This is a preprint of: “Z2-equivariant linear type bi-center cubic polynomial Hamiltonian vector fields”, Ting Chen, Shimin Li, Jaume Llibre, J. Differential Equations, vol. 269, 832–861, 2020. DOI: [10.1016/j.jde.2019.12.020] Z2 -EQUIVARIANT LINEAR TYPE BI-CENTER CUBIC POLYNOMIAL HAMILTONIAN VECTOR FIELDS TING CHEN1 , SHIMIN LI1,∗ AND JAUME LLIBRE2 Abstract. We study the global dynamical behavior of Z2 -equivariant cubic Hamiltonian vector fields with a linear type bi-center at (±1, 0). By using a series of symbolic computation tools, we obtain all possible phase portraits of these Z2 -equivariant Hamiltonian systems. 1. Introduction and statement of the main results In the qualitative theory of planar vector fields the analysis when an equilibrium point p is either a center or a focus is one of the classical problems. We called it the center problem or the center-focus problem. Poincaré [30] and Dulac [14] defined that an equilibrium point p of a vector field in R2 is a center if it has a neighborhood U filled with periodic orbits with the unique exception of this equilibrium point. And a center p is global if R2 \ {p} is filled with periodic orbits. For polynomial vector fields the equilibrium point p in R2 is called elementary if at least one of the eigenvalues of the Jacobian matrix (or the linear part) at this point is nonzero, otherwise it is called non-elementary. We say that a non- elementary equilibrium point is nilpotent if its two eigenvalues are zero but its Jacobian matrix is not identically zero, and degenerate when the Jacobian matrix is identically zero. Correspondingly, if a polynomial vector field in R2 has a center at the origin, after making a time rescaling and a linear change of variables, this system can be written as ∑n n ∑ (1) ẋ = Xk (x, y), ẏ = Yk (x, y), k=1 k=1 where {˙} := d/dt, t is the time,   (0, 0) ,  (2) (X1 (x, y), Y1 (x, y)) = (y, 0) ,   (−y, x) , Xk (x, y) and Yk (x, y) are real homogeneous polynomials of degree k (k ≥ 2) in x and y. Then the center at the origin is called a degenerate center, a nilpotent center and a linear type center when (X1 , Y1 ) satisfies the right hand side of (2) respectively. Some algorithms for the characterization of these three type centers have been studied in [5, 11, 17, 26, 31]. 2010 Mathematics Subject Classification. Primary: 34C07, 34C08. Key words and phrases. Z2 -equivariant, Hamiltonian, Phase portrait, Bi-center, Infinity. 1 2 TING CHEN, SHIMIN LI AND JAUME LLIBRE As we know, a great deal of work has been done for the classifications of centers for the vector fields in R2 during more than a century. The center problem of quadratic polynomial systems started with the results of Bautin [4], Kapteyn [18, 19] and Zoladek [38]. The classification of centers for some cubic polynomial vector fields were studied in [1, 8, 27, 34] and references therein. But it is very difficult to provide the complete characterization of the centers for all cubic polynomial vector fields, and it is unsolved problem. There has been partial results about the centers of the quartic and the quintic polynomial vector fields (see [6, 7]). Recently, the phase portraits of some cubic Hamiltonian vector fields with a nilpotent center or a linear type center at the origin were classified in [12, 13, 16]. However there are very few families of the vector fields about the existence of multiple centers. In this work we study a class of planar Zq -equivariant polynomial vector fields, whose phase portraits are unchanged by a rotation of 2π/q radians around one point, where q is a positive integer. The study of Zq -equivariant poly- nomial systems is very importance for the Hilbert’s 16th problem, for more details see [20, 21, 22]. Yu and Han [36, 37], Liu and Huang [24] studied the limit cycle bifurcations for several classes of Z2 -equivariant cubic polynomial systems. On the other hand we say that Zq -equivariant polynomial systems have a bi-center at the equilibrium points p1 and p2 if these two equilibrium points are both centers. We will call it the bi-center problem. The authors of [23, 32] studied the bi-center problem for some Z2 -equivariant polynomial systems. Chen et al. [9] investigated the local critical bifurcation in a class of Z2 -equivariant cubic polynomial systems with a bi-center. But they did not provide the corresponding phase portraits of these vector fields. To characterize the phase portraits of a planar polynomial vector field in the Poincaré disc, we need to know the local phase portraits of the infinite and fi- nite equilibria of these systems using the Poincaré compactification. And we will introduce a series of methods to characterize the equilibria of the polynomial sys- tems. Then we apply these methods to study the global dynamics of a class of Z2 - equivariant cubic Hamiltonian vector fields with a linear type bi-center at (±1, 0), described by ẋ = − (1 + a21 )y + a21 x2 y − 3b03 xy 2 + a03 y 3 , (3) ẏ = − x + x3 − a21 xy 2 + b03 y 3 . In Section 3 we will show how to obtain the Z2 -equivariant cubic Hamiltonian systems (3). For determining the global flow of these polynomial vector fields we need to characterize their separatrices. The separatrices in the Poincaré disc D2 include all the finite equilibria, all the infinite orbits, all the limit cycles, and all the separatrices of the hyperbolic sectors of the infinite and finite equilibria. If we denote by Σ the closed set formed by all the separatrices of the vector field, the components of D2 \Σ are called the canonical regions. The separatrix squeleton of a polynomial vector field is formed by Σ union one orbit for each canonical region. The phase portraits of two polynomial vector fields in the Poincaré disc are topological equivalent if and only if the two corresponding separatrix squeletons are topological equivalent, see for more details [29]. We denote by s the number of separatrices, by r the number of canonical regions. For more details about the Poincaré compactification Z2 -EQUIVARIANT LINEAR TYPE BI-CENTER CUBIC HAMILTONIAN SYSTEMS 3 see [10, 15]. And in the phase portraits of the following theorem a line of the form “ · · · · · · ” denotes that it is filled of equilibria of a Hamiltonian system (3). We state the main result of this paper. Theorem 1.1. The phase portraits in the Poincaré disc of the Z2 -equivariant cubic Hamiltonian systems (3) with a linear type bi-center at (±1, 0) are topologically equivalent to the 36 phase portraits described in Figures 1 and 2. In the next section we will give some preliminaries about the Poincaré compacti- fication for analyzing the equilibrium points. In Section 3 we provide how to obtain the Z2 -equivariant cubic Hamiltonian systems (3). In Section 4 we characterize the global phase portraits of systems (3) in the Poincaré disc, that is we prove Theorem 1.1. 2. Preliminaries In this section we introduce some preliminaries about the Poincaré compactifi- cation and the equilibria. These are described in Chapter 5 of [15]. This compact- ification is very helpful for studying the phase portraits of the planar polynomial vector fields. We denote by R2 the plane in R3 defined with the points s = (s1 , s2 , s3 ) = (x1 , x2 , 1). Let P (R2 ) be the set of the polynomial vector fields X in R2 of the form (4) (x˙1 , x˙2 ) = (X(x1 , x2 ), Y (x1 , x2 )). The degree d of X equals max{deg X, deg Y }. Let S2 = {s ∈ R3 : s21 + s22 + s23 = 1} be the Poincaré sphere. The plane R2 is tangent to S2 at the point (0, 0, 1). We denote by S1 = {s ∈ S2 : s3 = 0} the equator of S2 which corresponds to the infinity of R2 . For the Poincaré compactification p(X ) of X one close disc D2 = {s ∈ R2 : s21 + s22 ≤ 1} is called the Poincaré disc. The equilibria in D2 lying on S1 , i.e. the boundary of the disc D2 , are the corresponding infinite equilibria of X . The equilibria in the interior of the Poincaré disc, i.e. on D2 \ S1 , are the corresponding finite equilibria of X . Then we provide the expressions of the Poincaré compactification for the polynomial differential systems. In order to draw the phase portraits of the infinite equilibria of S1 we only need to consider the local charts Ui = {s ∈ D2 : si > 0} and Vi = {s ∈ D2 : si < 0}, for i = 1, 2, with the corresponding diffeomorphisms (5) φi : Ui → R2 , ψi : Vi → R2 , defined by sm sn φi (s) = −ψi (s) = ( , ) = (u, v) si si for m < n and m, n ̸= i. Thus (u, v) will play different roles in these two local charts. The expression of p(X ) in U1 is (6) (u̇, v̇) = (v d (Y1 − uX1 ), −v d+1 X1 ), where X1 = X(1/v, u/v) and Y1 = Y (1/v, u/v). In U2 it is (7) (u̇, v̇) = (v d (X2 − uY2 ), −v d+1 Y2 ), 4 TING CHEN, SHIMIN LI AND JAUME LLIBRE 1.1 s= , r=4; 1.2 s=21, r=7; 1.3 s=29, r=9; 1.4 s=29, r=8; 1.5 s=33, r=11; 1.6 s=33, r=10; 1.7 s=35, r=12; 1.8 s=37, r=11; 