero, hence we need to do a blow- up to characterize the local dynamics at this point. First we translate (1, 0) to the origin. Doing the blow-up (u, v) → (u, w) with w = v/u and eliminating the common factor u we get the system (21) u′ = −4u − 4u2 + 6auw − u3 + 9au2 w − uw 2 + 3au3 w − 2u2 w 2 v ′ = 2w + uw − 4w 2 a − 3w 2au + w 3 + w 3 u. When u = 0, since 2a2 − 1 < 0, the only singular point of system (21) is the origin, and it is a saddle. The blow-up of the origin gives the same information as in the case of systems (II), hence the point (1, 0) of U1 has two hyperbolic sectors, see Figure 4. It is easy to check that the origin of chart U2 is not a singular point. 4.6.3. Case 0 ≤ c < 1. When 0 < c < 1 there are 4 infinite singular points on the local chart U1 . It is easy to see that all of them are nodes (two attracting nodes and two repelling nodes). √ On the other hand, if c = 0 there are only two singular points (± 2/2, 0) on U1 and both are nodes. In U2 systems (V I) are given by (20). Therefore if 0 < c < 1 the origin is not a singular point in U2 . When c = 0 and b 6= 0 the origin of the chart U2 is a nilpotent singular point, using Theorem 3.5 of [11] we see that locally the origin of U2 consists of one hyperbolic, one elliptic and two parabolic sectors. On the other hand, when b = c = 0 the origin of the chart U2 is an infinite singular point of the system, whose linear part is zero, hence we need to do a blow-up to characterize the local dynamics at this point. Doing the blow-up (u, v) 7→ (u, w) with w = v/u and eliminating the common factor u we get the system (22) u′ = −2u + u(u2 + 3auw + w 2 ), w ′ = w(1 − auw − w 2 ). When u = 0 the singular points of system (22) are the roots of the polynomial w(1 − w)(1 + w). Therefore we have three singular points. In this case the origin is a saddle and the other singular points are nodes. Consequently the origin of the local chart U2 has two elliptic and two parabolic sectors, see Figure 6. 4.6.4. Case c < 0. In this case the systems (V I) have only two singular points q √ c(1 − 1 − c) ± . c 19 Figure 6. Local phase portrait at the origin of system (22). These points are nodes. Finally, the origin of chart U2 is not a singular point. In short we have Tables 4 and 5. Parameters Infinite singular points in chart U1 c>1 There are no infinite singular points c = 1 b 6= −3a Two singular points with one hyperbolic one elliptic and two parabolic sectors b = −3a Two singular points with two hyperbolic sectors 01 c = 1 b 6= −3a There are no singular points b = −3a 0 0 and c = 0. The singular points, on the y-axis, of systems (V I) are the origin and (0, −1/b). The singular point (0, −1/b) is a saddle if ab < −1/2 and a center if ab > −1/2. When ab = −1/2 this point is nilpotent, using Theorem 3.5 of [11] we obtain that (0, −1/b) is a saddle. When b2 − 4c = 0 and b 6= 0. Besides the origin, system (V I) have, on the y-axis, one singular point, namely (0, −2/b). When ab 6= −1 the singular point (0, −2/b) is nilpotent, using Theorem 3.5 of [11] we obtain that (0, −2/b) is a cusp. When ab = −1, on the other hand, the point (0, −2/b) is a finite singular point, whose linear part is zero, hence we need to do a blow-up to characterize the local dynamics at this point. First we translate (0, −2/b) to the origin. The systems (V I) become 1 b b2 x′ = x2 − y 2 − x2 y + y 3 (23) b 2 4 ′ 2 3 2 y = − xy − x + xy . b We perform the directional blow-up (u, v) → (u, w) with w = v/u and we have 1 b b2 u′ = u2 − u2 w(u + w − uw 2) (24) b 2 4 3 b b2 w = −u − uw + uw (2u + w − uw 2 ). ′ 2 2 b 2 4 21 We eliminate the common factor u between u′ and w ′ , and get vector field 1 b b2 u′ = u − uw(u + w − uw 2 ) (25) b 2 4 3 b b2 w ′ = −u − w + w 2 (2u + w − uw 2). b 2 4 √ When u = 0 the singular points of system (25) are (0, 0) and (0, ± 6/b) and all of them are saddles. Consequently the local phase portrait at the point (0, −2/b) of systems (V I), with ab = −1, has six hyperbolic, see Figure 7. w w v u u u Systems (25) Systems (24) with Systems (23) the common factor u Figure 7. Local phase portrait at the point (0, −2/b) of systems (V I), with ab = −1. When b2 − 4c = 0 and b = 0, that is, b = c = 0 it is easy to check that the origin is the only finite singular on the y-axis. In short we have Table 6. Again the next step is to count the indices of the finite and infinite singular points of systems (V I) on the Poincaré sphere. First we need the following proposition. We denote 4 + a2 + 6ab + b2 c∗ = − . 