This is a preprint of: “Nilpotent bi-center in continuous piecewise Z2 -equivariant cubic polynomial Hamiltonian systems”, Ting Chen, Shimin Li, Jaume Llibre, Nonlinear Dynam., vol. 110, 705–721, 2022. DOI: [10.1007/s11071-022-07631-z] NILPOTENT BI-CENTER IN CONTINUOUS PIECEWISE Z2 -EQUIVARIANT CUBIC POLYNOMIAL HAMILTONIAN SYSTEMS (I) TING CHEN1 , SHIMIN LI1 AND JAUME LLIBRE2 Abstract. One of the classical and difficult problems in the theory of planar differential systems is to classify their centers. Here we classify the global phase portraits in the Poincaré disc of the class continuous piecewise differential systems separated by one straight line and formed by two Z2 -equivariant cubic Hamiltonian systems with nilpotent bi-centers at (±1, 0). 1. Introduction and statement of the main results The problem in the qualitative theory of planar differential equations of distinguishing be- tween a focus and a center is known as the center-focus problem. This classical problem started with Poincaré [36] in 1881 and Dulac [15] in 1908, and nowadays the center-focus problem remains as one of main subjects in the qualitative theory of planar polynomial differential systems. We say that a singular point p of a planar differential system is a center if it has a neighbor- hood U filled with periodic orbits with the unique exception of this singular point. If a planar polynomial differential system has a linear type center, or a nilpotent center, or a degenerate center at the origin of coordinates, after making a time rescaling and a linear change of variables, this differential system can be written as    (−y, x)   (1) (ẋ, ẏ) = (y, 0) + (f (x, y), g(x, y)),     (0, 0) respectively. Here the dot denotes derivative with respect to the time t, and f (x, y) and g(x, y) are real polynomials without constant and linear terms. The focus-center problem for the quadratic polynomial differential systems has been solved see [3, 7, 15, 23, 24, 37, 40, 43]. There are partial results in the classification of the centers for the cubic polynomial differential systems, see for instance [9, 11, 32, 41, 44, 45], but the focus-center problem for the cubic polynomial differential systems still remains open. Recently Colak el at. [12, 13] studied the phase portraits of some cubic Hamiltonian dif- ferential systems with a linear type center and a nilpotent center at the origin, respectively. Liu and Li [30] investigated the linear type bi-center problem for Z2 -equivariant differential systems. Here we shall study the Z2 -equivariant polynomial systems having two centers at the singular points. The study of Zq -equivariant polynomial systems, whose phase portraits are unchanged by a rotation of 2q (q ∈ Z + ) radians around one point, is closely related to the well-known Hilbert 16th problem, for more details see [22, 27, 28]. Chen el at. [10] provided all possible phase portraits of Z2 -equivariant cubic polynomial Hamiltonian vector fields with a linear type bi- center. Li el at. [25, 26] studied the bi-center and isochronous bi-center problems in some Z2 -equivariant cubic systems. 2010 Mathematics Subject Classification. Primary: 34C07, 34C08. Key words and phrases. Nilpotent, Bi-center, Hamiltonian, Phase portrait. 1 2 TING CHEN, SHIMIN LI AND JAUME LLIBRE However many natural phenomena have been modeled more accurately by dynamical systems whose differential systems are non-smooth (see for instance [1, 6, 31]), increasing contributions have been attracted to the qualitative analysis of non-smooth systems, see [2, 17]. In this paper we deal with the following family of piecewise smooth systems, ( +  f (x, y), g + (x, y) if S(x, y) > 0, (2) (ẋ, ẏ) = − −  f (x, y), g (x, y) if S(x, y) < 0, where S : R2 → R is a C ∞ function and (f ± (x, y), g ± (x, y)) are smooth vector fields. In fact, systems (2) have two different regions Γ± = {(x, y) ∈ R2 : ±S(x, y) > 0} separated by the discontinuity line Γ = S −1 (0). The focus-center problem in piecewise smooth systems becomes much more difficult and complicated than for the smooth systems. For example, a singular point of systems (2) on the discontinuous curve S(x, y) = 0 can be a center even it is neither a center for the first system, nor a center for the second system of (2). Some methods have been developed for studying the linear type focus-center problem of the piecewise smooth systems (2). Thus Gasull and Torregrosa [20] developed an efficient method for computing the Lyapunov constants of switching polynomial systems, which can be used to determine the center conditions for a linear type singular point. By computing the Lyapunov constants, the authors of [8, 39] gave a complete classification on the linear type center conditions of the origin in several classes of Bautin switching systems. For more results on the focus-center problem of the piecewise smooth systems with the linear type singular points, see [14, 21]. The focus-center problem for the nilpotent singular points is much more challenging com- pared to the study for the linear type singular points. Computationally efficient methods have been developed for studying the focus-center problem of the planar smooth systems with nilpo- tent singular points, see [18, 29, 30, 38]. However there are no work for studying the nilpotent focus-center problem in piecewise smooth polynomial systems. In this paper we will study the global dynamics of a class of piecewise Z2 -equivariant differ- ential systems formed by two cubic Hamiltonian systems separated by the straigh line y = 0, and having nilpotent bi-centers at the points (±1, 0). In section 3 we prove that such class of piecewise differential systems can be written as     − a21 y + 3b03 y 2 + a21 x2 y − 3b03 xy 2 − 2(1 + a221 )y 3    1 1  if y > 0,      − x + x3 − a21 xy 2 + b03 y 3 ẋ (3) =  