1.9 s=37, r=10; 1.10 s=39, r=12; 1.11 s=13, r=3; 1.12 s=9, r=3; 1.13 s=6, r=3; 1.14 s=10, r=4; 1.15 s=18, r=6; 1.16 s=21, r=7; 1.17 s=25, r=7; 1.18 s=41, r=14; 1.19 s=39, r=12; 1.20 s=41, r=14; 1.21 s=23, r=8; 1.22 s=27, r=9; 1.23 s=27, r=8; 1.24 s=37, r=14. Figure 1. The topological phase portraits 1.1-1.24 in the Poincaré disc of Theorem 1.1. Z2 -EQUIVARIANT LINEAR TYPE BI-CENTER CUBIC HAMILTONIAN SYSTEMS 5 1.25 s=35, r=12; 1.26 s=37, r=14; 1.27 s=21, r=6; 1.28 s=31, r=11; 1.29 s=31, r=10; 1.30 s=33, r=12; 1.31 s=13, r=6; 1.32 s=19, r=8; 1.33 s=14, r=7; 1.34 s=10, r=5; 1.35 s=23, r=9; 1.36 s=18, r=8. Figure 2. The topological phase portraits 1.25-1.36 in the Poincaré disc of Theorem 1.1. where X2 = X(u/v, 1/v) and Y2 = Y (u/v, 1/v). It is sufficient to study the equi- libria at U1 |v=0 and at the origin of U2 for studying all the infinite equilibria of X . The expressions of p(X ) in Vi , i = 1, 2 are those in Ui multiplied by (−1)d−1 . Note that if s ∈ S1 is an infinite equilibrium point, −s ∈ S1 is also an infinite equilibrium point. Hence the number of infinite equilibria is even. Next we introduce the topological index of an equilibrium point, which is one useful tool to determine the type of the equilibria. Here we will present two impor- tant theorems, the Index Poincaré Formula and the Poincaré–Hopf Theorem, for more details see Chapter 6 of [15]. Theorem 2.1. We denote by p an isolated equilibrium point with the finite sectorial decomposition property. Let q, h and e be the number of parabolic, hyperbolic and elliptic sectors of p, respectively. Then the topological index of the equilibrium point p equals 1 + (e − h)/2. Corollary 2.2. The topological indices of a center, a cusp, a saddle and a node equal 1, 0, −1 and 1, respectively. Theorem 2.3. For any continuous vector field on the sphere S2 with finitely many equilibria, the sum of their topological indices is 2. 6 TING CHEN, SHIMIN LI AND JAUME LLIBRE 3. Obtaining systems (3) Without loss of generality the Z2 -equivariant cubic Hamiltonian systems here considered are obtained from the differential systems ẋ =a10 x + a01 y + a03 y 3 + a12 xy 2 + a21 x2 y + a30 x3 = X(x, y), (8) ẏ =b10 x + b01 y + b03 y 3 + b12 xy 2 + b21 x2 y + b30 x3 = Y (x, y). Suppose (±1, 0) are two equilibrium points of systems (8). Then we have (9) a10 = −a30 , b10 = −b30 , and the Jacobian matrix of systems (8) at (±1, 0) is given by ( ) 2a30 a01 + a21 (10) J= . 2b30 b01 + b21 Proposition 3.1. Assume a10 = −a30 and b10 = −b30 . The equilibrium points (±1, 0) are two linear type centers of Hamiltonian systems (8) if and only if a12 + 3b03 = a21 + b12 = 3a30 + b21 = b01 − a30 = 0, and 2a230 + (a01 + a21 )b30 < 0. Proof. If the equilibrium points (±1, 0) are two linear type centers of systems (8), we have the trace and the determinant of (10) at (±1, 0) are zero and positive, respectively. Thus, we obtain b01 = −b21 − 2a30 and 2a230 + (a01 + a21 )b30 < 0. Next, let H(x, y) be the quartic Hamiltonian of systems (8). To find this Hamil- tonian, we integrate X(x, y) of (8) with respect to y and obtain ∫ H1 (x, y) =f (x) + X(x, y)dy (11) a01 2 a21 2 2 a12 3 a03 4 =f (x) − a30 xy + a30 x3 y + y + x y + xy + y , 2 2 3 4 for some real polynomials f . And we integrate Y (x, y) of (8) with respect to x and obtain ∫ H2 (x, y) =g(y) − Y (x, y)dx b30 2 b30 4 b21 3 (12) =g(y) + x − x + (2a30 + b21 )xy − x y 2 4 3 b12 2 2 − x y − b03 xy 3 , 2 for some real polynomials g. Equating H1 (x, y) to H2 (x, y) we obtain (13) a12 + 3b03 = a21 + b12 = 3a30 + b21 = 0, f (x) = b30 x2 /2−b30 x4 /4 and g(y) = a01 y 2 /2+a03 y 4 /4. Therefore the statement of this proposition holds if and only if (13), b01 = a30 and 2a230 +(a01 +a21 )b30 < 0.  Then we take a30 = 0, a01 + a21 = −1 and b30 = 1, so that systems (8) become the Z2 -equivariant cubic Hamiltonian systems (3) having the Hamiltonian 1 2 1 4 1 + a21 2 a03 4 a21 2 2 (14) H(x, y) = x − x − y + y + x y − b03 xy 3 . 2 4 2 4 2 Z2 -EQUIVARIANT LINEAR TYPE BI-CENTER CUBIC HAMILTONIAN SYSTEMS 7 Doing the transformation √ 2x1 √ (15) x= + 1, y = y1 , t= 2τ 2 systems (3) can be written as √ √ a21 2 3 2b03 x′1= − y1 + 2a21 x1 y1 + 2 x1 y1 − 3b03 y1 − x1 y12 + a03 y13 , (16) √ 2 √ 2 √ ′ 3 2 2 1 3 2a21 2 a21 2 2b03 3 y1 =x1 + x1 + x1 − y1 − x1 y1 + y1 , 4 4 2 2 2 where {′ } := d/dτ . Note that the equilibrium point (1, 0) of systems (3) corresponds to the origin of systems (16), which is a center-focus type point. Since the flow of Hamiltonian systems preserves the area, any finite equilibrium point of Hamiltonian systems must be either a center, or a union of an even number of hyperbolic sectors. In particular, any finite nilpotent equilibrium point of Hamil- tonian systems is either a cusp, a center or a saddle, for more details see Theorem 3.5 of [15]. Then the equilibrium points (±1, 0) of systems (3) cannot be foci, so that they are two linear type centers. In addition we do not need to determine the limit cycles of systems (3) because the existence of Hamiltonian prevents the existence of limit cycles. 4. Global phase portraits of systems (3) Note that we can assume b03 ≥ 0 because the Z2 -equivariant cubic Hamiltonian systems (3) are invariant under the following transformation (x, y, t, a21 , a03 , b03 ) → (−x, y, −t, a21 , a03 , −b03 ). Thus we study the phase portraits of systems (3) in two different cases b03 = 0 and b03 > 0. 4.1. Phase portraits when b03 = 0. Systems (3) are ẋ = − (1 + a21 )y + a21 x2 y + a03 y 3 , (17) ẏ = − x + x3 − a21 xy 2 . It is easy to see that systems (17) are symmetric with respect to the x-axis and the y-axis. We calculate the resultant of ẋ and ẏ with respect to x, and obtain (18) Res[ẋ, ẏ, x] = y 3 (−1 − a21 + a03 y 2 )(−1 + a03 y 2 + a221 y 2 )2 . For (18) to be identically zero, it needs a21 = −1 and a03 = 0. Then systems (17) have non-isolated equilibrium points. Thus, we consider the following two subcases a203 + (a21 + 1)2 = 0 and a203 + (a21 + 1)2 ̸= 0. 4.1.1. Subcase a203 + (a21 + 1)2 = 0. Then systems (17) become ẋ = − x2 y, (19) ẏ = − x + x3 + xy 2 . 8 TING CHEN, SHIMIN LI AND JAUME LLIBRE Figure 3. The phase portrait of system (20). And the y-axis is filled of equilibrium points of system (19). By the rescaling time dt = xdτ system (19) is given by (20) x′ = −xy, y ′ = −1 + x2 + y 2 , which has one invariant straight line x = 0. Thus, we can analyze the phase portrait of (20) for studying the associated Hamiltonian system (19). First we study the infinite equilibria of (20) using the Poincaré compactification. In the local chart U1 system (20) becomes (21) u′ = 1 + 2u2 − v 2 , v ′ = uv. When v = 0 system (21) has no equilibrium points. In the local chart U2 system (20) writes (22) u′ = −u(2 + u2 − v 2 ), v ′ = v(−1 − u2 + v 2 ). The origin of (22) is an equilibrium point, which is an attracting node. Hence system (20) has only two infinite equilibrium points, the origins of the local charts U2 and V2 . Since the degree of system (20) is 2, the flow in a neighborhood of the origin of V2 has the opposite sense with respect to the flow in a neighborhood of the origin of U2 . This completes the local phase portraits at the infinite equilibria of system (20). Next we study the finite equilibria of system (20). System (20) has two equilibria p3,4 = (0, ±1) different from the bi-center at p1,2 = (±1, 0). Since the eigenvalues of the Jacobian matrix of (20) at p3 are λ1 = 2 and λ2 = −1 the equilibrium point p3 is a saddle. Similarly we have that p4 is also a saddle. Since the y-axis is an invariant straight line and system (19) has no more finite equilibrium points, the saddles p3,4 must be on the boundary of the region formed by all the period annuluses surrounding the center p1 , this region is called the period annulus of the center p1 . In short, by the symmetries with respect the x-axis and the y-axis we obtain that the phase portrait of system (20) is the one shown in Figure 3. Going back to the Z2 -equivariant cubic Hamiltonian system (19), the y-axis is filled of equilibrium points of system (19), and due to the rescaling time dt = xdτ the orbits of system (19) has the opposite orientation with respect to the ones of system (20) in the region x < 0. Consequently, the global phase portrait of system (19) in Poincaré disc is topologically equivalent to the one 1.1 of Figure 1. Z2 -EQUIVARIANT LINEAR TYPE BI-CENTER CUBIC HAMILTONIAN SYSTEMS 9 4.1.2. Subcase a203 + (a21 + 1)2 ̸= 0. Using (6) systems (17) in the local chart U1 become u̇ =1 − 2a21 u2 − v 2 − a03 u4 + (1 + a21 )u2 v 2 , (23) v̇ =uv(−a21 − a03 u2 + v 2 + a21 v 2 ). Then the infinite equilibria (u, 0) of systems (23) in the local chart U1 must satisfy 1 − 2a21 u2 − a03 u4 = 0. Moreover the Jacobian matrix of systems (23) at (u, 0) is ( ) −4u(a21 + a03 u2 ) 0 (24) M= . 