4(2a2 − 1) Proposition 8. System (VI) has non-elementary points with x 6= 0 if and only if c = c∗ . They exist when 3a + b 6= 0, 1 + a2 + ab > 0 and they are two cusps. Proof. We compute the Groebner basis between x′ , y ′ and the deter- minant of the Jacobian matrix. The first component of the Groebner basis is (2a2 − 1)2 (b2 − 4c)(1 + 2ab + 4a2 c)2 (4 + a2 + 6ab + b2 − 4c + 8a2 c)y. 22 F.S. DIAS, J. LLIBRE AND C. VALLS Parameters Finite singular points on y–axis except the origin. b2 − 4c < 0 There are no singular points 2 ∗ b − 4c > 0 c > 0 a < a < a One center and one saddle a < a∗ < a a∗ ≤ a ≤ a Two saddles c < 0 a < a∗ ≤ a One center and one saddle a ≤ a < a∗ a < a < a∗ Two centers c = 0 ab ≤ −1/2 One saddle ab > −1/2 One center 2 b − 4c = 0 b 6= 0 ab 6= −1 One cusp ab = −1 One singular point with six hy- perbolic sectors b=0 There are no singular points Table 6. Finite singular points, on the √ y–axis, of sys- −b+ b2 −4c tems √ (V I). In this table a = 4c and a∗ = 2 −b− b −4c 4c . Note that if y =√ 0 the√unique singular point is the origin. Moreover, since a ∈ (−1/ 2, 1/ 2) we have that 2a2 − 1 6= 0. When c = b2 /4 the singular point which is not hyperbolic is (0, −2/b) and belongs to the y-axis. If 1 + 2ab + 4a2 c = 0, we obtain the values a and a∗ given in Table 6 and the non-hyperbolic singular point is (0, 2a) which again lyes in the y-axis. Finally, when 4 + a2 + 6ab + b2 − 4c + 8a2 c = 0, i.e c = c∗ we get that the non-hyperbolic finite singular points are, whenever they exist, p   2 (1 − 2a2 )(1 + a2 + ab) 2(2a2 − 1) ± , . 3a + b 3a + b They exist when 3a + b 6= 0 and 1 + a2 + ab ≥ 0. Note that when 2 −b2 +2 1 + a2 + ab = 0 then c∗ = 5a4(2a2 −1) and both points lye on the y-axis. Hence, in order that they exist and are outside the y-axis we must have 3a + b 6= 0 and 1 + a2 + ab > 0. Computing the eigenvalues of the Jacobian matrix at these points we see that they are nilpotent and using Theorem 3.5 in [11] we conclude that they are cusps. This concludes the proof of the proposition.  23 We will study the different global phase portraits on the Poincaré sphere as follows: c = 1 (that we will distinguish between b + 3a = 0 and b + 3a 6= 0); c = 0 (that we will distinguish between the cases b = 0 and b 6= 0); b2 = 4c with ab = −1; c = c∗ with a + 3b 6= 0 (otherwise c∗ = 1) and 1 + a2 + ab > 0 (otherwise there are no non-hyperbolic singular points outside the y-axis), will be studied in full detail. We recall that c∗ > 1 and c = b2 /4 > 0. Later, we will study the cases c > 1 (so that if c = c∗ then 1 + a2 + ab ≤ 0 and if b2 − 4c = 0 then ab 6= 1), c ∈ (0, 1) (with c 6= c∗ and if b2 − 4c = 0 then ab 6= −1) and c < 0. These last three cases will be studied using the information on the index taking into account the detailed information on the infinite singular points and on the finite singular points lying in the y-axis done above. In the following subsections we will determine the different local phase portraits in the different cases mentioned above. 4.6.6. Case c = 1 and b + 3a = 0. If a ∈ (−2/3, 0) ∪ (0, 2/3) there are no finite singular points among the origin. Taking into account the local information on the infinite singular points given in Tables 4 and 5 we conclude that the global phase portrait is topologically equivalent to 15 in Figure 1. √ √ If a ∈ (−1/ 2, −2/3) ∪ (2/3, 1/ 2) there are two finite singular points (both of them in the y-axis) which are a saddle and a center. Taking into account the local information on the infinite singular points given in Table 5 we conclude that the global phase portrait is topolog- ically equivalent to 16 in Figure 1. Finally, if a = ±2/3 among the origin, there is a cusp. Again, taking into account the local information on the infinite singular points, we conclude that the global phase portrait is topologically equivalent to 17 in Figure 1. 4.6.7. Case c = 1 and b + 3a 6= 0. If 1 + 4a2 + 2ab ≤ 0 and b > 2, among the origin there are two saddles (both on the y-axis). Taking into account the local information on the infinite singular points given in Table 5 we conclude that the global phase portrait is topologically equivalent to 18 in Figure 1 (it is attained for example when b = 3 and a = −1/2). If 1 + 4a2 + 2ab = 0 and b = 2 among the origin, there is a finite singular point in the y-axis formed by six hyperbolic sectors. Hence, the global phase portrait is topologically equivalent to 19 in Figure 1. 24 F.S. DIAS, J. LLIBRE AND C. VALLS If 1 + 4a2 + 2ab > 0 and b > 2 among the origin, there is a center and a saddle on the y-axis which are are a center and a saddle and two saddles outside the y-axis. Taking into account the information on the infinite singular points given in Table 5 together with using the first integral H to obtain possible saddle connections between the saddle at the y-axis with the other two saddles (always connected by symmetry), we conclude that the global phase portraits are topologically equivalent to 20 in Figure 1 (attained, for instance, when b = 3 and a = 1/2). If 1 + 4a2 + 2ab > 0 and b = 2 then among the origin there are a cusp in the y-axis and two saddles outside the y-axis. The global phase portrait is topologically equivalent to 21 in Figure 1 (attained for example when a = 1/2). If 1 + 4a2 + 2ab > 0 and b < 2 then among the origin there are two saddles outside the y-axis. The global phase portrait is topologically equivalent to 22 in Figure 1 (attained, for instance, when b = 1 and a = 1/2). 