2 2  ẏ   − a 21 y − 3b03 y 2 + a21 x2 y − 3b03 xy 2 − 2(1 + a221 )y 3       if y < 0,  − 1 x + 1 x3 − a21 xy 2 + b03 y 3  2 2 where b03 < 0 and the singular point (1, 0) of the first system of (3) is a third-order singular point, see section 3 for the definition of third-order singular point. The Hamiltonian functions for these two Hamiltonian systems are 1 2 1 4 1 1 1 H(x, y)+ = x − x − a21 y 2 + b03 y 3 + a21 x2 y 2 − b03 xy 3 − (1 + a221 )y 4 , 4 8 2 2 2 for the Hamiltonian system in y > 0, and 1 2 1 4 1 1 1 H(x, y)− = x − x − a21 y 2 − b03 y 3 + a21 x2 y 2 − b03 xy 3 − (1 + a221 )y 4 , 4 8 2 2 2 for the Hamiltonian system in y < 0. Note that the piecewise differential systems (3) only are continuous on the straight line y = 0, so they are non-smooth piecewise differential systems. We also remark that the piecewise differential system (3) only depends on two parameters a21 and b03 . NILPOTENT BI-CENTER IN PIECEWISE Z2 -EQUIVARIANT CUBIC 3 1.1 S = 25, R = 8, 1.2 S = 27, R = 9, 1.3 S = 25, R = 7, 1.4 S = 21, R = 7, 1.5 S = 23, R = 8, 1.6 S = 21, R = 4, 1.7 S = 13, R = 4, 1.8 S = 9, R = 3, 1.9 S = 10, R = 4, 1.10 S = 6, R = 3. Figure 1. The ten topological phase portraits in the Poincaré disc of Theorem 1.1. Theorem 1.1. In the Poincaré disc the phase portraits of the continuous piecewise Z2 -equivariant cubic Hamiltonian systems (3) with a nilpotent bi-center at (±1, 0) are topologically equivalent to one of the 10 phase portraits showed in Figure 1. In section 2 we provide a brief introduction to the Poincaré compactification, a summary on how to determine the phase portrait using the separatrix skeleton, and some basic results on the topological indices that we shall need for proving Theorem 1.1. In section 3 we show how 4 TING CHEN, SHIMIN LI AND JAUME LLIBRE to obtain the continuous piecewise Z2 -equivariant cubic Hamiltonian systems (3). Finally 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 2.1. Poincaré compactification. In order to classify the global dynamics of the piecewise differential systems (3), the first crucial step is to characterize their finite and infinite singular points in the Poincaré compactification, as we shall see such compactification is possible due to the fact that our Hamiltonian systems are polynomial. This tool is described in chapter 5 of [16]. The second main step for determining the global flow in the Poincaré disc of polynomial differential systems is the characterization of their separatrices. For the polynomial differential systems in the Poincaré disc it is known that the separatrices are all the infinite orbits, all the finite singular points, the separatrices of the hyperbolic sectors of the finite and infinite singular points, and the limit cycles. If Σ denotes the set of all separatrices in the Poincaré disc D2 , Σ is a closed set and the components of D2 \ Σ are called the canonical regions. We denote by S and R the number of separatrices and canonical regions, respectively. Roughly speaking this compactification identifies the plane R2 with the interior of the closed unit disc D2 centered at the origin of R2 , and extends analytically the differential system to its boundary, usually called the circle of the infinity. Now we shall describe the equations of the Poincaré compactification for a polynomial differential system in R2 . We consider the set of all polynomial vector fields in R2 of the form (4) (x˙1 , x˙2 ) = X(x1 , x2 ) = (P (x1 , x2 ), Q(x1 , x2 )), where P and Q are real polynomials in the variables x1 and x2 of degree d1 and d2 , respectively. Taking d = max{d1 , d2 }. Denote by Tp S2 be the tangent space to the 2-dimensional sphere S2 = {s = (s1 , s2 , s3 ) ∈ R : s21 + s22 + s23 = 1} at the point p. Assume that X is defined in the plane T(0,0,1) S2 = R2 . 3 Consider the central projection f : T(0,0,1) S2 → S2 . This map defines two copies of X, one in the open northern hemisphere and other in the open southern hemisphere. Denote by X0 the vector field Df ◦ X defined on S2 except on its equator S1 = {y ∈ S2 : y3 = 0}. Clearly S1 is identified to the infinity of R2 . If X is a planar polynomial vector field of degree d, then p(X) is the only analytic extension of y3d−1 X0 to S2 , the vector field p(X) is called the Poincaré compactification of the vector field X, for more details see Chapter 5 of [16]. On the Poincaré sphere S2 we use the following six local charts to do the calculations, which are given by Ui = {s ∈ S2 : si > 0} and Vi = {s ∈ S2 : si < 0}, for i = 1, 2, 3, with the corresponding diffeomorphisms (5) ϕi : Ui → R2 , ψi : Vi → R2 , defined by ϕi (s) = −ψi (s) = (sm /si , sn /si ) = (u, v) for m < n and m, n 6= i. Thus (u, v) will play different roles in the distinct local charts. The expression of the vector field p(X) are         1 u 1 u 1 u (u̇, v̇) = v d Q , − uP , , −v d+1 P , in U1 , v v v v v v         d u 1 u 1 d+1 u 1 (u̇, v̇) = v P , − uQ , , −v Q , in U2 , v v v v v v (u̇, v̇) = (P (u, v), Q(u, v)) in U3 . We note that the expressions of the vector field p(X) in the local chart (Vi , ψi ) is equal to the expression in the local chart (Ui , φi ) multiplied by (−1)d−1 for i = 1, 2, 3. The orthogonal projection under π(y1 , y2 , y3 ) = (y1 , y2 ) of the closed northern hemisphere of S2 onto the plane s3 = 0 is a closed disc D2 of radius one centered at the origin of coordinates called the Poincaré disc. Since a copy of the vector field X on the plane R2 is in the open northern hemisphere of S2 , the interior of the Poincaré disc D2 is identified with R2 and the NILPOTENT BI-CENTER IN PIECEWISE Z2 -EQUIVARIANT CUBIC 5 boundary of D2 , the equator of S2 , is identified with the infinity of R2 . Consequently the phase portrait of the vector field X extended to the infinity corresponds to the projection of the phase portrait of