0 −u(a21 + a03 u2 ) Proposition 4.1. For systems (23) with a203 + (a21 + 1)2 ̸= 0 the following state- ments hold. (I) If a03 < −a221 and a21 > 0, or a03 ≤ 0 and a21 ≤ 0 systems (23) have no infinite equilibrium points. (II) If a03 > 0 systems (23) have two infinite equilibrium points A1,2 , which are an attracting node and a repelling node, respectively. (III) If a√03 = 0 and a21 > 0 systems (23) have two infinite equilibrium points (± 2a21 /2a21 , 0), which are an attracting node and a repelling node, re- spectively. (IV) If 0 > a03 > −a221 and a21 > 0 systems (23) have four infinite equilib- rium points A1,2,3,4 , where A1,4 are two attracting nodes and A2,3 are two repelling nodes. (V) If a03 = −a221 and a21 > 0 systems (23) have two infinite equilibrium points √ (± a21 /a21 , 0), whose local phase portraits both consist of two hyperbolic sectors, see Figure 4(c). Here we have √ √ A1,2 = (± (−a21 + a03 + a221 )/a03 , 0) and √ √ A3,4 = (± (−a21 − a03 + a221 )/a03 , 0). Proof. Statements (I) to (IV) are easy to prove. They just need to check the roots of 1 − 2a21 u2 − a03 u4 = 0 and compute the traces and the determinants of (24) at these equilibrium points. Now we consider case (V). If a03 = −a221 and a21 > 0 the infinite equilibrium √ points (± a21 /a21 , 0) are degenerate. In order to study the local phase portraits of these two infinite equilibria we need to do some changes of variables called blow-up’s, for a detailed introduction to the blow-up’s changes see for instance [2]. Due to the symmetries of the systems we only need to analyze the local phase at the equilibrium √ √ point ( a21 /a21 , 0). Doing the change of variables (u, v) → (u + a21 /a21 , v) the √ equilibrium point ( a21 /a21 , 0) is translated to the origin of coordinates, and we obtain the systems √ √ u̇ =1 − 2a21 (u + a21 /a21 )2 + a221 (u + a21 /a21 )4 − v 2 √ √ (25) + (u + a21 /a21 )2 v 2 + a21 (u + a21 /a21 )2 v 2 , √ √ v̇ =v(u + a21 /a21 )(−a21 + a221 (u + a21 /a21 )2 + v 2 + a21 v 2 ). 10 TING CHEN, SHIMIN LI AND JAUME LLIBRE w w v u u u (a) (b) (c) Figure 4. Blow-up at the origin of (25). (a) Systems (27); (b) Systems (26); (c) Systems (25). We do the blow-up (u, v) → (u, w) with w = v/u for analyzing the local phase portrait at the origin of systems (25). And we obtain the systems 1 2 2 √ √ √ u̇ = u (4a21 + w2 + 4a221 a21 u + 2 a21 w2 u + 2a21 a21 w2 u + a321 u2 a21 (26) + a21 w2 u2 + a221 w2 u2 ), 1 √ √ √ ẇ = − wu2 (2a221 + w2 + a221 a21 u + a21 w2 u + a21 a21 w2 u). a21 By the rescaling of the time dt = a21 dτ /u2 we have √ √ √ u′ =4a221 + w2 + 4a221 a21 u + 2 a21 w2 u + 2a21 a21 w2 u + a321 u2 (27) + a21 w2 u2 + a221 w2 u2 , √ √ √ w′ = − w(2a221 + w2 + a221 a21 u + a21 w2 u + a21 a21 w2 u). Systems (27) have no equilibrium points for u = 0. Thus we study the local phase portrait in the neighborhood of u = 0. Note that u′ |w=0 = 4a221 + O(u) showing that the flow is increasing in the direction u, and w′ |u=0 = −2a221 w+O(w2 ) showing that the flow is decreasing and increasing in the direction w when w > 0 and w < 0 (see Figure 4(a)). Going back through the changes of variables until systems (25), as in the pass from systems (27) to systems (26) does not change the orientation of flows of systems (26) except that u = 0 is filled with equilibrium points, see Figure 4(b). And taking into account the behavior of the flows on the √ √ axes u̇|v=0 = a21 u2 (2 + a21 u)2 > 0 and v̇|u=0 = a21 (1 + a21 )v 3 /a21 , we obtain that the phase portrait at the origin of systems (25) consists of two hyperbolic sectors (see Figure 4(c)).  We proceed to study when the origin of the local chart U2 is an infinite equilib- rium. Systems (17) in the local chart U2 acquire the form (28) u̇ = a03 + 2a21 u2 − u4 − (1 + a21 )v 2 + u2 v 2 , v̇ = uv(a21 − u2 + v 2 ). Proposition 4.2. For systems (28) with a203 + (a21 + 1)2 ̸= 0 the following state- ments hold. (I) If a03 ̸= 0 systems (28) have no infinite equilibrium points. (II) If a03 = 0 systems (28) have an infinite equilibrium point at the origin. Furthermore: Z2 -EQUIVARIANT LINEAR TYPE BI-CENTER CUBIC HAMILTONIAN SYSTEMS 11 w w u u (a) v v v u u u (b) Figure 5. Blow-up at the origin of (28) with a21 ̸= 0. (a) Systems (29) with a21 > 0 and a21 < 0 respectively; (b) Systems (28) with a21 > 0, −1 < a21 < 0 and a21 < −1 respectively. (II.1) If a21 > 0 or a21 < −1 the local phase portrait at the origin of systems (28) consists of two parabolic sectors and two elliptic sectors, see the first or the third picture in Figure 5(b), respectively. (II.2) If −1 < a21 ≤ 0 the local phase portrait at the origin of systems (28) consists of two hyperbolic sectors, see the second picture in Figure 5(b). Proof. Obviously the origin of systems (28) is not an infinite equilibrium point when a03 ̸= 0. If a03 = 0 the origin of systems (28) is a degenerate equilibrium point. In order to study its local phase portrait we apply the directional blow-up (u, v) → (u, w) with w = v/u and the rescaling of the time dt = dτ /u2 , then systems (28) become (29) u′ = 2a21 − u2 − w2 − a21 w2 + u2 w2 , w′ = w(−a21 + w2 + a21 w2 ). When a21 ̸= 0 systems (29) have no equilibrium points for u = 0, therefore we study the local phase portrait in a neighborhood of the origin. Since u′ |w=0 = 2a21 −u2 and w′ |u=0 = −a21 w+O(w3 ), the local phase portrait at the origin of (29) is given by Figure 5(a) when a21 > 0 and a21 < 0. Going back through the change of variables until systems (28) and taking into account the flow of systems (28) on the axes we have u̇|u=0 = −(1 + a21 )v 2 , u̇|v=0 = 2a21 u2 + O(u4 ) and v̇|u=0 = 0. Then we obtain that the local phase portrait at the origin of the chart U2 consists of two parabolic and two elliptic sectors when a21 > 0 or a21 < −1, and it consists of two hyperbolic sectors when −1 < a21 < 0, see Figures 5(b). When a21 = 0 systems (29) have one equilibrium point (0, 0) for u = 0, which is degenerate. Again we apply the blow-up (u, w) → (u, V ) with V = w/u and obtain (30) u′ = −1 − V 2 + u2 V 2 , V ′ = V (1 + V 2 + uV 2 − u2 V 2 ), 12 TING CHEN, SHIMIN LI AND JAUME LLIBRE V w v u u u (a) (b) (c) Figure 6. Blow-up at the origin of (28) with a21 = 0. (a) System (30); (b) Systems (29); (c) Systems (28). a03 I IV II 0 a21 −1 V III VII VI Figure 7. The bifurcation diagram of the infinite equilibrium points of systems (17) with a203 + (a21 + 1)2 ̸= 0. by a rescaling of the time dt = dτ /u2 . Taking u = 0 system (30) has no equilibrium points. We reconstruct the flow through these changes of variables to obtain the local phase portrait of the origin of systems (28), and obtain that it consists of two hyperbolic sectors, see Figure 6.  In summary the polynomial differential systems (17) have at most eight equilib- rium points at infinity. And the bifurcation diagram of these infinite equilibria is shown in Figure 7, and the results for the infinite equilibria are listed in Table 1. Here we denote by C, S, N, H and E the center, the saddle, the node, the equilib- rium point whose local phase portrait consists of two hyperbolic sectors, and the equilibrium point whose local phase portrait consists of two parabolic sectors and two elliptic sectors, respectively. Note that the degree of systems (17) is odd, the flow in the local chart Vi (i = 1, 2) has the same sense as in the local chart Ui . Now we shall study the finite equilibria of systems (17). These systems have the third finite equilibrium point p3 = (0, 0), which is a center or a saddle when a21 < −1 or a21 > −1, respectively. If a21 = −1 the origin p3 is nilpotent. Doing Z2 -EQUIVARIANT LINEAR TYPE BI-CENTER CUBIC HAMILTONIAN SYSTEMS 13 Table 1. The results for the infinite equilibrium points of systems (17) with a203 + (a21 + 1)2 ̸= 0. Regions Isolated infinite equilibria Conditions I 2 attracting N, 2 repelling N a03 > 0 II 2 attracting N, 2 repelling N, 2 E a03 = 0, a21 > 0 III 2H a03 = 0, −1 < a21 ≤ 0 IV 2E a03 = 0, a21 < −1 V 4 attracting N, 4 repelling N a21 > 0, −a221 < a03 < 0 VI 4H a21 > 0, a03 = −a221 VII 0 a03 < 0, a21 < 0 or a21 > 0, a03 < −a221 the change of time t → −t systems (17) become ẋ =(x2 − a03 y 2 )y = H(x, y), (31) ẏ =x(1 − x2 − y 2 ) = x + G(x, y). Therefore x = 0 is one solution of x + G(x, y) = 0 in a neighborhood of the origin of systems (31). Then we have H(0, y) = −a03 y 3 and ( ) ∂H ∂G + = 0. ∂x ∂y (0,y) From Theorem 3.5 of [15] we have that p3 is a center or a saddle when a03 > 0 or a03 < 0, respectively. In addition if (1 + a21 )a03 > √ 0 systems (17) have the fourth and the fifth finite equilibrium points p4,5 = (0, ± (1 + a21 )a03 /a03 ), which are two centers or two saddles when a03 + a21 + a221 > 0 or a03 + a21 + a221 < 0, respectively. And they are nilpotent when a03 + a21 + a221 = 0. We again apply Theorem 3.5 of [15] for these nilpotent equilibria and obtain that they are two saddles. Now we consider the equilibrium points which are not on the x-axis or on the y-axis. Solving −1 − a21 + a21 x2 + a03 y 2 = 0 and −1 + x2 − a21 y 2 = 0, we obtain four possible equilibria ( √ √ ) (a03 + a221 )(a03 + a21 + a221 ) a03 + a221 (xi , yi ) = ± ,± , a03 + a221 a03 + a221 for i = 6, 7, 8, 9. Therefore systems (17) have four equilibrium points pi = (xi , yi ) (i = 6, 7, 8, 9) different from the above five ones when a03 + a221 > 0 and a03 + a21 + a221 > 0, otherwise they have no these equilibrium points. And they are saddles because the traces and the determinants of the Jacobian matrix of (17) at pi = (xi , yi ) (i = 6, 7, 8, 9) are zero and −4(a03 + a21 + a221 )/(a03 + a221 ) < 0, respectively. Proposition 4.3. The phase portraits of systems (17) are topologically equivalent to 14 TING CHEN, SHIMIN LI AND JAUME LLIBRE a21 = −1 a03 a03 = 1 − a221 1.17 1.3 1.4 1.16 1.2 1.1 1.5 1.6 1.7 a21 1.15 0 1 1.8 1.10 1.12 1.9 1.13 1.14 1.11 a03 = −a221 a03 = −a21 − a221 Figure 8. The bifurcation diagram of the phase portraits of sys- tems (17). 