4.6.8. Case c = 0 and b = 0. In this case there are four saddles outside the y-axis. Taking into account the information on the infinite singular points given in Table 5 and using the first integral H to obtain all the saddle connections, we conclude that the global phase portraits are topologically equivalent to 23 in Figure 1 (it is attained, for example, when a = 1/2) and to 24 in Figure 1 (attained when a = 0). 4.6.9. Case c = 0 and b 6= 0. In this case, among the origin, there exists always the finite singular point (0, −1/b) studied in Table 6. If b < −1/(2a) then there are four saddles outside the y-axis and if b ≥ −1/(2a) there are two saddles outside the y-axis. Taking this into account, the information on the infinite singular points given in Table 5 and using the first integral H to obtain possible saddle connections, we conclude that the global phase portraits are topologically equivalent to the following ones of Figure 1 (we have included in parenthesis the possible values of a and b for which these global phase portraits are attained): 25 (a = −1/2, b = 1/2); 26 (a = −1/2, b = 2); 27 (a = −1/2, b = 1); 28 (a = −1/3, b = 1); and 29 (a = −7/30, b = 1). 4.6.10. Case b2 − 4c = 0 and ab = −1. In this case among the origin there√always exists the finite singular point (0, −2/b) (see Table 6). If b ≤ 2 or b = 2 there are no more finite singular points. √ On the other hand, if b > 2 with b 6= 2, there are two√finite singular points outside the y-axis which are: two saddles if b ∈ ( 2, 2) and two 25 centers if b > 2. Taking this into account, the information on the infinite singular points given in Table 5 and using the first integral H to obtain possible saddle connections, we conclude that the possible global phase portraits are topologically equivalent to the following ones of Figure 1 (again, we have included in parenthesis the possible values of a and b for which √ these global phase portraits are attained): 30 (b = 3/2); 31 (b = 3); 32 (b = 18/10); 19 (b = 2); 33 (b = 3). 4.6.11. Case c = c∗ with b+3a 6= 0 and 1+a2 +ab > 0. Note that c∗ > 0 and there are two cusps outside the y-axis. If 4 + a(a + 6b + 2ab2 ) > 0 there are no more finite singular points. If 4 + a(a + 6b + 2ab2 ) = 0 there is a cusp in the y-axis while if 4 + a(a + 6b + 2ab2 ) < 0 there is a saddle and a center. Taking this into account, the information on the infinite singular points given in Table 5, we conclude that the possible global phase portraits are topologically equivalent to the following ones of Figure 1 (we have included in parenthesis the possible values of a and b for which these global phase portraits √ are attained): 7 (a = −1/2, b = 2); 34 (a = −1/2, b = 3 − 1/ 2) and 35 (a = −1/2, b = 24/10). 4.6.12. Case c > 1 so that if c = c∗ then 1 + a2 + ab ≤ 0 and if b2 − 4c = 0 then ab 6= 1. We recall that in this case there are no infinite singular points. First we consider the case in which b2 − 4c < 0. Among the finite singular points we only know that the origin is a center. Hence the known singular points have total index 2 on the Poincaré sphere. By Theorem 6, the remaining finite singular points must have total index 0. Since systems (V I) have at most four finite singular points in addition to the origin outside the y-axis, and since c 6= c∗ , none of them are non-elementary (see Proposition 8), we have the following possibilities: (i) no more finite singular points, or (ii) two saddles and two centers. In case (i) the global phase portrait is topologically equivalent to 2 in Figure 1 and is attained for instance for c = 2, b = 1 and a = −1/2. In case (ii) the global phase portrait is topologically equivalent to 8 in Figure 1 and is attained for instance for c = 2, b = 1 and a = 1/2. Now we consider the case in which b2 − 4c = 0. Among the finite singular points we only know that the origin is a center and there is a cusp in the y-axis. Hence the known singular points have total index 2 on the Poincaré sphere. By Theorem 6, the remaining finite singular points must have total index 0. Since systems (V I) have at most four finite singular points in addition to the origin, and none of them are non-elementary, we have the following possibilities: (i) no more finite singular points, or (ii) two saddles and two centers. In case (i) the 26 F.S. DIAS, J. LLIBRE AND C. VALLS global phase portrait is topologically equivalent to 9 in Figure 1 and is attained for instance for c = 4, b = 4 and a = 1/2. In case (ii) the global phase portrait is topologically equivalent to 13 in Figure 1 and is attained for instance for c = 4, b = 4 and a = −1/2. Finally, we consider the case in which b2 −4c > 0. If a∗ ≤ a ≤ a there are two saddles in the y-axis. Hence the known singular points have