the vector field p(X) on the Poincaré disc D2 . The singular points of p(X) in the Poincaré disc lying on S1 are the infinite singular points of the corresponding vector field X. The singular points of p(X) in the interior of the Poincaré disc, i.e. on S2 \ S1 , are the finite singular points. For polynomial vector fields (4) if s ∈ S1 is an infinite singular point, then −s ∈ S1 is another infinite singular point. Thus the number of infinite singular points is even and the local behavior of one is that of the other multiplied by (−1)d+1 . This symmetry property in general does not hold for piecewise smooth differential systems (2) because the singular points at infinity are not diametrically opposite. But in our case systems (3) are symmetry with respect to the origin, so we just need to analyze the phase portraits of the infinite singular points in the local chart U1 |v=0 and at the origin of the local chart U2 . 2.2. Separatrix skeleton. Given a flow (D2 , φ) by the separatrix skeleton we mean the union of all the separatries of the flow together with one orbit from each one of the canonical regions. Let C1 and C2 be the separatrix skeletons of the flows (D2 , φ1 ) and (D2 , φ2 ) respectively. We say that C1 and C2 are topologically equivalent if there exists a homeomorphism h : D2 → D2 which sends orbits to orbits preserving or reversing the direction of all orbits. From Markus [33], Neumann [34] and Peixoto [35] it follows the next theorem which shows that is enough to describe the separatrix skeleton in order to determine the topological equivalence class of a differential system in the Poincaré disc D2 . Theorem 2.1 (Markus–Neumann–Peixoto Theorem). Assume that (D2 , φ1 ) and (D2 , φ2 ) are two continuous flows with only isolated singular points. Then these flows are topologically equivalent if and only if their separatrix skeletons are equivalent. 2.3. Topological index. Next we introduce the topological index of the singular points, which is one useful tool to determine the type of the singular points. Here we will present two important theorems, the Index Poincaré Formula and the Poincaré–Hopf Theorem, for more details see Chapter 6 of [16]. Theorem 2.2. We denote by p an isolated singular point with the finite sectorial decomposi- tion property. Let q, h and e be the number of parabolic, hyperbolic and elliptic sectors of p, respectively. Then the topological index of the singular point p equals 1 + (e − h)/2. Corollary 2.3. The topological indices of a center, a cusp, a saddle and a node equal 1, 0, −1 and 1, respectively. Theorem 2.4. For any continuous vector field on the sphere S2 with finitely many singular points, the sum of their topological indices is 2. Remark 2.5. Since the flow of Hamiltonian smooth systems preserves the area, we have that any finite singular point of a Hamiltonian smooth system must be either a center, or union of an even number of hyperbolic sectors. In particular, the finite nilpotent singular points of Hamiltonian planar differential systems are either saddles, centers, or cusps, for more details see Theorem 3.5 of [16]. 3. Obtaining systems (3) Here a vector field X(x, y) is Z2 -equivariant if −X(x, y) = X(−x, −y). Then Z2 -equivariant piecewise cubic polynomial differential systems (3) separated by the straight line y = 0 are 6 TING CHEN, SHIMIN LI AND JAUME LLIBRE differential systems of the form    a00 + a10 x + a01 y + a20 x2 + a11 xy + a02 y 2 + a30 x3          + a21 x2 y + a12 xy 2 + a03 y 3      if y > 0,    2 2 3    b00 + b10 x + b01 y + b20 x + b11 xy + b02 y + b30 x       ẋ + b21 x2 y + b12 xy 2 + b03 y 3 (6) =   ẏ   − a00 + a10 x + a01 y − a20 x2 − a11 xy − a02 y 2 + a30 x3          + a21 x2 y + a12 xy 2 + a03 y 3      if y < 0.    2 2 3     − b00 + b10 x + b01 y − b20 x − b11 xy − b02 y + b30 x    + b21 x2 y + b12 xy 2 + b03 y 3 Assuming that (±1, 0) are two singular points of systems (6), we have (7) a00 = −a20 , a10 = −a30 , b00 = −b20 , b10 = −b30 . The Jacobian matrices of the first and second systems of (6) evaluated at (1, 0) are   ± ±2a20 + 2a30 a01 ± a11 + a21 (8) J = . ±2b20 + 2b30 b01 ± b11 + b21 It follows from J + = J − that (9) a20 = a11 = b20 = b11 = 0. If we assume that b30 = 0, then J + = J − yields a triangular matrix having the two charac- teristic roots λ1 = 2a30 , λ2 = b01 + b21 . Furthermore we take λ1 = λ2 = 0, because we want that the singular points (±1, 0) of the piecewise differential systems (6) are nilpotent, so we obtain (10) a30 = 0, b01 = −b21 . From (7), (9) and (10) the piecewise differential systems (6) become  !  y(a01 + a21 x2 + a02 y + a12 xy + a03 y 2 )       if y > 0, ẋ y(−b21 + b21 x2 + b02 y + b12 xy + b03 y 2 ) (11) = ! ẏ   y(a01 + a21 x2 − a02 y + a12 xy + a03 y 2 )   if y < 0.  y(−b21 + b21 x2 − b02 y + b12 xy + b03 y 2 ) Since the polynomials in (11) have a common factor y, the singular points (±1, 0) are not isolated singular points. Hence in order to make the singular points (±1, 0) isolated nilpotent singular points of system (11) we force that   0 0 J± = , 1 0 consequently b30 = 21 . Then systems (6) can be rewritten as (12)     − a21 y + a21 x2 y + a02 y 2 + a12 xy 2 + a03 y 3 = X + (x, y)       x x3 if y > 0,     − + − b 21 y + b 21 x 2 y + b 02 y 2 + b 12 xy 2 + b03 y 3 = Y + (x, y) ẋ 2 2 =   ẏ   − a21 y + a21 x2 y − a02 y 2 + a12 xy 2 + a03 y 3 = X − (x, y)       if y < 0.  3  − x + x − b21 y + b21 x2 y − b02 y 2 + b12 xy 2 + b03 y 3 = Y − (x, y)  2 2 NILPOTENT BI-CENTER IN PIECEWISE Z2 -EQUIVARIANT CUBIC 7 Next, let H + (x, y) be the Hamiltonian of the first system of systems (12). To find this Hamiltonian, we integrate X + (x, y) of (12) with respect to y and obtain Z H1+ (x, y) =f (x) + X + (x, y)dy (13) 1 1 1 1 1 =f (x) − a21 y 2 + a21 x2 y 2 + a02 y 3 + a12 xy 3 + a03 y 4 , 2 2 3 3 4 for some real polynomials f (x). And we integrate Y + (x, y) of (12) with respect to x and obtain Z + H2 (x, y) =g(y) − Y + (x, y)dx (14) 1 1 1 1 =g(y) + x2 − x4 + b21 xy − b21 x2 y − b02 xy 2 − b12 x2 y 2 − b03 xy 3 , 4 8 3 2 for some real polynomials g(y). Equating H1+ (x, y) to H2+ (x, y) we obtain (15) b12 = −a21 , a12 = −3b03 , b02 = b21 = 0, f (x) = x /4 − x /8 and g(y) = −a21 y /2 + a02 y /3 + a03 y 4 /4. 