1.1 if a03 = 0 and a21 = −1; 1.2 if a03 > 0, a21 > −1 and a03 + a21 + a221 ≤ 0; 1.3 if a03 > 0, a21 > −1, a03 + a21 + a221 > 0 and a03 ̸= 1 − a221 ; 1.4 if a03 > 0, a21 > −1, a03 + a21 + a221 > 0 and a03 = 1 − a221 ; 1.5 if a03 = 0 and 0 < a21 < 1; 1.6 if a03 = 0 and a21 = 1; 1.7 if a03 = 0 and a21 > 1; 1.8 if 0 > a03 > −a221 , a21 > 0 and a03 < 1 − a221 ; 1.9 if 0 > a03 > −a221 , a21 > 0 and a03 = 1 − a221 ; 1.10 if 0 > a03 > 1 − a221 and a21 > 0; 1.11 if a03 = −a221 and a21 > 0; 1.12 if a03 = 0 and 0 ≥ a21 > −1; 1.13 if a03 < −a221 and a21 > 0, or a03 < 0 and −1 ≤ a21 ≤ 0; 1.14 if a03 ≤ −a21 − a221 and a21 < −1; 1.15 if 0 > a03 > −a21 − a221 and a21 < −1; 1.16 if a03 = 0 and a21 < −1; 1.17 if a03 > 0 and a21 ≤ −1. Moreover the corresponding bifurcation diagram is shown in Figure 8. Proof. We just prove the phase portraits from 1.3 to 1.10 because the proof of the others follows from the previous analysis and Propositions 4.1 and 4.2 directly. 1) If a21 > −1 and a03 + a21 + a221 > 0 we obtain that systems (17) have four infinite equilibrium points on the Poincaré sphere by Propositions 4.1 and 4.2. The equilibrium points A1,2 are in U1 , and the corresponding points in V1 , which are all nodes. And there are nine finite equilibrium points, where p1,2,4,5 are four centers and p3,6,7,8,9 are five saddles. Hence systems (17) have two bi-centers, one at p1,2 and one at p4,5 . We have that the local phase portraits at these equilibria in the Poincaré disc are shown in Figure 9(a). And the Hamiltonian H(x, y) has the same value at saddles p6,7,8,9 because it is an even function. Z2 -EQUIVARIANT LINEAR TYPE BI-CENTER CUBIC HAMILTONIAN SYSTEMS 15 (a) (b) (c) Figure 9. The local phase portraits at all equilibria of systems (17). (a) If a21 > −1 and a03 + a21 + a221 > 0; (b) If a03 = 0 and a21 > 0; (c) If 0 > a03 > −a221 and a21 > 0. Figure 10. A eight–figure loop. Since the finite equilibrium points are either centers or saddles, there must be at least one saddle on the boundary of the period annulus of each center. If only saddle p3 is on the boundary of period annulus of p1 , taking into account the symmetry, p3 must be also on the boundary of period annulus of p2 . Thus it creates one eight– figure loop, see Figure 10. Then there are two saddles on the boundary of period annulus of p4 . We suppose that these two saddles are p6,8 . So the other two saddles p7,9 are on the boundary of period annulus of p5 . Furthermore the phase portraits of systems (17) in this case are topologically equivalent to the 1.3 of Figure 1. If two saddles p6 and p7 are on the boundary of period annulus of p1 . Taking into account the symmetry the saddles p8 and p9 must be on the boundary of period annulus of p2 . Then the saddle p3 must be on the boundary of the period annulus of the period annulus of p4 , by the symmetry, p3 is also on the boundary of period annulus of p5 . And it creates one eight–figure loop. Hence in this case the global phase portrait in the Poincaré disc is shown in Figure 11. Since the separatrix skeletons of the phase portrait in Figure 11 and of the phase portrait 1.3 are equivalent, these two phase portraits are topologically equivalent. From the phase portrait 1.3 to the phase portrait of Figure 11, and the continuity of the phase portraits with respect to the parameters there is a bifurcation curve a03 = f (a21 ) which corresponds to the phase portrait 1.4 of Figure 1. In order to obtain this phase portrait it requires to find a set of explicit values a03 and a21 , which is not an easy task. But in this case we can find this bifurcation curve by computing the invariant straight lines of systems (17). In fact, assuming that x = ky is an invariant line of systems (17), we have ẋ − k ẏ = (−1 − a21 + a21 k 2 y 2 + a03 y 2 )y − k 2 y(−1 + k 2 y 2 − a21 y 2 ) (32) = (−1 − a21 + k 2 )y + (a03 + 2a21 k 2 − k 4 )y 3 . 16 TING CHEN, SHIMIN LI AND JAUME LLIBRE Figure 11. The phase portrait of systems (17) with a21 > −1, a03 + a21 + a221 > 0 and a03 > 1 − a221 . √ Hence when (32) is identically zero we have k = ± 1 + a21 and a√03 = 1 − a221 . Furthermore the saddles p6,9 are on the invariant straight√line x = 1 + a21 y and the saddles p7,8 are on the invariant straight line x = −√ 1 + a21 y. And we have that the infinite √ equilibrium points A1,2 √ = (y/x, 0) = (± 1 + a21 /(1 + a21 ), 0) are on the lines x = 1 + a21 y and x = − 1 + a21 y, respectively. Then there are three saddles p3 , p6 and p7 on the boundary of period annulus of p1 . In fact we obtain that H(x, y) is zero at these three saddles. By the symmetry we obtain the phase portrait 1.4 of Figure 1. 2) Assume that a03 = √ 0 and a21 > 0, by Propositions 4.1 and 4.2, the infinite equilibrium points are (± 2a21 /2a21 , 0) in U1 and the origin in U2 , and also the corresponding points in V1,2 . The four infinite equilibrium points in U1 and V1 are nodes. The phase portraits at the origins of U2 and V2 both consist of two parabolic and two elliptic sectors. And systems (17) have seven finite equilibrium points, where p1,2 are two centers and p3,6,7,8,9 are five saddles. We have that the local phase portraits at these equilibria in Poincaré disc are shown in Figure 9(b). If only the saddle p3 is on the boundary of period annulus of p1 , taking into account the symmetry, this saddle is also on the boundary of period annulus of p2 . It creates one eight–figure loop. Then one attracting separatrix and one repelling separatrix of saddle p6 connect with one repelling separatrix of the saddle p7 and one attracting separatrix of the saddle p8 , respectively. By the symmetry we have that the phase portrait in the Poincaré disc is topologically equivalent to 1.5 of Figure 1, which is realized when a21 = 0.5. In a similar way to the analysis of the phase√ portrait 1.4, we have that if a03 = 1 − a221 , i.e. a21 = 1 the straight lines x = ± 2y are two invariant lines of√systems (17). √ Then the saddles p6,9 and p7,8 are on the invariant lines √ x = 2y and x = − 2y, respectively. And the infinite equilibrium points (± 2/2, 0) are also on these two invariant lines, respectively. Hence the saddles p3 , p6 and p7 are on the boundary of the period annulus of p1 . Taking into account the symmetry the phase portraits of systems (17) are topologically equivalent to 1.6 of Figure 1. If the two saddles p6 and p7 are on the boundary of the period annulus of p1 . The separatrices of the saddle p3 must go to the infinity. Then this phase portrait is topologically equivalent to 1.7 of Figure 1. When a21 = 2 this phase portrait is realized. Moreover we have that the phase portraits of systems (17) in the Poincaré disc are topologically equivalent to 1.5, 1.6 and 1.7 of Figure 1 when 0 < a21 < 1, a21 = 1 and a21 > 1, respectively. Z2 -EQUIVARIANT LINEAR TYPE BI-CENTER CUBIC HAMILTONIAN SYSTEMS 17 3) Assume that 0 > a03 > −a221 and a21 > 0, the infinite equilibrium points are A1,2,3,4 in U1 , and also the corresponding points in V1 . And all of them are nodes. In addition there are seven finite equilibrium points for systems (17), where p1,2 are two centers and p3,6,7,8,9 are five saddles. We obtain that the local phase portraits at these equilibria in the Poincaré disc are shown in Figure 9(c). Similarly to the above cases now the phase portraits of systems (17) are topologically equivalent to 1.8, 1.9 and 1.10 of Figure 1 when a03 < 1 − a221 , a03 = 1 − a221 and a03 > 1 − a221 , respectively. Further, all results for the isolated equilibrium points in the above cases are listed in Table 2.  