total index -2 on the Poincaré sphere. By Theorem 6, the remaining finite singular points must have total index 4. Since the systems (V I) have at most four finite singular points in addition to the origin, and non of them are non-elementary, we have that they must be two centers. The global phase portrait is topologically equivalent to 36 in Figure 1 and it is attained, for instance, when c = 4, b = 5 and a = −1/4. If a > a∗ or a < a, there is one saddle and one center in the y-axis. So, the known singular points have total index 2 on the Poincaré sphere. By Theorem 6, the remaining finite singular points must have total index 0. Since systems (V I) have at most four finite singular points in addition to the origin, and none of them are non-elementary, we have the following possibilities: (i) no more finite singular points, or (ii) two saddles and two centers. In case (i) the global phase portraits are topologically equivalent to 11 in Figure 1 for instance when a = −1/2, b = 3, c = 2. In case (ii) using the first integral H to obtain possible saddle connections, we conclude that the possible global phase portraits are topologically equivalent to the following ones of Figure 1 (we have included in parenthesis the possible values of a, b and c for which these global phase portraits are attained): 37 (a = 1/2, b = 35/10, c = 3); 38 (a = 1/2, b = 3.66965166.., c = 3 here b is a root of the polynomial −1411344 − 256608λ + 2485107λ2 + 181412λ3 − 1427428λ4 −883552λ5 − 21744λ6 + 59200λ7 + 8512λ8); and 10 (a = 1/2, b = 4 and c = 3). 4.6.13. Case c ∈ (0, 1) with c 6= c∗ and if b2 − 4c = 0 then ab 6= 1. We recall that in this case there are four nodes in the local chart U1 . First we consider the case in which b2 − 4c < 0. Among the finite singular points we only know that the origin is a center. Hence the known singular points have total index 10 on the Poincaré sphere. By Theorem 6, the remaining finite singular points must have total index -8. Since systems (V I) have at most four finite singular points in addition to the origin outside the y-axis, and none of them are non- elementary, they must be four saddles. Using the first integral H to obtain possible saddle connections, we conclude that the possible global phase portraits are topologically equivalent to the following ones of 27 Figure 1: 39 (a = 1/2, b = 1, c = 1/2) and 40 (a = −1/3, b = 1, c = 1/2). Now assume that b2 − 4c = 0. Among the finite singular points we only know that the origin is a center and there is a cusp in the y-axis. Hence the known singular points have total index 10 on the Poincaré sphere. By Theorem 6, the remaining finite singular points must have total index -8. Since the systems (V I) have at most four finite singular points in addition to the origin outside the y-axis, and none of them are non-elementary, they must be four saddles. Using the first integral H to obtain possible saddle connections, we conclude that the possible global phase portraits are topologically √ equivalent to the√ following ones √ of Figure 1: 41 (a = −1/2, b = √2, c = 1/2); 42 (a = − 2/3, b = 2, c = 1/2) and 43 (a = −1/3, b = 2, c = 1/2). Finally, we consider the case in which b2 −4c > 0. If a∗ ≤ a ≤ a there are two saddles in the y-axis. Hence the known singular points have total index 6 on the Poincaré sphere. By Theorem 6, the remaining finite singular points must have total index -4. Since systems (V I) have at most four finite singular points in addition to the origin, and none of them are non-elementary, they must be two saddles. Using the first integral H to obtain possible saddle connections, we conclude that the possible global phase portraits are topologically equivalent to the following ones of Figure 1: 44 (a = −63/100, b = 2, c = 1/2); 45 (a = −1/3, b = 2, c = 1/2); 46 (a = −0.61646555.., b = 2, c = 1/2 here a is a root of the polynomial √ √ √ √ 54 √ 2 − 59 + (142 2 √ − 148)λ + (132 2 − 168)λ2 − (20 2 + 32)λ3 −(6 2 + 17)λ4 + (18 2 + 12)λ5 + 18λ6 ). If a < a∗ < a or a∗ < a > a there is one saddle and one center in the y-axis. So, the known singular points have total index 10 on the Poincaré sphere. By Theorem 6, the remaining finite singular points must have total index -8. Since systems (V I) have at most four finite singular points in addition to the origin, and none of them are non- elementary, they must be four saddles. Using the first integral H to obtain possible saddle connections, we conclude that the possible global phase portraits are topologically equivalent to the following ones of Figure 1: 47 (a = −7/30, b = 7/10, c = 1/10); 48 (a = −1/5, b = 7/10, c = 1/10); 49 (a = −4/15, b = 7/10, c = 1/10); 50 (a = −1/4, b = 1, c = 0.23213904.. here c is a root of the polynomial −2064384 + 28744704λ − 148561920λ2 + 344888192λ3 −34882836λ4 + 115789500λ5 + 4100625λ6); 28 F.S. DIAS, J. LLIBRE AND C. VALLS 51 (a = −1/4, b = 1, c = 6/25); 52 (a = −1/2, b = 1/2, c = 0.05730830.. here c is a root of the polynomial 333 − 12564λ + 132318λ2 − 266296λ3 + 245881λ4 −121500λ5 + 26244λ6); 53 (a = −1/2, b = 1/2, c = 3/50); 54 (a = −71/150, b = 71/50, c = 1/2). 