2 4 2 3 Then systems (12) become the piecewise Hamiltonian systems     − a21 y + a21 x2 y + a02 y 2 − 3b03 xy 2 + a03 y 3     if y > 0,   x x3     − +  − a21 xy 2 + b03 y 3 ẋ 2 2 (16) =   ẏ   − a21 y + a21 x2 y − a02 y 2 − 3b03 xy 2 + a03 y 3      if y < 0,  3  − x + x − a21 xy 2 + b03 y 3  2 2 where systems (16) have the Hamiltonian 1 1 1 1 1 1 (17) H(x, y)+ = x2 − x4 − a21 y 2 + a21 x2 y 2 + a02 y 3 − b03 xy 3 + a03 y 4 , 4 8 2 2 3 4 for the Hamiltonian system in y > 0, and the Hamiltonian 1 1 1 1 1 1 (18) H(x, y)− = x2 − x4 − a21 y 2 + a21 x2 y 2 − a02 y 3 − b03 xy 3 + a03 y 4 , 4 8 2 2 3 4 for the Hamiltonian system in y < 0, respectively. Introducing the transformation x → x + 1 into systems (16) we get (19)     2a21 xy + a21 x2 y + (a02 − 3b03 )y 2 + a12 xy 2 + a03 y 3 = Ψ+ (x, y)     3x2 x3  if y > 0,      x + + + a 21 y 2 − a21 xy 2 + b 03 y 3 = x + Φ + (x, y) ẋ 2 2 =   ẏ   2a21 xy + a21 x2 y − (a02 + 3b03 )y 2 + a12 xy 2 + a03 y 3 = Ψ− (x, y)       if y < 0,  2 3  x + 3x + x + a21 y 2 − a21 xy 2 + b03 y 3 = x + Φ− (x, y)  2 2 and so the singular point (1, 0) of systems (16) is moved to the origin of systems (19). Then we assume that X∞ ± f (y) = c± ky k k=2 are the unique solutions of the implicit function equations x + Φ± (x, y) = 0 in a neighborhood of the origin, respectively. In order to determine the local phase portraits of the nilpotent points (±1, 0) we write ∞ X Ψ± (f ± (y), y) = αk± y k , k=2 (20)   ∞ ∂Ψ± ∂Φ ± X + = βk± y k , ∂x ∂y (f ± (y),y) k=1 8 TING CHEN, SHIMIN LI AND JAUME LLIBRE where (21) βn± ≡ 0, α2± = ±a02 − 3b03 , α3± = a03 + 2a221 . For polynomial differential systems if α2 = α3 = · · · = αk−1 = 0 and αk 6= 0, then the multiplicity of the nilpotent singular point is exactly k, for more detail see [30]. It follows from Theorem 3.5 in [16] that if βn = 0 and αm 6= 0 this nilpotent singular point is a  a cusp if m = 2k,  (22) a saddle if m = 2k + 1 and αm > 0,  a center or a focus if m = 2k + 1 and α < 0. m Since the multiplicity of a nilpotent center or focus (i.e. of a monodromic singular point) of a differential system is an odd positive integer greater than one, it follows that the smallest multiplicity of (1, 0) must be 3 if the singular point (1, 0) is a nilpotent focus or a center in the first system of (16). For convenience, we will call this singular point a third-order singular point. More precisely, we have the following statement: The singular point (1, 0) of the first system of (16) is a monodromic critical point with multiplicity 3 if and only if α2+ = 0, α3+ < 0, namely, a02 = 3b03 , a03 + 2a221 < 0. Setting α3+ = −2 yields a03 = −2a221 − 2. Then we have that the singular point (1, 0) of the first system of (16) is monodromic. Therefore we obtain systems (3) and we have α2− = −6b03 , α3− = −2. If b03 = 0, i.e., α2− = 0, then the piecewise differential systems (3) are smooth. If b03 6= 0, i.e.,α2− 6= 0, then the singular point (1, 0) of the second differential system of (3) is a cusp. But the singular points (±1, 0) of the piecewise differential systems (3) cannot be monodromic when b03 > 0, so we only consider b03 < 0. In summary we have obtained the continuous piecewise differential system (3). Furthermore from Proposition 2.1 of [8] we have that the Hamiltonians of the first and second systems of (3) satisfy with H + (x, 0) ≡ H − (x, 0). Hence systems (3) have nilpotent bi-centers at (±1, 0). Remark that when α2+ 6= 0, i.e. a03 6= 3b03 systems (3) can also have nilpotent bi-centers at (±1, 0), but this case becomes more complicated we do not provide its analysis in this paper. 4. Global phase portraits of systems (3) Now we consider the finite singular points of systems (3). The singular points p1,2 = (±1, 0) are two centers, the origin p3 of systems (3) is also a singular point, whose Jacobian matrix is   0 −a21 (23) . − 21 0 From (23) we have that the origin is a saddle when a21 ≥ 0 (a nilpotent saddle when a21 = 0), or a center when a21 < 0. Now we need to study if there are additional singular points. Since systems (3) are symmetric with respect to the origin of coordinates p3 , we just need to study the phase portrait of the first system in (3). The Jacobian matrix of the first system of systems (3) at a finite singular point (x, y) is   y(2a21 x − 3b03 y) N1 (24) 1 2 2 , 2 (−1 + 3x − 2a21 y ) −y(2a21 x − 3b03 y) where N1 = −a21 + a21 x2 + 6b03 y − 6b03 xy − 6y 2 − 6a221 y 2 . NILPOTENT BI-CENTER IN PIECEWISE Z2 -EQUIVARIANT CUBIC 9 We claim that there are no finite singular points for the first system of system (3) whose linear part be identically zero. Indeed, we obtain that −1 + 3x2 − 2a21 y 2 and y(2a21 x − 3b03 y) have no common solutions, because the Gröbner basis for the polynomials ẋ, ẏ, −1 + 3x2 − 2a21 y 2 and y(2a21 x − 3b03 y) is 1. We again calculate the Gröbner basis for four polynomials ẋ, ẏ, y(2a21 x − 3b03 y) and N1 , then we obtain seven polynomials a21 y, b03 y 2 , y 3 , −a21 + a21 x2 + 6b03 y −6b03 xy −6y 2 , xy −b03 +b03 x+y, x(1+x)y 2 and (−1+x)x(1+x). It means that there are no other nilpotent singular points different from pk for k = 1, 2, 3, such these four polynomials be zero. Hence all the remaining finite singular points are hyperbolic, or semi-hyperbolic, or centers and by Theorems 2.15 and 2.19 of [16] the remaining finite singular points must be saddles or centers because the system is Hamiltonian. The explicit expressions of the finite singular