Table 2. The isolated equilibrium points corresponding to the topological phase portraits 1.1-1.17. Phase portraits Isolated finite equilibria Isolated infinite equilibria 1.1 2C 0 1.2 2 C, 3 S 2 attracting N, 2 repelling N 1.3,1.4 4 C, 5 S 2 attracting N, 2 repelling N 1.5,1.6,1.7 2 C, 5 S 2 attracting N, 2 repelling N, 2 E 1.8,1.9,1.10 2 C, 5 S 4 attracting N, 4 repelling N 1.11 2 C, 1 S 4H 1.12 2 C, 1 S 2H 1.13 2 C, 1 S 0 1.14 3 C, 2 S 0 1.15 5 C, 4 S 0 1.16 3 C, 4 S 2E 1.17 3 C, 4 S 2 attracting N, 2 repelling N 4.2. Phase portraits when b03 > 0. Without loss of generality we can assume that b03 = 1 and systems (3) become ẋ = − (1 + a21 )y + a21 x2 y − 3xy 2 + a03 y 3 , (33) ẏ = − x + x3 − a21 xy 2 + y 3 . Then we obtain that systems (33) have finitely many equilibrium points. Indeed, by computing the resultant of ẋ and ẏ with respect to x we have the polynomial (34) (−1 − a21 )y 3 + F1 y 5 + F2 y 7 + F3 y 9 , where F1 =9 + 3a03 + 9a21 + 2a03 a21 + 2a221 + 2a321 , F2 = − 9a03 − 3a203 + 18a21 − a203 a21 + 15a221 − 4a03 a221 (35) − 2a03 a321 − a421 − a521 , F3 =27 + a303 − 18a03 a21 + 2a203 a221 − 2a321 + a03 a421 . 18 TING CHEN, SHIMIN LI AND JAUME LLIBRE For polynomial (34) to be identically zero, it needs a21 = −1 so that the coefficient of y 3 is zero. Then (34) simplifies to (36) a03 y 5 + (−3 − 11a03 − 2a203 )y 7 + (29 + 19a03 + 2a203 + a303 )y 9 . It is easy to see that the coefficients of y 5 and y 7 in polynomial (36) cannot be zero at the same time. Hence polynomial (34) is not identically zero. On the other hand, the Jacobian matrix of systems (33) at one finite equilibrium point (x, y) is ( ) (2a21 x − 3y)y M1 (37) , M2 −(2a21 x − 3y)y where M1 = −1 − a21 + a21 x2 − 6xy + 3a03 y 2 , M2 = −1 + 3x2 − a21 y 2 . We claim that there is no degenerate finite equilibrium points for systems (33). Indeed, we obtain that (2a21 x − 3y)y and M2 have no common solutions, because the Gröbner basis for the polynomials ẋ, ẏ, (2a21 x − 3y)y and M2 is 1. We again calculate the Gröbner basis for ẋ, ẏ, (2a21 x − 3y)y and M1 , then we obtain four polynomials 1 + a21 , (9 + 4a03 )y 2 , y 2 (−12 + 31y 2 ) and 8x + 31y 3 . It means that if a21 = −1 the origin p3 is nilpotent, and√ if a21 = −1 √ and a03 = −9/4 systems (33) have two nilpotent equilibrium points (3 3/31, ±2 3/31) additional √ to the√origin. By Theorem 3.5 of [15] we have that the origin is a saddle and (3 3/31, ±2 3/31) are two cusps. Moreover the other finite equilibrium points must be saddles or centers. Now we study the infinite equilibrium points of systems (33). In U1 systems (33) become u̇ =1 − 2a21 u2 + 4u3 − a03 u4 − v 2 + u2 v 2 + a21 u2 v 2 , (38) ẏ =uv(−a21 + 3u − a03 u2 + v 2 + a21 v 2 ). And the Jacobian matrix of systems (38) at (u, 0) is ( ) −4u(a21 − 3u + a03 u2 ) 0 (39) . 0 −u(a21 − 3u + a03 u2 ) Obviously the origin is not an equilibrium point of systems (38). And if a03 = 0 systems (38) have at most three equilibrium points on v = 0, and if a03 ̸= 0 they have at most four equilibrium points on v = 0. From (39) it is easy to compute that these infinite equilibrium points in U1 must be nodes or degenerate equilibria. If (40) f1 (u) = 1 − 2a21 u2 + 4u3 − a03 u4 , f2 (u) = a21 − 3u + a03 u2 . we compute the Gröbner basis for f1 (u) and f2 (u), then we obtain one polynomial (41) 12a03 − a203 a21 − a221 − a03 a321 + (3a21 − 4a203 )u. Hence systems (38) have at most one degenerate infinite equilibrium point for the value of u which vanishes the previous polynomial. In order to provide a complete analysis of the global phase portraits of (33), it is necessary to study separately in a03 = 0 and a03 ̸= 0. Z2 -EQUIVARIANT LINEAR TYPE BI-CENTER CUBIC HAMILTONIAN SYSTEMS 19 4.2.1. Subcase a03 = 0. By the above analysis we obtain √ that there are √ one and three infinite equilibrium points in U1 when a21 < 3/ 3 2 and√a21 > 3/ 3 2, respec- tively. And they are all nodes. On the √other hand, if a21 √ = 3/ 3 2 systems (36) have 3 3 two infinite equilibrium points (−1/(2 2), 0) and (1/ 2, 0), which are a repelling node and a degenerate equilibrium point.√ We apply the blow-up technique and obtain that the local phase portrait of (1/ 3 2, 0) consists of two hyperbolic sectors, as it is shown in Figure 6(c). We proceed to analyze the origin of the local chart U2 . Systems (33) in this chart acquire the form u̇ = − 4u + 2a21 u2 − u4 − v 2 − a21 v 2 + u2 v 2 , (42) v̇ =v(−1 + a21 u − u3 + uv 2 ). Obviously the origin of systems (42) is an equilibrium point, which is an attracting node. We will now study the finite equilibria. We get that the third equilibrium point of systems (33) is p3 = (0, 0), which is a center or a saddle when a21 < −1 or a21 > −1, respectively. If a21 = −1, p3 is a nilpotent saddle. Next we consider the finite equilibria different from p1,2,3 . We compute the Gröbner basis for −1 − a21 + a21 x2 − 3xy and −x + x3 − a21 xy 2 + y 3 , and obtain four polynomials, the following two (4a21 + 3a221 )x + (3 + 12a21 + 9a221 + a321 + a421 )y + 9xy 2 (43) + (16a221 + 15a321 − a521 − a621 )y 3 + (27a21 − 2a421 )y 5 = 0 and f (y) =1 + a21 + (−9 − 9a21 − 2a221 − 2a321 )y 2 + (−18a21 − 15a221 (44) + a421 + a521 )y 4 + (−27 + 2a321 )y 6 = 0, will be enough for our analysis. Then we obtain that there are at most six finite equilibrium points additional to p1,2,3 . But the explicit expressions of these finite equilibria and their eigenvalues in terms of parameter a21 are complicated, it is hard to study their existence and their types. Hence we need to present more algebraic tools for solving this problem. Now we provide an useful way to obtain the information about the number of real roots of an arbitrary polynomial f (x). We need to calculate the discriminant sequence {S1 , S2 , · · · , Sn } of f (x) (for more details see [10, 35]) and determine its sign list [sign(S1 ), sign(S2 ), · · · , sign(Sn )], where the sign function is   −1, if x < 0,  (45) sign(x) = 0, if x = 0,   1, if x > 0. For any sign list (SL) [l1 , l2 , · · · , ln ], we obtain the revised sign list (RSL) [r1 , r2 , · · · , rn ] as follows: 1. If lk ̸= 0 we write rk = lk . 20 TING CHEN, SHIMIN LI AND JAUME LLIBRE 2. If a section [li , li+1 , · · · , li+j ] of the given sign list satisfies with li+1 = · · · = li+j−1 = 0 and li li+j ̸= 0, we replace the subsection [li+1 , li+2 , · · · , li+j−1 ] with [−li , −li , li , li , −li , −li , li , li , −li , · · · ] keeping the number of terms. Then we obtain that the RSL [r1 , r2 , · · · , rn ] has no zeros between two nonzero members. From [35], we have the following theorem for computing the number of real roots of f (x). Theorem 4.4. For any polynomial (46) f (x) = a0 + a1 x + · · · + ak xk with real coefficients, if the number of the sign changes of the RSL [r1 , r2 , · · · , rn ] of (46) equals m, and the number of nonzero elements of this RSL equals n, the number of the distinct real roots of (46) is n − 2m. Thus we can determine the number and the type of the remaining finite equi- librium points by Theorems 4.4, 2.1 and 2.3. And we do not need to calculate the coordinates of these equilibrium points. Assume that −27 + 2a321 ̸= 0 and 1 + a21 ̸= 0, we compute the discriminant sequence {S1 , S2 , S3 , S4 , S5 , S6 } of f (y) from (44), and obtain S1 = 6M12 , S2 = −12a21 M13 M2 , (47) S3 = −16a21 M13 M2 M3 , S4 = −32(1 + a21 )M13 M3 M4 , S5 = −32(1 + a21 )2 M13 M4 M52 M6 , S6 = −64(1 + a21 )3 M13 M54 M62 , where M1 = − 27 + 2a321 , M2 = −18 − 15a21 + a321 + a421 , M3 = − 729 − 729a21 + 162a221 + 432a321 + 279a421 − 24a521 − 54a621 − 30a721 + a821 + 2a921 + a10 21 , (48) M4 = − 8748 − 10206a21 − 2187a221 + 1620a321 + 3078a421 + 783a521 + 144a621 − 316a721 − 97a821 − 42a921 + 11a10 11 12 21 + 4a21 + 2a21 , M5 = − 27 − 27a21 + 4a321 + 3a421 , M6 = − 135 − 81a21 − 45a221 + 13a321 + 8a421 + 4a521 . From the above analysis of the infinite equilibrium points of systems (33), we sep- arate the study in the following three cases. √ 1) If a21 > 3/ 3 2 we have S1 > 0, S4 < 0, S5 < 0, S6 < 0 and S2 S3 ≥ 0. We denote by Root[f (α), i] the i-th real root of the polynomial f (α) with respect to α, √and it has Root[f (α), i] < Root[f (α), j] when i < j. We have S2,3 > 0 when 3 3/ 2 < a21 < Root[M2 , 2]. And we have S2,3 = 0 when a21 = Root[M2 , 2], and S2,3 < 0 when a21 > Root[M2 , 2]. Therefore the corresponding RSL of the first case is [1, 1, 1, −1, −1, −1], and the RSL of the last two cases is [1, −1, −1, −1, −1, −1]. Thus, by Theorem 4.4, systems (33) have four finite equilibrium points p6,7,8,9 other than p1,2,3 , where p1,2 are two centers and p3 is a saddle. On the other hand, systems (33) have three infinite equilibrium points Ai = (ui , 0) (i = 1, 2, 3) in U1 , one in the U2 , and the corresponding points in V1,2 . Denote that u1 > u2 > u3 . The eight infinite equilibria are nodes. Hence, on the Poincaré sphere the sum of the indices of the known equilibria is 10. By Theorem 2.3, the Z2 -EQUIVARIANT LINEAR TYPE BI-CENTER CUBIC HAMILTONIAN SYSTEMS 21 (a) (b) (c) (d) Figure 12. The local phase portraits √ at all equilibria of systems √ (33) with a03 = 0. (a) If a21√> 3/ 3 2; (b) If −1 ≤ a21 < 3/ 3 2; (c) If a21 < −1; (d) If a21 = 3/ 3 2. Figure 13. A center-loop. sum of the indices of the remaining finite equilibria must be −8. Furthermore the remaining equilibrium points in the interior of the Poincaré disc are four saddles, where two must be symmetry to the other ones with respect to the origin. Then we obtain that the local phase portraits at these equilibria in the Poincaré disc are shown in Figure 12(a). We claim that in this case systems (33) have at most two saddles on the boundary of period annulus of the bi-center at p1,2 . By computing the Gröbner basis for the three polynomials ẋ/y, ẏ and H(x, y) − h with some h ∈ R/{0}, we get that H(x, y) has the same non-zero values at no more than two equilibrium points. And we obtain four even polynomials Fi (a21 , y, h) (i = 1, 2, 3, 4) in y from this Gröbner basis, where F2,3 (a21 , y, h) are quadratic and F4 (a21 , y, h) is quartic. There is a polynomial F5 (a21 , x, y, h) which is linear in x with nonzero coefficients. We calculate the resultant of the coefficients of y 2 in F2 (a21 , y, h) and F3 (a21 , y, h) with respect to h, and obtain A(1+a21 )M6 > 0, where A is a constant. Thus at least one coefficient of y 2 of the polynomials F2,3 (a21 , y, h) is nonzero. It means that there are at most two distinct real roots for these three polynomials, then our claim is proved. By the symmetry of systems (33) there must be only one saddle on the boundary of period annulus of each center, creating a center-loop, see Figure 13. Here we assume that these two saddles are p7,8 . If one repelling and one attracting separatrix of the origin of systems (33) connect with the origin of U2 and with the infinite equilibrium point A1 of U1 , respectively, then we have that this phase portrait in the Poincaré disc is topologically equivalent 22 TING CHEN, SHIMIN LI AND JAUME LLIBRE Table 3. The√conditions of the revised sign lists of f (y) when −1 < a21 < 3/ 3 2. RSL SL Conditions [1, 1, 1, 1, −1, 1] [1, 1, 1, 1, −1, 1] −1 < a21 < Root[M3 , 1]; [1, 1, −1, 1, 1, 1] [1, 1, −1, 1, 1, 1] Root[M4 , 1] < a21 < 0; [1, 1, −1, 0, 0, 1] a21 = Root[M4 , 1]; [1, 1, −1, −1, −1, 1] [1, 1, −1, −1, −1, 1] Root[M3 , 1] < a21 < Root[M4 , 1]; [1, 1, 0, 0, −1, 1] a21 = Root[M3 , 1]; √ [1, −1, −1, 1, 1, 1] [1, −1, −1, 1, 1, 1] Root[M4 , 2] < a21 < 3/ 3 2; [1, −1, −1, 0, 0, 1] a21 = Root[M4 , 2]; [1, −1, −1, −1, 1, 1] [1, −1, −1, −1, 1, 1] Root[M3 , 2] < a21 < Root[M5 , 2] or Root[M5 , 2] < a21 < Root[M6 , 1]; [1, −1, −1, −1, −1, 1] [1, −1, −1, −1, −1, 1] Root[M6 , 1] < a21 < Root[M4 , 2]; [1, −1, 1, 1, 1, 1] [1, −1, 1, 1, 1, 1] 0 < a21 < Root[M3 , 2]; [1, −1, 0, 0, 1, 1] a21 = Root[M3 , 2]; [1, −1, −1, −1, 0, 0] [1, −1, −1, −1, 0, 0] a21 = Root[M5 , 2] or a21 = Root[M6 , 1]. to 1.18 of Figure 1, which can be realized when a21 = 4. If one repelling and one attracting separatrix of the origin of systems (33) connect with the origin U2 and with the infinite equilibrium point A3 of U1 , respectively, then the phase portraits of systems (33) are topologically equivalent to 1.20 of Figure 1, which can be realized when a21 = 2.5. From the phase portraits 1.18 to 1.20 it follows by the continuity of the phase portraits with respect to the parameters that there must exist one phase portrait that one attracting separatix of the origin of systems (33) connects with the finite equilibrium point p6 , which corresponds to the phase portrait 1.19 of Figure 1. In fact we can find this bifurcation point a21 by computing the invariant straight lines x = ky of systems√ (33). √Similarly to (32) we obtain a21 = 3 and k = 2. Then we have p6,9 = (±2 6/3, ± 6/3) and the infinite equilibrium point A2 = (1/2, 0) are on the invariant line x = 2y. Moreover we obtain that the phase portrait in the Poincaré √ disc is topologically equivalent to 1.18 or 1.20 of Figure 1 when a21 > 3 or 3/ 3 2 < a21 < 3, respectively. √ √ 2) Assume that −1 ≤ a21 < 3/ 3 2. If −1 < a21 < 3/ 3 2 we obtain the possible RSL of f (y) and their associated SL as it is shown in Table 3. By Theorem 4.4 systems (33) have two finite equilibrium points p7,8 additional to p1,2,3 , where p1,2 are two centers and p3 is a saddle. Then on the Poincaré sphere the sum of the indices of the known equilibria is 6. From Theorem 2.3 the sum of the indices of the remaining equilibria must be −4. Thus, the two remaining finite equilibrium points p7,8 are two saddles. The local phase portraits of systems (33) at these equilibria in the Poincaré disc √are shown√ in Figure 12(b). On the other hand, if a21 = −1 we have p7,8 = (±3 3/29, ∓ 3/29), which also are two saddles. And the y-axis is invariant by the flow of systems (33). Similarly to the above case the saddles p7,8 Z2 -EQUIVARIANT LINEAR TYPE BI-CENTER CUBIC HAMILTONIAN SYSTEMS 23 Table 4. The conditions of the revised sign lists of f (y) when a21 < −1. RSL SL Conditions [1, 1, 1, −1, −1, −1] [1, 1, 1, −1, −1, −1] Root[M2 , 1] < a21 < Root[M5 , 1] or Root[M5 , 1] < a21 < −1; [1, −1, −1, −1, −1, −1] [1, −1, −1, −1, −1, −1] a21 < Root[M2 , 1]; [1, 0, 0, −1, −1, −1] a21 = Root[M2 , 1]; [1, 1, 1, −1, 0, 0] [1, 1, 1, −1, 0, 0] a21 = Root[M5 , 1]. must be on the boundary of period annulus of the bi-center at p1,2 , respectively. And they create two center-loops. √ Then we have that the phase portraits of systems (33) with −1 ≤ a21 < 3/ 3 2 are topologically equivalent to 1.21 of Figure 1. 3) If a21 < −1 then the RSL of f (y) and their associated SL are given in Table 4. From Theorem 4.4, if a21 ̸= Root[M5 , 1] the polynomial f (y) has four distinct real roots. Thus systems (33) have four finite equilibrium points p6,7,8,9 additional to p1,2,3 , where p1,2,3 are three centers. And systems (33) have four infinite equilibria, which are nodes. Hence the sum of the indices of the known equilibria is 10 on the Poincaré sphere. By Theorem 2.3, the sum of the indices of the remaining equilibria must be −8. Thus, they are four saddles in Poincaré disc. If a21 = Root[M5 , 1] there two distinct roots and two repeated real roots for f (y). In fact systems (33) also have four finite equilibrium points p6,7,8,9 different from p1,2,3 , and they are four saddles. Then we have that the local phase portraits at all equilibria in the Poincaré disc are shown in Figure 12(c). From the previous proof we have that H(x, y) has the same non-zero values at no more than two equilibrium points. Hence there must be one saddle on the boundary of the period annulus of each center, and every one of these saddles creates a center-loop. Here we assume that these two previous saddles are p7,8 . And the other two saddles p6,9 must be on the boundary of the period annulus of the center at the origin. Then the phase portraits of systems (33) in the Poincaré disc are topologically equivalent to 1.22 of Figure 1. √ Now we consider the case a21 = 3/ 3 2, then systems (33) have two finite equi- librium points p7,8 ≈ (±1.06093, ∓0.22024) additional to p1,2,3 , and they are two saddles. And systems (33) have six infinite equilibrium points, three in U1,2 and three in V1,2 . We have that the local phase portraits of all equilibria in the Poincaré disc are shown in Figure 12(d). Similarly to the above case, there are no more than two equilibrium points in the same energy level. Hence the saddles p7,8 must be on the boundary of the period annulus of the centers p1,2 , respectively. Then these two saddles create two center-loops. Therefore the phase portrait of systems (33) now is topologically equivalent to 1.23 of Figure 1. In summary all results for the isolated equilibrium points of the phase portraits 1.18-1.23 are listed in Table 5. Further, we have the following result. 24 TING CHEN, SHIMIN LI AND JAUME LLIBRE Table 5. The isolated equilibrium points corresponding to the topological phase portraits 1.18-1.23. Phase portraits Isolated finite equilibria Isolated infinite equilibria 1.18,1.19,1.20 2 C, 5 S 4 attracting N, 4 repelling N 1.21 2 C, 3 S 2 attracting N, 2 repelling N 1.22 3 C, 4 S 2 attracting N, 2 repelling N 1.23 2 C, 3 S 2 attracting N, 2 repelling N, 2 H 1.22 1.21 1.23 1.19 a21 √ −1 3/ 3 2 3 1.20 1.18 Figure 14. The bifurcation diagram of the phase portraits of sys- tems (33) with a03 = 0. Proposition 4.5. The phase portraits of systems (33) in the Poincaré disc with a03 = 0 are topologically equivalent to 1.18 if a21 > 3; 1.19 if a21√= 3; 1.20 if 3/ 3 2 < a21 < √ 3; 1.21 if −1 ≤ a21 < 3/ 3 2; 1.22 if a21 < −1;√ 1.23 if a21 = 3/ 3 2. Moreover the bifurcation diagram is shown in Figure 14. 