4.6.14. Case c < 0. We note that there are two nodes in the local chart U1 and that b2 − 4c > 0. If a < a < a∗ then among the origin there are two centers. So, the known singular points have total index 10 on the Poincaré sphere. By Theorem 6, the remaining finite singular points must have total index -8. Since systems (V I) have at most four finite singular points in addition to the origin, and none of them are non-elementary, they must be four saddles. Using the first integral H to obtain possible saddle connections, we conclude that the possible global phase portraits are topologically equivalent to the following ones of Figure 1: 55 (a = −1/4, b = 3/4, c = −1); 56 (a = −1/4, b = 17/20, c = −1). If a < a or a > a∗ then among the origin there is a center and a saddle. So, the known singular points have total index 6 on the Poincaré sphere. By Theorem 6, the remaining finite singular points must have total index -4. Since systems (V I) have at most four finite singular points in addition to the origin, and none of them are non- elementary, they must be two saddles. Using the first integral H to obtain possible saddle connections, we conclude that the possible global phase portraits are topologically equivalent to the following ones of Figure 1: 3 (a = −40/100, b = 1, c = −1); 57 (a = −1/3, b = 1, c = −1); 14 (a = −48/150, b = 1, c = −1). 4.7. Global phase portrait of systems (V II). Without loss of gen- erality we can assume that b ≥ 0 (otherwise changing (x, y, a, b, c) → (−x, −y, −a, −b, c) we get the same system with b ≥ 0). We recall that the Hamiltonian of system (V II) is 1 b 1 1 c H = y 2 + ax2 y + y 3 + x4 + x2 y 2 + y 4 . 2 3 4 2 4 We now study the infinite singular points of these systems. We distinguish between the cases c > 0, c = 0 and c < 0. 29 4.7.1. Infinite singular points. In the local chart U1 systems (V II) can be written as u′ = −1 − 2u2 − cu4 − v(3au + u2 v + bu3 ), v ′ = −v(u + av + uv 2 + bu2 v + cu3 ). When v = 0, the candidates for singular points are the roots of the polynomial −1 − 2u2 − cu4 , i.e. q √ q √ c(−1 + 1 − c) c(−1 − 1 − c) ± and ± , c 6= 0. c c In the local chart U2 systems (V II) can be written as u′ = c + bv + v 2 + 3au2 v + 2u2 + u4 , v ′ = uv(2av + u2 + 1). 4.7.2. Case c ≥ 0. When c > 0 there are no infinite singular points on the local charts U1 and U2 . When c = 0 there are no infinite singular points on the local chart U1 . Hence, the origin of U2 is a singular point. When b 6= 0 the origin is nilpotent, using Theorem 3.5 of [11] together with blow-up techniques we obtain that the phase portrait of the origin consists of one hyperbolic, one elliptic and two parabolic sectors, see Figure 5. When b = 0, on the other hand, the origin is an infinite singular point, whose linear part is zero, hence we need to do a blow- up to characterize the local dynamics at this point. Doing the blow-up (u, v) → (u, w) with w = v/u and eliminating the common factor u we get the system u′ = 2u + u(u2 + 3auw + w 2 ) (26) w ′ = −w − w(auw + w 2 ). When u = 0 the only singular point of system (26) is the origin, and it is a saddle. The blow-up of the origin gives the same information as in the case of systems (II), hence the origin of U2 has two hyperbolic sectors, see Figure 4. 4.7.3. Case c < 0. In this case the systems (V II) have only two sin- gular points q √ c(−1 − 1 − c) ± . c These points are nodes. Finally, the origin of chart U2 is not a singular point. In short we have Table 7. 30 F.S. DIAS, J. LLIBRE AND C. VALLS Parameters Infinite singular points in chart U1 and U2 c>0 There are no infinite singular points c<0 In the chart U1 there are one attracting node and one repelling node. In the chart U2 there are no infinite singular points c = 0 b 6= 0 In the chart U1 there are no infinite singular points. In the chart U2 the origin has a one hyperbolic, one elliptic and two parabolic sectors b = 0 In the chart U1 there are no infinite singular points. In the chart U2 the origin has two hyperbolic sectors Table 7. Infinite singular points in the charts U1 and U2 . 4.7.4. Finite singular points. The candidates for singular points of sys- tems (V II) other than its origin are  √  −b ± b2 − 4c E1,2 = 0, , c 6= 0 2c     p p E3,4 = ± −y3 (y3 + 2a), y3 , E5,6 = ± −y5 (y5 + 2a), y5 , c 6= 1 where p (−b + 3a) ± (b − 3a)2 + 4(1 − c)(1 − 2a2 ) y3,5 = . 