points different from pk for k = 1, 2, 3, and their eigenvalues in terms of parameters a21 and b03 are complicated, it is hard to study their existence and their types. Thus we need to present more algebraic tools for solving this problem. From the first system in (3) we compute the Gröbner basis for ẋ and ẏ and we obtain eight polynomials, where the following two polynomials   y 2 3a21 b03 − 3a21 b03 x − 9b203 y + (2a21 + 9b203 )xy + 2b03 (3 + 4a221 )y 2 and  y 6a21 b03 + (−2a21 − 18b203 + 15a221 b203 )y + (18b03 + 12a221 b03 − 54a21 b303 )y 2 3 (25)  + (−4 − 4a221 + 36a21 b203 + 32a321 b203 + 27b403 )y 3 = y 3 f (y). are enough for our analysis. We note that polynomial (25) is not identically zero, because in order that it be identically zero we need that a21 = b03 = 0, but then the resultant reduces to −4y 6 6= 0. Now in order to study the number of the real roots of the polynomial f (y) we shall use the method of the discriminant sequence associated to f (y) developed in [42]. We associate to the polynomial (26) f (y) = a0 + a1 y + · · · + ak y k the (2k + 1) × (2k + 1) matrix   a0 a1 a2 ··· ak  0 ka0 (k − 1)a1 ··· ak−1     a0 a1 ··· ak−1 ak     na ··· 2ak−2 ak−1   0  M =  ··· ··· .   ··· ···     a0 a1 ··· ak     0 ka0 ··· ak−1  a0 a1 ak−1 ak We define dj as the determinant of the submatrix of M constructed with the first j rows and columns of the matrix M for j = 1, . . . , 2k + 1. Thus we have the sequence (27) {d1 , d2 , . . . , d2k+1 }. Consider the discriminant sequence {d2 , d4 , · · · , d2k } and the sequence of its signs [sign(d2 ), sign(d4 ), · · · , sign(d2k )], called sign list, where as usual the sign function is  −1  if x < 0, (28) sign(x) = 0 if x = 0,   1 if x > 0. For a sign list [s1 , s2 , · · · , sn ] of f (y) we define its revised sign list [l1 , l2 , · · · , ln ] as follows: 10 TING CHEN, SHIMIN LI AND JAUME LLIBRE 1. If sk 6= 0 we write lk = sk . 2. If subsection [si , si+1 , · · · , si+j ] of this sign list, which satisfies with si+1 = · · · = si+j−1 = 0 and si si+j 6= 0, we replace the subsection [si+1 , si+2 , · · · , si+j−1 ] with [−si , −si , si , si , −si , −si , si , si , −si , · · · ] keeping the number of terms. For convenient we denote by RSL and SL the revised sign list and the sign list of the discriminant sequence, respectively. Then the RSL [l1 , l2 , · · · , ln ] has no zeros between two nonzero members. From Theorems 2.1 and 3.3 of [42] we obtain the following two theorems. Theorem 4.1. Let f (y) be the polynomial (26) with real coefficient. If the number of the sign changes of the RSL [d2 , d4 , . . . , d2k ] is equal to m, and the number of nonzero elements of this RSL is equal to `, then the number of the distinct real roots of the polynomial f (y) is ` − 2m. Theorem 4.2. Let f (y) be the polynomial (26) with real coefficient such that f (0) 6= 0. If the number of the sign changes of the RSL [d1 d2 , d2 d3 , · · · , d2k d2k+1 ] is equal to m, and the number of nonzero elements of this RSL is equal to l, the number of the negative roots of the polynomial f (y) is l/2 − m. We separate the study of the roots of the polynomial f (y) of (25) in two cases. Case 1: The coefficient of the cubic term of f (y) in (25) is zero, i.e. N2 = −4 − 4a221 + 36a21 b203 + 32a321 b203 + 27b403 = 0. Then we have  q  2 (29) b203 = − 9a21 − 8a321 + (3 + 4a421 )3 . 27 Now we calculate the resultant of the coefficient of y 2 of f (y) in (25) with N2 with respect to the variable a21 and obtain 6912b203 (1 + 27b403 )3 6= 0. Thus the coefficient of y 2 in f (y) is nonzero when the coefficient of y 3 is zero. Multiply this quadratic coefficient and the constant term of f (y) we obtain  q  8 a21 − 9a21 − 8a321 + 27 + 108a221 + 144a421 + 64a621 × 3 (30)   q  2 3 + 2a221 − a21 − 9a21 − 8a321 + 27 + 108a221 + 144a421 + 64a621 ≤ 0. 3 √ Hence f (y) has at most one positive root. Actually if a21 = 0 we have b03 = − 2/(33/4 ), and f (y) has no positive roots. Case 2: The coefficient of the cubic term of f (y) in (25) is nonzero, i.e. N2 6= 0. Then finding the number of the positive roots of f (y) in (25) is equivalent to find the number of the negative roots of −f (−y). Now we shall compute the negative roots of the polynomial −f (−y) using Theorem 4.2. So we consider the sequence (31) {d1 d2 , d2 d3 , d3 d4 , d4 d5 , d5 d6 , d6 d7 } associated to −f (−y), and we have d1 = N2 , d2 = 3N22 , d3 = −6b03 (3 + 2a221 − 9a21 b203 )N22 , d4 = −6N22 N3 , d5 = −4N22 N4 , (32) d6 = −4(a21 + 9b203 + 6a221 b203 )2 N22 N5 , d7 = 24a21 b03 (a21 + 9b203 + 6a221 b203 )2 N22 N5 , NILPOTENT BI-CENTER IN PIECEWISE Z2 -EQUIVARIANT CUBIC 11 where N3 =8a21 + 8a321 − 36b203 − 204a221 b203 − 172a421 b203 − 54a21 b403 + 396a321 b403 + 480a521 b403 − 486b603 − 567a221 b603 , N4 = − 16a221 − 16a421 + 198a21 b203 + 852a321 b203 + 656a521 b203 + 162b403 (33) − 567a221 b403 − 5724a421 b403 − 5088a621 b403 + 2673a21 b603 − 486a321 b603 + 3456a521 b603 + 7200a721 b603 + 8748b803 + 5103a221 b803 − 4860a421 b803 , N5 =32a21 + 32a321 − 36b203 − 240a221 b203 − 208a421 b203 + 2052a21 b403 + 5040a321 b403 + 3000a521 b403 − 1944b603 − 2025a221 b603 . From Theorem 4.2 we obtain that the polynomial −f (−y) has three distinct negative roots if and only if the revised sign list of (31) is [1, 1, 1, 1, 1, 1] or [−1, −1, −1, −1, −1, −1], which we cannot be obtained varying the parameters a21 and b03 . Therefore the polynomial −f (−y) has at most two negative roots. Now we study the case when the polynomial −f (−y) has distinct negative roots. We denote by R[f (α), i] the i-th real root of the polynomial f (α) with respect to α, and these roots are ordered as follows R[f (α), i] < R[f (α), j] if and only if i < j. We describe the possible revised sign lists of the associated discriminant sequences as we show in Tables 1,2,3, when the poly- nomial −f (−y) has two negative roots, one negative root and no negative roots, respectively, where N6 = − 36 − 21a221 + 20a421 , N7 = − 108 − 27a221 + 99a421 + 5a621 , N8 = − 81 + 1026a221 + 3429a421 + 3498a621 + 1175a821 , N9 = − 648 + 837a221 + 3942a421 + 3468a621 + 1000a821 , (34) N10 = − 128490624 − 62227804500a221 − 515496116628a421 − 916466231925a621 + 210402679464a821 + 1653908444856a10 21 + 863216641008a12 14 16 21 − 432308074320a21 − 139867591104a21 + 313035878400a18 20 21 + 137815040000a21 , N11 = − 243 − 4536a221 − 9180a421 − 3168a621 + 4260a821 + 2540a10 21 . Table 1. The conditions in order that the revised sign list (RSL) of (31) has two distinct negative roots. RSL Conditions [1, 1, 1, −1, −1, −1] R[N6 , 1] ≈ −1.40204 < a21 < R[N7 , 1] ≈ −1.07347, b03 ≤ R[N4 , 1], or a21 = R[N7 , 1], b03 < R[N4 , 1], or R[N7 , 1] < a21 < 0, b03 < R[N2 , 1]; [1, 1, 1, 1, 1, −1] a21 ≤ R[N6 , 1], b03 < R[N2 , 1], or R[N6 , 1] < a21 < R[N7 , 1], R[N4 , 1] < b03 < R[N2 , 1]. In summary, the first system of the piecewise differential system (3) has at most two singular points different from pj for j = 1, 2, 3. Next we shall determine the local phase portraits of these additional finite singular points using the information provided by the infinite singular points. In the local chart U2 the first system of (3) becomes 1 u0 = (−4 − 4a221 − 8b03 u + 4a21 u2 − u4 + 6b03 v − 2a21 v 2 + u2 v 2 ), (35) 2 0 1 v = − v(2b03 − 2a21 u + u3 − uv 2 ). 2 12 TING CHEN, SHIMIN LI AND JAUME LLIBRE Table 2. The conditions in order that the revised sign list (RSL) of (31) has one distinct negative roots. RSL Conditions q 3 [1, −1, 1, 1, 1, 1] R[N11 , 2] < a21 ≤ 2 , R[N3 , 1] < b03 ≤ R[N3 , 2], q r 3 3+2a221 or a21 > 2 , R[N3 , 1] < b03 < − 9a21 ; q r 3+2a212 3 [1, 1, −1, 1, 1, 1] 2 < a21 ≤ R[N6 , 2], − 9a21 < b03 ≤ R[N4 , 1], r 3+2a2 21 or a21 > R[N6 , 2], − 9a21 < b03 ≤ R[N5 , 2]; [1, 1, 1, −1, 1, 1] R[N8 , 2] < a21 ≤ R[N9 , 2], R[N5 ,r 1] < b03 < R[N2 , 1], 3+2a2 21 < b or R[N9 , 2] < a21 < R[N10 , 2], − 9a21 03 < R[N2 , 1], q r 3+2a 2 3 21 < b or R[N10 , 2] ≤ a21 < 2, − 9a21 03 < R[N2 , 1], b03 6= R[N4 , 2], q 3 or 2 ≤ a21 ≤ R[N6 , 2], R[N4 , 2] < b03 < R[N2 , 1], or a21 > R[N6 , 2], R[N4 , 3] < b03 < R[N2 , 1]; q 3 [1, 1, 1, 1, −1, 1] R[N10 , 2] < a21 ≤ 2 , R[N4 , 1] < b03 < R[N4 , 2], q 3 or 2 < a21 ≤ R[N6 , 2], R[N3 , 2] < b03 < R[N4 , 2], or a21 > [N6 , 2], R[N3 , 2] < b03 < R[N4 , 3]; q r 3 3+2a2 21 [1, −1, −1, 1, 1, 1] a21 > R[N6 , 2], b03 ≤ R[N4 , 1], or a21 > 2 , b 03 = − 9a21 ; r 3+2a2 21 [1, 1, 1, −1, −1, 1] 0 < a21 ≤ R[N8 , 2], − 9a21 < b03 < R[N2 , 1], r 3+2a2 21 < b or R[N8 , 2] < a21 < R[N9 , 2], − 9a21 03 < R[N5 , 1], q 3 or R[N10 , 2] ≤ a21 < 2 , b03 = R[N 6 , 1], or R[N10 , 2] < a21 ≤ R[N6 , 2], b03 = R[N6 , 2], or a21 > R[N6 , 2], b03 = R[N6 , 3]; r 3+2a2 21 [1, −1, −1, −1, 1, 1] R[N9 , 2] < a21 < R[N11 , 2] = 1.20891, R[N5 , 1] < b03 ≤ − 9a21 , r 3+2a2 21 or a21 = R[N11 , 2], R[N5 , 1] < b03 ≤ − 9a21 , b03 6= R[N3 , 1], or R[N11 , 2] < a21 , R[N5 , 1] < b03 < R[N3 , 1],r q 3+2a2 3 21 or R[N11 , 2] < a21 < 2 , R[N3 , 2] < b03 ≤ − 9a21 ; q 3 [1, 1, −1, −1, −1, 1] 2 < a21 ≤ R[N6 , 2], R[N4 , 1] < b03 ≤ R[N3 , 2], or a21 > R[N6 , 2], R[N4 , 2] < b03 ≤ R[N3 , 2]; r 3+2a2 21 [1, −1, −1, −1, −1, 1] 0 < a21 < R[N9 , 2], b03 ≤ − 9a21 , r 3+2a2 21 or a21 = R[N9 , 2], b03 < − 9a21 , or R[N9 , 2] < a21 < R[N6 , 2], b03 < R[N5 , 1], or a21 > R[N6 , 2], R[N4 , 1] < b03 < R[N5 , 1]; r 3+2a2 21 [1, −1, −1, −1, 0, 0] a21 = R[N9 , 2], b03 = − 9a21 , or R[N9 , 2] < a21 , b03 = R[N5 , 1]; [−1, 1, 1, −1, −1, −1] R[N7 , 1] < a21 < 0, R[N2 , 1] < b03 < R[N4 , 1]; r 2a21 (1+a2 21 ) , [−1, 1, 1, 1, 1, −1] a21 ≤ R[N7 , 1], R[N2 , 1] < b03 < 0 and b03 6= 3 r 2a21 (1+a2 21 ) ; or R[N7 , 1] < a21 < 0, R[N4 , 1] < b03 < 0 and b03 6= 3 r 2a21 (1+a2 21 ) . [−1, 1, 1, 1, 0, 0] a21 < 0, b03 = 3 Clearly the origin of U2 is not a singular point. In U1 the first system of (3) has the form 1 u0 = (1 − 4a21 u2 + 8b03 u3 + 4u4 + 4a221 u4 − 6b03 u3 v − v 2 + 2a21 u2 v 2 ), (36) 2 v 0 =uv(−a21 + 3b03 u + 2u2 + 2a221 u2 − 3b03 uv + a21 v 2 ). The linear part of system (36) on v = 0 is   4uN12 −3b03 u3 (37) , 0 uN12 where N12 = −a21 + 3b03 u + 2u2 + 2a221 u2 . By computing the resultant of (38) g(u) = u0 |v=0 = 1 − 4a21 u2 + 8b03 u3 + (4 + 4a221 )u4 NILPOTENT BI-CENTER IN PIECEWISE Z2 -EQUIVARIANT CUBIC 13 Table 3. The conditions in order that the revised sign list (RSL) of (31) has no negative roots. RSL Conditions [−1, 1, −1, 1, 1, 1] 0 < a21 < R[N10 , 2], R[N3 , 1] < b03 ≤ R[N4 , 1], or R[N10 , 2] ≤ a21 < R[N11 , 2], R[N3 , 1] < b03 < R[N4 , 3], or R[N11 , 2] ≤ a21 < R[N6 , 2], R[N3 , 3] < b03 ≤ R[N4 , 3], or a21 > R[N6 , 2], R[N3 , 1] < b03 < R[N4 , 4]; [−1, 1, 1, −1, 1, 1] 0 < a21 ≤ R[N8 , 2], R[N5 , 1] < b03 < R[N3 , 1], or R[N10 , 2] ≤ a21 < R[N11 , 2], R[N2 , 1] < b03 < R[N3 , 1], or a21 ≥ R[N11 , 2], R[N2 , 1] < b03 < R[N3 , 3]; [−1, 1, −1, −1, 1, 1] 0 < a21 < R[N11 , 2], b03 = R[N3 , 1], or a21 ≥ R[N11 , 2], b03 = R[N3 , 3]; [−1, 1, 1, −1, −1, 1] 0 < a21 < R[N8 , 2], R[N2 , 1] < b03 < R[N5 , 1]; [−1, 1, −1, −1, −1, 1] 0 < a21 < R[N10 , 2], R[N4 , 1] < b03 < 0, or R[N10 , 2] ≤ a21 ≤ R[N6 , 2], R[N4 , 3] < b03 < 0, or a21 > R[N6 , 2], R[N4 , 4] < b03 < 0, [−1, 1, 1, −1, 0, 0] 0 < a21 < R[N8 , 2], b03 = R[N5 , 1]. and uN12 with respect to the variable u we obtain the polynomial −(1 + a221 )N2 . And the possible singular points in U1 are nilpotent when N2 = 0, or nodes when N2 6= 0. Now we shall determine the local phase portraits of the infinite singular points in the chart U1 . We need to find the real solutions in g(u) = 0. But we will be able to determine the number and the type of the remaining infinite singular points using Theorems 4.2 and 2.4. Then we do not need to calculate explicitly the coordinates of these singular points. Remark 4.3. When u < 0 the