4.2.2. Subcase a03 ̸= 0. In this subcase the characterization of the phase portraits of systems (33) become extremely difficult because the explicit expressions of some infinite and finite equilibrium points are complicated. To overcome these difficulties we do the following analysis. We compute the Gröbner basis for −1 − a21 + a21 x2 − 3xy + a03 y 2 and −x + x3 − a21 xy 2 + y 3 from (33), and obtain five polynomials. And we only need to consider two polynomials, one is linear in x and the other one is (49) − 1 − a21 + F1 y 2 + F2 y 4 + F3 y 6 , where F1,2,3 are shown in (35). Hence there are at most six finite equilibrium points additional to p1,2,3 . Then we compute the discriminant sequence {S1 , S2 , S3 , S4 } of polynomial f1 (u) from (40) for determining the infinite equilibrium points. And we have S1 = 4a203 , S2 = −16a203 (−3 + a03 a21 ), S3 = −64a203 F4 , (50) S4 = −256a203 F3 , where F3 is shown in (35) and F4 = −3a03 + a203 a21 − 2a221 + a03 a321 . Z2 -EQUIVARIANT LINEAR TYPE BI-CENTER CUBIC HAMILTONIAN SYSTEMS 25 We separate the study in the two cases a21 ≥ −1 or a21 < −1, then p3 is a saddle or a center, respectively. 1) Assume that a21 ≥ −1. 1.a) When the polynomial f1 (u) has four distinct real roots, the RSL of its discriminant sequence must be√[1, 1, 1, 1], which needs either a03 < 0 and a21 > Root[F3 , 2], or 0 < a03 < 3/ 3 16 and Root[F3 , 1] < a21 < Root[F3 , 2]. Then systems (38) have four equilibrium points in U1 , which are all nodes, and the origin of U2 is not an equilibrium point. The sum of the indices of the known equilibria is 10 on the Poincaré sphere. By Theorem 2.3, the sum of the indices of the remaining equilibria must be −8. From the previous analysis of (37) we have that the remaining finite equilibrium points are elementary and a pair of points are symmetric with respect to the origin. Thus they must be four saddles. Similarly to the subcase a03 = 0, by computing the Gröbner basis for ẋ, ẏ and H(x, y) − h we obtain that H(x, y) has the same h ∈ R/{0} at no more than two finite equilibrium points. Then we obtain that the global phase portraits of systems (33) in the Poincaré disc are topologically √ equivalent to 1.18 and 1.20 of Figure 1. In particular when a03 = 1 − a221 + 4 1 + a21 the phase portrait is topologically equivalent to 1.19 of Figure 1, which has an invariant straight line. Indeed, forcing that the straight line x = ky be this invariant line of systems (33) we obtain (51) ẋ − k ẏ = (−1 − a21 + k 2 )y + (a03 + 2a21 k 2 − k 4 − 4k)y 3 . Hence we need that G1 = −1 − a21 + k 2 = 0 and G2 = a03 + 2a21 k 2 − k 4 − 4k = 0 in order that (51) be identically zero. The resultant of G1 and G2 with respect to the variable k is −15 − 2a03 + a203 − 16a21 − 2a221 + 2a03 a221 + a421 . √ √ This resultant is zero if a03 = 1 − a221 + 4 1 + a21 or a03 = 1 − a221 − 4 1 + a21 , but the latter cannot occur in this case. Indeed, we just need to substitute it into (50) and check its RSL. 1.b) When the polynomial f1 (u) has three distinct real roots, the RSL √ of its discriminant sequence must be [1, 1,√1, 0], which needs either a03 < 3/ 3 16 and a21 = Root[F3 , 2], or 0 < a03 < 3/ 3 16 and a21 = Root[F3 , 1]. From these two cases we have F3 = 0, thus systems (33) have three infinite equilibrium points, where one is degenerate and the other two are nodes. Let A1 = (u1 , 0) be the degenerate infinite equilibrium point, and A2 = (u2 , 0) and A3 = (u3 , 0) be two nodes, for u1 > u2 > u3 . In addition the polynomial (49) becomes a quartic polynomial, hence there are at most four finite equilibrium points additional to p1,2,3 . √ When 0 < a03 < 3/ 3 16 and a21 = Root[F3 , 1] the local phase portrait of the degenerate point A1 consists of two hyperbolic sectors, see the second picture of Figure 5(b). By Theorem 2.3 the sum of the indices of the known equilibria of systems (33) is 6 on the Poincaré sphere. Then the sum of the indices of the remaining finite equilibria must be −4. By the symmetry of the equilibria the only possibility is that systems (33) have two saddles different from p1,2,3 in the Poincaré disc. Then we have that in this case the phase portrait in the Poincaré 26 TING CHEN, SHIMIN LI AND JAUME LLIBRE disc is topologically equivalent to 1.23 of Figure 1, which is realized when a03 = 1 and a21 = 2. √ When a03 < 3/ 3 16 and a21 = Root[F3 , 2] the local phase portrait of A1 consists of two parabolic and two elliptic sectors, see the first picture of Figure 5(b). More- over the remaining finite equilibria must have total index −8. By the symmetry the only possibility is that they are four saddles. We have that the local phase portraits at all equilibria in the Poincaré disc are topologically equivalent to Figure 9(b), which it is not symmetric with respect to the x-axis or the y-axis. On the other hand there must be one saddle on the boundary of the period annulus of each center, creating two center-loops. We assume that these two saddles are p7,8 . If one repelling and one attracting separatrices of the origin connect to the infinite equilibrium point A1 , we obtain the phase portrait 1.24 of Figure 1, and when a03 = 0.7 and a21 = 2.9198 this phase portrait is achieved. If one repelling and one attracting separatrices of the origin connect to the infinite equilibrium points A1 and A3 respectively, we have the phase portrait 1.26 of Figure 2, which is realized when a03 = 1 and a21 = 2.13488. From the phase portraits 1.24 and 1.26 it follows by the continuity of the phase portraits with respect to the parameters the existence of the phase portrait 1.25 of Figure 2, which has one invariant straight line. Indeed, by calculating F3 = G1 = G2 = 0 from (51) we obtain either a03 = 0.71794.. and a21 = 2.85173.., or a03 = −3 and a21 = 0. But the latter is not satisfied in this subcase. 1.c) When the polynomial f1 (u) has two distinct real roots, the RSL of its discriminant sequence must be either [1, −1, −1, −1], or [1, 1, √ −1, −1], or [1, 1, 1,√ −1], or [1, 1, 0, 0], which just needs either F3 > 0, or a21 = 3/ 3 4 and a03 = 3/ 3 16. Then there are four infinite equilibria for systems (33), and all of them are nodes. Furthermore we have that the sum of the indices of the remaining finite equilibria on the Poincaré sphere must be −4. By (37) we know that the remaining finite equilibria are elementary. Then additional to the equilibria p1,2,3 , there are either (i) two saddles p7,8 , or (ii) two centers p4,5 and four saddles p6,7,8,9 . In case (i) if the two saddles p7,8 are on the boundary of the period annulus of the bi-center, creating two center-loops. In this case we deduce that the global phase portrait is topologically equivalent to 1.21 of Figure 1. If the saddle at the origin is located on the boundary of the period annulus of p1 , by the symmetry it is also in the boundary of the period annulus of p2 . When the saddles p7,8 are outside the region enclosed by the separatrices of the origin the global phase portraits of systems (33) are topologically equivalent to 1.2 of Figure 1, which can be realized when a21 = 2 and a03 = −10. By the continuity of the phase portraits with respect to parameters a03 and a21 from the√ phase portrait 1.2 to 1.21, we obtain the bifurcation curve a03 = 1 − a221 − 4 1 + a21 , on this curve we have the phase portrait 1.27 of Figure √ 2, i.e. the saddles at origin and p7,8 are on the invariant straight line x = − 1 + a21 y. This phase portrait can be realized when a21 = 2 and a03 = −9.9282. And the invariant straight line can be obtained from (51). In case (ii) we have that the local phase portraits at all equilibrium points are shown in Figure 9(a). If only the two saddles p7,8 and the two centers p1,2 create Z2 -EQUIVARIANT LINEAR TYPE BI-CENTER CUBIC HAMILTONIAN SYSTEMS 27 (a) (b) Figure 15. The local phase portraits at all equilibria of systems (33) with a03 ̸= 0 and a21 ≥ −1 in case 1.d). (a) Case (i); (b) Case (ii). two center-loops, and if the origin p3 is on the boundary of the period annulus of p4 , by the symmetry, the origin also will be on the boundary of the period annulus of p5 . Then it creates one eight–figure loop. Furthermore the origin must be inside the region enclosed by the separatrices of the saddles p6 and p9 . Therefore the phase portrait is topologically equivalent to 1.28 of Figure 2, and this phase portrait can be obtained when a03 = 2 and a21 = −0.9. If there are four saddles p6,7,8,9 located on the boundary of the period annulus of each center, creating four center-loops, then the phase portrait is topologically equivalent to 1.30 of Figure 2, which can be realized when a03 = 2 and a21 = −0.8. From the phase portraits 1.28 and 1.30 it follows by the continuity of the phase portraits with respect to the parameters the existence of the phase portrait 1.29 of Figure 2, which can be realized when a21 = 1 and a03 = 5.65685. That is, the saddles p3,6,9 are on one invariant straight line, and the saddles p3,6 are on the boundary of the period annulus of p4 , by the symmetry the saddles p3,9 are on the boundary of the period annulus of p5 . Similarly √ to subcase 1.a) this phase portrait can be obtained when a03 = 1 − a221 + 4 1 + a21 from (51). 