2(c − 1) As in previous cases we need to study only the singularities E1,2 which are the singularities that are on the y-axis. These singularities occur only when b2 − 4c ≥ 0. The calculations in this case are very similar to the systems (V ) and systems (V I). So we will only present the final result in the Table 8. Lemma 9. We recall that when c < 0 there are at most two singular points are outside the y-axis. √ √ Proof. Since c < 0 and a ∈ (−1/ 2, 1/ 2) we have that y3 y5 < 0. In order that the four points E3,4 and E5,6 exist we must have y3 (y3 +2a) < 0 and y5 (y5 + 2a) < 0. If y3 > 0 then in order that E3,4 exist we must have y3 + 2a < 0 which implies a < 0. But then since y3 y5 < 0 we have that y5 < 0 and since a < 0 then also y5 + 2a < 0. This yields y5 (y5 + 2a) > 0 and so E5,6 do not exist. 31 On the other hand if y3 < 0 then in order that E3,4 exist we must have y3 + 2a > 0 which implies a > 0. But then since y3 y5 < 0 we have that y5 > 0 and since a > 0 then also y5 + 2a > 0. This yields y5 (y5 + 2a) > 0 and so E5,6 do not exist. This concludes the proof of the lemma.  Parameters Finite singular points on y–axis except the origin. b2 − 4c < 0 There are no singular points b − 4c > 0 c > 0 a∗1 < a1 < a One center and one saddle 2 a < a∗1 < a1 a∗1 ≤ a ≤ a1 Two centers c < 0 a1 < a∗1 < a One center and one saddle a < a1 < a∗1 a1 ≤ a ≤ a∗1 Two saddles c = 0 ab ≤ 1/2 One saddle ab > 1/2 One center b2 − 4c = 0 b 6= 0 ab 6= 1 One cusp ab = 1 One singular point with 2 hyper- bolic sectors b=0 There are no singular points Table 8. Finite singular points, on the √ y–axis, of sys- ∗ b− b2 −4c tems √ (V II). In this table a1 = 4c and a1 = b+ b2 −4c 4c . Again the next step is to count the indices of the finite and infinite singular points of systems (V II) on the Poincaré sphere. First we need the following proposition. We take the notation 4 + a2 − 6ab + b2 c=− . 4(2a2 − 1) Proposition 10. System (VII) has non-elementary points with x 6= 0 if and only if c = c. They exist when b − 3a 6= 0, 1 + a2 − ab < 0 and they are two cusps. Proof. We compute the Groebner basis between x′ , y ′ and the deter- minant of the Jacobian matrix. The first component of the Groebner basis is (2a2 − 1)2 (b2 − 4c)(−1 + 2ab − 4a2 c)2 (4 + a2 − 6ab + b2 − 4c + 8a2 c)y. 32 F.S. DIAS, J. LLIBRE AND C. VALLS Note that if y √= 0 the √ unique singular point is the origin. Moreover, since a ∈ (−1/ 2, 1/ 2) we have that 2a − 1 6= 0. When c = b2 /4 the 2 singular point which is not hyperbolic is (0, −2/b) and belongs to the y-axis. If −1 + 2ab − 4a2 c = 0, that is, c = (1 − 2ab)/(4a2 ) we obtain the values a1 and a∗1 given in Table 8 and the non-hyperbolic singular point is (0, −2a) which again lyes in the y-axis. Finally, when 4 + a2 − 6ab + b2 − 4c + 8a2 c = 0, i.e c = c we get that the non-hyperbolic finite singular points are, whenever they exist, p   2 (2a2 − 1)(1 + a2 − ab) 2(2a2 − 1) ± , . 3a − b b − 3a They exist when b − 3a 6= 0 and 1 + a2 − ab ≤ 0. Note that when 2 −b2 +2 1 + a2 + ab = 0 then c = 5a 4(2a2 −1) and both points lye on the y-axis. Hence, in order that they exist and are outside the y-axis we must have b − 3a 6= 0 and 1 + a2 − ab < 0. Computing the eigenvalues of the Jacobian matrix at these points we see that they are nilpotent and using Theorem 3.5 in [11] we conclude that they are cusps. This concludes the proof of the proposition.  We will study the different global phase portraits on the Poincaré sphere as follows: c = 0 (that we will distinguish between the cases b = 0 and b 6= 0); b2 = 4c with ab = 1; c = c with b − 3a 6= 0 and 1 + a2 − ab < 0 (otherwise there are no non-hyperbolic singular points outside the y-axis), will be studied in full detail. We recall that c > 0 and c = b2 /4 > 0. Later, we will study the cases c > 0 (so that if c = c then either b − 3a = 0 or 1 + a2 − ab ≥ 0 and if b2 − 4c = 0 then ab 6= 1) and c < 0. These last two cases will be studied using the information on the index taking into account the detailed information on the infinite singular points and on the finite singular points lying in the y-axis done above. In the following subsections we will determine the different local phase portraits in the different cases mentioned above. 4.7.5. Case c = 0. If b = 0 the only finite singular point is the origin. Taking into account the information on the infinite singular points given in Table 7 we get that the global phase portrait is topologically equivalent to 1 in Figure 1. On the other hand, if b 6= 0 then among the origin there always exist the finite singular point (0, −1/b) studied in Table 8. If a ≤ 1/(2b) 33 there are no finite singular points outside the y-axis, and if a > 1/(2b) there are two saddles outside the y-axis. Taking this into account, the information on the infinite singular points given in Table 7 and the fact that the saddles outside the y-axis are connected by symmetry, we conclude that the global phase portraits are topologically equivalent to the following ones of Figure 1 (we have included in parenthesis the possible values of a and b for which these global phase portraits are attained): 6 (a = 1/2, b = 1) and 5 (a = 1/2, b = 2). 