infinite singular points of the first system of (3) in U1 are virtual points, but there are corresponding infinite singular points in V1 with u > 0 by the symmetry. And there are no infinite singular points in U2 and V2 in our cases. Hence we can study all real solutions in g(u) = 0 to study the infinite singular points. We compute the sequence {de2 , de4 , de6 , de8 } of g(u) from (38), and have de2 = 64(1 + a221 )2 , de4 = 1024(1 + a221 )2 (2a21 + 2a321 − 3b203 ), (39) de6 = −16384(1 + a221 )2 (2a21 + 2a321 + 3b203 + a221 b203 ), de8 = −65536(1 + a221 )2 N2 . We cannot find the parameter values such that the corresponding RSL be [1, 1, 1, 1], [−1, −1, −1, −1], [1, 1, 1, −1], [1, −1, −1, −1], [−1, −1, −1, 1], [1, −1, −1, −1], [−1, −1, −1, 0] or [1, 1, 1, 0], but we know that there are at most two distinct positive roots of (38), i.e., there are at most two infinite singular points in U1 . (a) When the polynomial g(u) has two distinct roots, we obtain that the possible RSL of g(u) is [1, 1, −1, −1], whose condition is b03 < R[N2 , 1], i.e., q p 2(−9a21 − 8a321 + (3 + 4a221 )3 ) b03 < − √ . (3 3) Since N2 6= 0, from (37) we have that the two remaining infinite singular points are two nodes in U1 . Since the piecewise differential systems (3) are symmetric with respect to the origin, systems (3) have two corresponding infinite singular points in V1 . On the other hand, systems (3) are continuous becasue (f + (x, 0), g + (x, 0)) = (f − (x, 0), g − (x, 0)) in these systems. (a.1) If a21 < 0, the origin p3 is a center. Hence on the Poincaré sphere the sum of the indices of the known singular points is 10. By Theorem 2.4, the sum of the indices of the remaining finite singular points must be −8. From the previous analysis, systems (3) have at most two finite singular points in y > 0, which are different from pj for j = 1, 2, 3. Due to the symmetry with respect to the origin of coordinates the remaining finite singular points are four saddles pj for j = 4, 5, 6, 7, where the two saddles p4 and p6 are in y > 0, and two saddles p5 and p7 are in y < 0. From (35) and (36) we have u0 |u=0 = −2(1 + a221 ) < 0 in U2 and u0 |u=0 = 2 > 0 in 14 TING CHEN, SHIMIN LI AND JAUME LLIBRE A2 A2 p6 p4 p4 A1 A1 p2 p3 p1 p2 p1 B1 B1 p3 p5 p7 p5 B2 B2 (a) (b) Figure 2. The local phase portraits at all finite and infinite singular points of the piecewise differential system (3) when g(u) has two distinct roots. U1 . Then we obtain that the local phase portraits at these singular points in the Poincaré disc are shown in Figure 2(a). Since the finite singular points of the piecewise differential systems (3) are saddles or centers there must be one saddle on the boundary of period annulus of the center. (a.1.1) Assume that saddle p6 is on the boundary of period annulus of center p3 , by the symmetry the saddle p7 must be on the boundary of period annulus of the center p3 . If one repelling and one attracting separatrices of saddle p4 of systems (3) connect with saddle p5 and with the infinite singular point A1 and A2 of U1 (see Figure 2(a)), respectively, then the saddles p6 and p7 are also on the boundary of the period annulus of the center p1 . By the symmetry we have that this phase portrait in the Poincaré disc is topologically equivalent to the phase portrait 1.1 of Figure 1, which for instance can be realized when a21 = −1 and b03 = −2. (a.1.2) If the saddle p4 is on the boundary of period annulus of the center p1 , creating a center-loop. By the symmetry the phase portrait of the piecewise differential systems (3) is topologically equivalent to the phase portrait 1.2 of Figure 1. This phase portrait for instance can be realized when a21 = −1 and b03 = −5. (a.1.3) From the phase portraits 1.1 to 1.2 it follows by the continuity of the phase portraits with respect to the parameters that there must exist one phase portrait that the saddles p4 and p6 are on the boundary of period annulus of the center p1 . We have the phase portrait 1.3 of Figure 1. (a.2) If a21 ≥ 0 then the origin p3 is a saddle. Hence on the Poincaré sphere the sum of the indices of the known singular points is 6. By Theorem 2.4 the sum of the indices of the remaining finite singular points must be −2. Hence the finite singular points other than pj for j = 1, 2, 3, can be either two saddles, or four saddles and two centers. From the previous analysis we know that when a12 ≥ 0 the piecewise differential systems (3) have at most one finite singular point in y > 0, see Tables 2 and 3. Hence the remaining finite singular points are two saddles p4 and p5 , where p4 is in y > 0 and p5 is in y < 0. Then we obtain that the local phase portraits at these singular points in the Poincaré disc are shown in Figure 2(b). (a.2.1) If only the saddle p3 is on the boundary of the period annulus of the center p1 , taking into account the symmetry p3 is also on the boundary of the period annulus of p2 , creating one eight-figure loop. Then one repelling and one attracting separatrices of the saddle p4 of system (3) connect with the saddle p5 and with the infinite singular point A1 and A2 of the local chart U1 (see Figure 2(b)), respectively. Hence we have that this phase portrait in the Poincaré disc is topologically equivalent to the phase portrait 1.4 of Figure 1, which for instance can be realized when a21 = 1 and b03 = −0.4. (a.2.2) The saddle p4 on the boundary of period annulus of the center p1 creates a center- loop. Then one repelling and one attracting separatrices of the saddle p3 connect with the infinite singular point A1 and A2 of the local chart U1 . By the symmetry the phase portrait NILPOTENT BI-CENTER IN PIECEWISE Z2 -EQUIVARIANT CUBIC 15 of the piecewise differential systems (3) is topologically equivalent to the phase portrait 1.5 of Figure 1. This phase portrait can be