1.d) When the polynomial f1 (u) has one real root, the RSL of its discriminant sequence must be either [1, 0, 0, 0], or [1, 1, −1, 0], or [1, −1, −1, 0], which needs a03 ≤ Root[29 + 19a03 + 2a203 + a303 , 1] = −1.58141.. and a21 = Root[F3 , 1]. From F3 = 0 we obtain that systems (33) have one degenerate infinite equilibrium point in U1 . The phase portrait of this infinite equilibrium point consists of either (i) two hyperbolic sectors, or (ii) two parabolic sectors and two elliptic sectors because we have u̇|v=0 = 1 + O(u) and v̇|u=0 = 0 from (38). Assume (i). By Theorem 2.3 the sum of the indices of the known equilibria of (33) is 2 on the Poincaré sphere. Thus the finite equilibrium points of systems (33) different from p1,2,3 can be either (i.a) no equilibrium points, or (i.b) two centers and two saddles, but the latter cannot occur. Indeed, when F3 = 0 the polynomial (49) simplifies to (52) −1 − a21 + F1 y 2 + F2 y 4 , where F1,2 are shown in (35). If we replace y 2 by z we get the quadratic polynomial (53) −1 − a21 + F1 z + F2 z 2 . 28 TING CHEN, SHIMIN LI AND JAUME LLIBRE Hence (52) has four real roots when (53) has two positive roots. Then we can use Descartes’ rule of signs for studying the number of positive roots of (53) given in the following theorem. Theorem 4.6. The number of positive roots of a real polynomial (46) is either the number of sign differences between consecutive nonzero coefficients, or less than it by a multiple of 2. When (53) has two positive roots there must be two changes of sign between the coefficients of (53). Assume that a21 > −1 we have −1 − a21 < 0. Then the coefficients of z and z 2 must be positive and negative, √ respectively. Hence it must satisfy with F1 > 0 and F2 < 0, i.e. 0 < a03 < 3/ 3 16 and a21 = Root[F3 , 2], which is in contradiction with this case. Therefore systems (33) have at most two finite equilibria different from p1,2,3 . Consider case (i.a). We obtain that the local phase portraits at all equilibria in the Poincaré disc are shown in Figure 15(a). If the origin is on the boundary of the period annulus of p1 , by the symmetry it is also on the boundary of the period annulus of p2 , creating one eight–figure loop. Then in this case the phase portrait is topologically equivalent to 1.12 of Figure 1, which can be realized when a03 = −10 and a21 = 1.98273. If the origin is not on the boundary of the period annulus of the bi-center at p1,2 , the separatrices of p3 must connect to some infinite equilibrium points. And the bi-center at p1,2 must be inside the region enclosed by the separatrices of the origin, then we have the phase portrait 1.31 of Figure 2. In fact we can obtain that p3 and the infinite equilibria are on the invariant straight line x = −y, see (51). This phase portrait can be realized when a03 = −3 and a21 = 0. Assume (ii). Then the sum of the indices of the remaining finite equilibria must be −4 on the Poincaré sphere. Hence systems (33) have two saddles p7,8 different from p1,2,3 . The local phase portraits of systems (33) at these equilibria in Poincaré disc are shown in Figure 15(b). We claim that the saddles p7,8 neither are connected with the origin, nor are outside the region enclosed by the boundary of the period annulus of p3 , as it is shown in the first three pictures of Figure 16. The first two pictures of Figure 16 cannot occur, otherwise there would have six intersection points with the separatrices of p7,8 on a straight line l through p3 . That is, there would be six points on l with same energy level, but it is not possible for a quartic Hamiltonian H(x, y). In the third picture of Figure 16 the flow at infinity is clockwise, in contradiction with the fact that u̇|u=0,v=0 = 1 > 0 at the origin of the chart U1 . Then the phase portrait in Poincaé disc is topologically equivalent to 1.32 of Figure 2. This phase portrait can be realized by taking a03 = −2 and a21 = −0.61752. 1.e) The polynomial f1 (u) has no real roots if and only if either S2 > 0, S3 ≤ 0 and S4 > 0, or S2 ≤ 0 and S4 > 0, i.e. a03 < Root[29 + 19a03 + 2a203 + a303 , 1] and −1 ≤ a21 < Root[F3 , 1]. By Theorem 2.3, on the Poincaré sphere the known equilibrium points of (33) have total index 2. Then the sum of indices of the remaining finite equilibria must be 0. Thus we have the following possibilities: (i) no equilibrium points, (ii) two cusps, (iii) two centers and two saddles. Z2 -EQUIVARIANT LINEAR TYPE BI-CENTER CUBIC HAMILTONIAN SYSTEMS 29 (a) (b) (c) (d) (e) (f) Figure 16. The possible phase portraits for the separatrices con- figurations of two saddles. Case (i) can occur and we get that the phase portrait in the Poincaré disc is topologically equivalent to 1.13 of Figure 1, which can be realized when a21 = −0.7 and a03 = −3. Consider case (iii), then the two additional saddles cannot be outside the region enclosed by the separatrices of the saddle p3 and they cannot be connected with p3 . That is, the forth and fifth phase portraits of Figure 16 are not possible because a straight line through p3 would intersect the separatrices of these two saddles six times. For the sixth picture of Figure 16 the flow at infinity is clockwise and this is not possible. Indeed, we have u̇|u=0,v=0 = 1 > 0 at the origin of the chart U1 . Then we deduce that the phase portrait of systems (33) is topologically equivalent to 1.33 of Figure 2, which is achieved when a21 = −0.7 and a03 = −2. And it has two eight–figure loops due to the centers p1,2 and the four remaining finite equilibrium points. By the analysis of √(37) when √ a03 = −9/4 and a21 = −1 systems (33) have two nilpotent equilibria (3 3/31, ±2 3/31), which are two cusps. Then we will obtain the phase portrait in case (ii). Since there is no more equilibrium points these cusps must be on the boundary of the period annulus of each center. Hence we have that the phase portrait in the Poincaé disc is topologically equivalent to 1.34 of Figure 2. 2) Assume that a21 < −1. The polynomial f1 (u) has at most two distinct real roots because there are no parameters such that the RSL of its discriminant sequence is either [1, 1, 1, 1], or [1, 1, 1, 0]. Since a03 = −9/4 and a21 = −1 are not in this case, the remaining finite equilibria are elementary, i.e. they are saddles or centers. Similarly to the analysis of case 1.c) the polynomial f1 (u) has two distinct real roots if and only if S4 < 0, i.e. either Root[29 + 19a03 + 2a203 + a303 , 1] < a03 < 0 and Root[F3 , 1] < a21 < −1, or a03 > 0 and a02 < −1. Then there are four infinite 30 TING CHEN, SHIMIN LI AND JAUME LLIBRE equilibria for systems (33), which are nodes. Hence on the Poincaré sphere the sum of the indices of the remaining finite equilibria must be −8. Thus there are four saddles additional to the equilibria p1,2,3 . Since the Hamiltonian H(x, y) has the same non-zero value at no more than two equilibria, there are at most two saddles on the boundary of the period annulus of p3 . And by the symmetry the only way to have two centers and the two other saddles is to have two center-loops. Therefore the global phase portrait in the Poincaé disc is topologically equivalent to 1.22 of Figure 1. Similarly to cases 1.d) and 1.e) we have the phase portrait 1.35 when Root[29 + 19a03 + 2a203 + a303 , 1] < a03 < 0 and a21 = Root[F3 , 1], and the phase portrait 1.14 or 1.36 when a03 < 0 and a21 ≤ Root[F3 , 1]. On the other hand the phase portraits 1.14, or 1.35, or 1.36 can be realized when a03 = −10 and a21 = −2, or a03 = −0.9491 and a21 = −2, or a03 = −2 and a21 = −2, respectively. Then, all results for the isolated equilibrium points of the phase portraits 1.24- 1.36 are listed in Table 6. Table 6. The isolated equilibrium points corresponding to the topological phase portraits 1.24-1.36. Phase portraits Isolated finite equilibria Isolated infinite equilibria 1.24,1.25,1.26 2 C, 5 S 2 attracting N, 2 repelling N, 2 E 1.27 2 C, 3 S 2 attracting N, 2 repelling N 1.28,1.29,1.30 4 C, 5 S 2 attracting N, 2 repelling N 1.31 2 C, 1 S 2H 1.32 2 C, 3 S 2E 1.33 4 C, 3 S 0 1.34 2 C, 1 S, 2 cusp 0 1.35 3 C, 4 S 2E 1.36 5 C, 4 S 0 Thus we have obtained all the phase portraits of the Z2 -equivariant cubic Hamil- tonian systems (3) with a linear type bi-center, which are provided in Theorem 1.1. Acknowledgments The first author is partially supported by National Natural Science Foundation of China (Nos. 11771059 and 11801089). The second author is partially supported by Science and Technology Program of Guangzhou (No. 201707010426), Natural Science Foundation of Guangdong Province (2017A030313010, 2019A1515011885). 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