4.7.6. Case b2 − 4c = 0 and ab = 1. In this case among the origin there only exists the finite singular point (0, −2/b) which is formed by two hyperbolic sectors. Since there are no infinite singular points the global phaser portrait is topologically equivalent to 13 of Figure 1. 4.7.7. Case c = c with b − 3a 6= 0 and 1 + a2 − ab < 0. Note that c > 0 and so there are two cusps outside the y-axis. Moreover there are no infinite singular points. If 4 + a(a − 6b + 2ab2 ) > 0 there are no more finite singular points. If 4 + a(a − 6b + 2ab2 ) = 0 there is a cusp in the y-axis while if 4 + a(a − 6b + 2ab2 ) < 0 there is a saddle and a center. Taking this into account, the information on the infinite singular points and the symmetry, we conclude that the global phase portraits are topologically equivalent to the ones of Figures √ 7 (when a = 1/2 and b = 4), 58 (when a = 1/2 and b = 3 + 1/ 2); and 59 (when a = 7/10 and b = 2.22438). 4.7.8. Case c > 0 so that if c = c then either b−3a = 0 or 1+a2 −ab ≥ 0 and if b2 − 4c = 0 then ab 6= 1 or c 6= c. We recall that in this case there are no infinite singular points. First we consider the case in which b2 − 4c < 0. Among the finite singular points we only know that the origin is a center. Hence the known singular points have total index 2 on the Poincaré sphere. By Theorem 6, the remaining finite singular points must have total index 0. Since systems (V II) have at most four finite singular points in addition to the origin outside the y-axis (see Proposition 10), none of them are non-elementary. Note that in this case we have exactly the same infinite and finite singular points as in the case c > 1 for system (VI) and so we get the same topologically equivalent global phase portraits: 2 (when a = 1/2 and b = c = 1); and 8 (when a = 1/2, b = 5 and c = 7). Now we consider the case in which b2 − 4c = 0. Among the finite singular points we only know that the origin is a center and there is a cusp in the y-axis. Hence the known singular points have total index 2 on the Poincaré sphere. By Theorem 6, the remaining finite singular 34 F.S. DIAS, J. LLIBRE AND C. VALLS points must have total index 0. Since systems (V II) have at most four finite singular points in addition to the origin, and none of them are non-elementary, we have exactly the same infinite and finite singular points as in the case c > 1 for system (VI) and so we get the same topologically equivalent global phase portraits: 9 (when a = 1/2 and b = c = 4); and 13 (when a = 1/2, b = 3 and c = 9/4). Finally, we consider the case in which b2 − 4c > 0. If a > a∗1 or a < a1 , there is one saddle and one center in the y-axis. So, the known singular points have total index 2 on the Poincaré sphere. By Theorem 6, the remaining finite singular points must have total index 0. Since systems (V II) have at most four finite singular points in addition to the origin, and none of them are non-elementary, we have the same finite singular points as in the case c > 1 for system (VI) and so we get the same topologically equivalent global phase portraits: 11 (when a = −7/10, b = 1 and c = 1/5); 60 (when a = 1/2, b = 2.9 and c = 2); 10 (when a = 1/5, b = 7 and c = 11.5); 38 (when a = 1/5, b = 7 and c = 11.99722984.. here c is the root of the polynomial −3333909112734375 + 10575417764843750λ − 11748870462834375λ2 +2482717923483000λ3 − 192679718006000λ4 + 5112897914880λ5 +429981696λ6); and 37 (when a = 1/5, b = 7 and c = 12). On the other hand, if a∗1 ≤ a ≤ a1 there are two centers in the y-axis. Hence the known singular points have total index 6 on the Poincaré sphere. By Theorem 6, the remaining finite singular points must have total index -4. Since the systems (V II) have at most four finite singular points in addition to the origin, and non of them are non-elementary, we have that they must be two saddles. Hence, the possible global phase portraits are topologically equivalent to 61 of Figure 1 and it is attained for example when a = 1/2, b = 3 and c = 1. 4.7.9. Case c < 0. We note that there are two nodes in the local chart U1 and that b2 − 4c > 0. If a1 ≤ a ≤ a∗1 then among the origin there are two saddles. So, the known singular points have total index 2 on the Poincaré sphere. By Theorem 6, the remaining finite singular points must have total index 0. In view of Lemma 9 and Proposition 10 we know that system (V II) has at most two finite singular points outside the y-axis, and none of them are non-elementary. By symmetry they cannot exist. Using the first integral H to obtain possible saddle connections, we conclude that the possible global phase portraits are topologically equivalent to the following ones of Figure 1: 12 (a = 0, b = 1, c = −1); 4 (a = 0, b = 0, c = −1). 