realized by taking a21 = 1 and b03 = −1. (a.2.3) From the phase portraits 1.4 to 1.5 it follows by the continuity of the phase portraits with respect to the parameters that there must exist one phase portrait that the saddles p3 and p4 are on the boundary of the period annulus of the center p1 . Then this phase portrait is topologically equivalent to the phase portrait 1.6 of Figure 1. (b) When the polynomial g(u) has one distinct negative root, it shows that the possible RSL of g(u) is [1, 1, −1, 0], whose condition is b03 = R[N2 , 1]. Since N2 = 0, from (37) we have that the remaining infinite singular point in U1 is nilpotent. From the previous analysis we know that the infinite singular point are nilpotent. The linear part of the two systems of (3) at an infinite singular point (u, 0) of the local chart U1 is     4u 2 a221 + 1 u2 + 3b03 u − a21 ∓3b  03 u 3  , 0 u 2 a221 + 1 u2 + 3b03 u − a21 where in ∓3b03 u3 we have the minus sign for the first system of (3) and the positive sign for the second system. In order that the point (u, 0) ∈ U1 be a nilpotent infinite singular point the two following equations must be satisfied:   (40) 4u 2 a221 + 1 u2 + 3b03 u − a21 = 0, 1 − 4a21 u2 + 8b03 u3 + 4u4 + 4a221 u4 = 0. The second equation comes from imposing that (u, 0) be an infinite singular point in the local chart U1 for both systems forming the piecewise differential system (3). This system has only one real solution s s  q ! 3 1 2 2 (u, b03 ) = p , p −2a21 + 4a221 + 3a21 − 3 . 2 4a221 + 3 + 2a21 3 3 4a221 + 3 + 3a21 Then the nilpotent infinite singular point is s ! 3 (u, 0) = p ,0 , 2 4a221 + 3 + 2a21 since its u-coordinate is positive it belongs to the first system of (3) defined in y > 0. Applying Theorem 3.5 of [16] such a nilpotent infinite singular point is formed by an elliptic sector and a hyperbolic sector, the hyperbolic sector has its two separatrices contained in the straight line of the infinity and its elliptic sector is outside the Poincaré disc, so it does not appear in the phase portrait of our piecewise differential system. (b.1): a21 < 0. Then the origin is a center. From the previous analysis the piecewise differential systems (3) have at most two finite singular points, which are different from the three centers pj for j = 1, 2, 3. (b.1.1): We can assume that the remaining finite singular points are two saddles p4 and p5 , where the saddle p4 is in y > 0 and the saddle p5 is in y < 0. These piecewise differential systems have the phase portrait p 1.7 of Figure 1. For instance the piecewise differential system 1 √ √  (3) with a21 = −1 and b03 = − 9 1 + 7 5 + 7 realizes such a phase portrait. (b.1.2): Assume the remaining finite singular points are two centers p4 and p5 , where p4 is in y > 0 and p5 is in y < 0. Then on the Poincaré sphere the sum of the indices of the known singular points is 10. By Theorem 2.4 the sum of the indices of the remaining two nilpotent infinite singular points must be −8. But the nilpotent singular point formed by one elliptic and one hyperbolic sector has index zero. In summary, it follows that the remaining finite singular points p4 and p5 cannot be two centers. (b.2): a21 ≥ 0. Then the origin p3 is a saddle. Recall that p1 and p2 are centers, and the two nilpotent infinite singular points inside the Poincaré disc have a hyperbolic sector with their two separatrices contained in the straight line of the infinity. Then these piecewise differential 16 TING CHEN, SHIMIN LI AND JAUME LLIBRE systems have the phase portrait 1.8 of Figure 1. For instance the piecewise differential system √ q √  (3) with a21 = 1 and b03 = 7−5 2/(3 1 + 7 )/3 realizes such a phase portrait. (c) When the polynomial −g(−u) has no distinct negative roots, its possible RSL of g(u) are described in Table 4, i.e. b03 > R[N2 , 1]. In this case we have that there are no infinite singular points in U1 with u > 0. Table 4. The conditions of the revised sign lists of g(u) in (38) without real roots. RSL Conditionsr −2a21 (1+a221 ) < b [1, −1, 1, 1] a21 < 0, − 03 < 0; 3+a2 21 r 2a21 (1+a2 21 ) , [1, 1, −1, 1] a21 ≤ 0, R[N2 , 1] < b03 < − 3 or a21 > 0, R[N2 , 1] < b03 < 0; r r 2a21 (1+a2 21 ) −2a21 (1+a221 ) . [1, −1, −1, 1] a21 < 0, − < b03 < − 3 3+a2 21 (c.1) If a21 < 0 then the origin p3 is a center. Hence on the Poincaré sphere the sum of the indices of the known singular points is 6. By Theorem 2.4 the sum of the indices of the remaining finite singular points must be −4. From the previous analysis, when a21 < 0 the piecewise differential systems (3) have at most two finite singular points in y > 0, which are different from the p1 , p2 and the origin. By the symmetry the remaining finite singular points are two saddles, where one saddle is in y > 0 and the other is in y < 0. Similarly to case (b.1.1) we obtain the phase portrait 1.9 of Figure 1, which is achieved when a21 = −1 and b03 = −1. (c.2) If a21 ≥ 0 then the origin p3 is a saddle. And we know that when a21 ≥ 0 the piecewise differential systems (3) have at most one finite singular point in y > 0, which is a center or a saddle. In fact, by the symmetry and the sum of the indices we obtain that the piecewise differential systems (3) have no other finite singular points different from the pj , j = 1, 2, 3. Similarly to case (b.1.2) we obtain the phase portrait 1.10 of Figure 1, which can be realized when a21 = 1 and b03 = −0.2. 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Methods Nonlinear Anal. 8 (1996), 335–342. 1 School of Statistics and Mathematics, Guangdong University of Finance and Economics, Guangzhou, 510320, PR China 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain Email address: chenting0715@126.com (T.Chen), Email address: lism1983@126.com (S.Li), Email address: jllibre@mat.uab.cat (J.Llibre). ∗ Corresponding author