35 On the other hand, if a < a∗1 < a1 or a∗1 < a1 < a then among the origin there is a center and a saddle. So, the known singular points have total index 6 on the Poincaré sphere. By Theorem 6, the remaining finite singular points must have total index -4. Since systems (V II) have at most four finite singular points in addition to the origin, and none of them are non-elementary, we have the same finite singular points as in the case c < 0 for system (VI) and so we get the same topologically equivalent global phase portraits: 3 (when a = −1/2, b = 1 and c = −25); 57 (when a = −1/2, b = 1 and c = −17.86185044.. here c is the root of the polynomial −7200 + 106560λ − 619776λ2 + 1436966λ3 − 691358λ4 +73872λ5 + 6561λ6 ); and 14 (when a = −1/2, b = 1 and c = −14). Acknowledgements The first author is partially supported by FAPEMIG Grants APQ- 01086-15 and APQ-01158-17. The second author is partially supported by the Ministerio de Economı́a, Industria y Competitividad, Agen- cia Estatal de Investigacin grant MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017 SGR 1617, and the European project Dynamics-H2020-MSCA-RISE-2017- 777911. The third author is partially supported by FCT/Portugal through UID/MAT/04459/2013. References [1] V.I. Arnol’d and Y. Il’Yashenko, Ordinary differential equations, pages 1– 148. Enciclopedia Math. Sci. 1, Dynamical Systems I. Springer, Berlin Heildel- berg, New York, 1988. [2] N.N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Mat. Sb. 30 (1952), 181–196; Mer. Math. Soc. Transl. 100 (1954) 119. [3] J. Chavarriga and J. Giné, Integrability of a linear center perturbed by a fourth degree homogeneous polynomial, Publ. Mat. 40 (1996), 21–39. [4] J. Chavarriga and J. Giné, Integrability of a linear center perturbed by a fifth degree homogeneous polynomial, Publ. Mat. 41 (1997), 335–356. [5] I.E. Colak, J. Llibre, and C. Valls, Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields, J. Differential Equa- tions 257 (2014), 1623 – 1661. [6] I.E. Colak, J. Llibre, and C. Valls, Hamiltonian nilpotent centers of lin- ear plus cubic homogeneous polynomial vector fields, Advances in Mathematics 259 (2014), 655 – 687. 36 F.S. DIAS, J. LLIBRE AND C. VALLS [7] I.E. Colak, J. Llibre, and C. Valls, Bifurcation diagrams for Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields, J. of Differential Equations 258 (2015), 846–879. [8] I.E. Colak, J. Llibre, and C. Valls, Bifurcation diagrams for Hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields, J. of Differential Equations 262 (2016), 5518–5533. [9] 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. [10] H. Dulac, Détermination et integration d’une certaine classe d”equations différentielle ayant par point singulier un centre, Bull. Sci. Math. 32 (1908), 230–252. [11] F. Dumortier, J. Llibre and J.C. Artés, Qualitative Theory of Planar Differential Systems, Springer Verlag, New York, 2006. [12] J. Giné, The center problem for a linear center perturbed by homogeneous polynomials. Acta Math. Sin. (Engl. Ser.) 22 (2006), 1613–1620. [13] W. Kapteyn, On the midpoints of integral curves of differential equations of the first Degree, Nederl. Akad. Wetensch. Verslag Afd. Natuurk. Konikl. Nederland (1911), 1446–1457 (in Dutch). [14] W. Kapteyn, New investigations on the midpoints of integrals of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag Afd. Natuurk. 20 (1912), 1354–1365; Nederl. Akad. Wetensch. Verslag Afd. Natuurk. 21 (1913) 27–33 (in Dutch). [15] J. Llibre and C. Valls, Global phase portraits for the Abel quadratic polyno- mial differential equations of second kind with Z2 -symmetries, Bull. Canadian Math. Soc. 61 (2018), 149–165. [16] J. Li and Y. Liu, Global bifurcation in a perturbed cubic system with Z2 - symmetry, Acta Math. Appl. Sinica 8 (1992), 131–143. [17] J. Li and Y. Liu, Z2 -equivariant cubic system which yields 13 limit cycles, Acta Math. Appl. Sinica 30 (2014), 781–800. [18] L. Markus, Global structure of ordinary differential equations in the plane: Trans. Amer. Math Soc. 76 (1954), 127–148. [19] D. A. Neumann, Classification of continuous flows on 2–manifolds, Proc. Amer. Math. Soc. 48 (1975), 73–81. [20] M.M. Peixoto, Dynamical Systems. Proccedings of a Symposium held at the University of Bahia, 389–420, Acad. Press, New York, 1973. [21] H. Poincaré, Mémoire sur les courbes définies par les équations différentielles, Oeuvreus de Henri Poincaré, vol. 1, Gauthier Villars, Paris, 1951, pp. 95–114. [22] N.I. Vulpe, Affine-invariant conditions for topological distinction of quadratic systems in the presence of a center, (Russian) Differential Equations 19 (1983), 371–379. [23] P. Yu and M. Han, On limit cycles of the Liénard equation with Z2 symme- try, Chaos, Solitons & Fractals 31 (2007), 617–630. 37 1 Instituto de Matemática e Computação, Universidade Federal de Itajubá, Avenida BPS 1303, Pinheirinho, CEP 37.500–903, Itajubá, MG, Brazil. E-mail address: scalco@unifei.edu.br 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain. E-mail address: jllibre@mat.uab.cat 3 Departamento de Matemàtica, Instituto Superior Técnico, Univer- sidade de Lisboa, 1049-001 Lisboa, Portugal. E-mail address: cvalls@math.ist.utl.pt