This is a preprint of: “Nilpotent center in continuous piecewise quadratic polynomial Hamiltonian vector fields”, Ting Chen, Jaume Llibre, Internat. J. Bifur. Chaos Appl. Sci. Engrg., vol. 32(8), 2250116:1–23, 2022. DOI: [10.1142/S0218127422501164] Nilpotent center in continuous piecewise quadratic polynomial Hamiltonian vector fields Ting Chen School of Statistics and Mathematics, Guangdong University of Finance and Economics, Guangzhou, 510320, P.R. China, chenting0715@126.com Jaume Llibre Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain jllibre@mat.uab.cat In this paper we study the global dynamics of continuous piecewise quadratic Hamiltonian systems separated by the straight line x = 0, where these kinds of systems have a nilpotent center at (0, 0), which comes from the combination of two cusps of both Hamiltonian systems. By the Poincaré compactification we classify the global phase portraits of these systems. We must mention that it is extremely rare to find works studying the center-focus problem in piecewise smooth systems with non-elementary singular points as we did here. Keywords: Nilpotent, Bi-center, Hamiltonian, Phase portrait. 1. Introduction and statement of the main results From the works of Poincaré [Poincaré, 1881] and Dulac [Dulac, 1908] the center-focus problem, i.e the problem of distinguishing when a singular point is either a focus or a center, has been one of main problems in the qualitative theory of the planar differential equations. In the planar polynomial vector fields a singular point is either elementary or non-elementary. The linear part of the elementary singular point has at least one nonzero eigenvalue, and the non-elementary one has two zero eigenvalues. And a non-elementary singular point is called nilpotent if its linear part is not identically zero, otherwise is called degenerate. Moreover, if a planar polynomial differential equation has a linear type center, a nilpotent center and a degenerate center at the origin, after affine change of time and variables, this differential equation can be written as     (−y, x)   (ẋ, ẏ) = (y, 0) + (F (x, y), G(x, y)), (1)   (0, 0)   respectively, where F (x, y) and G(x, y) are real polynomials without constant and linear terms. The center-focus problem in the quadratic polynomial differential equations has been solved see [Artés & Llibre, 1994; Bautin, 1952; Kapteyn, 1911, 1912; Schlomiuk, 1993; Żola̧dek, 1994]. For the cubic polyno- 1 2 mial differential equations there are partial works in the center-focus problem, see [Aziz, Llibre & Pantazi, 2014; Cima & Llibre, 1990; Malkin, 1964; Vulpe & Sibirskii, 1988; Żola̧dek, 1994, 1996] for an update of these works. Recently, Colak el at. [Colak et al., 2014] studied the phase portraits of some cubic Hamil- tonian differential equations with a linear type center at the origin. The authors of [Chen et al., 2020] investigated the Z2 -equivariant linear type bi-center problem for cubic polynomial Hamiltonian systems. It is well known that the center-focus problem for non-elementary singular points becomes more d- ifficult. Computationally efficient methods have been developed to find the center conditions for planar systems with nilpotent singular points, see [Garcia et al., 2016; Liu & Li, 2015, 2011; Stróżzyna et al., 2012]. Note that quadratic polynomial equations do not have nilpotent centers, i.e. the simplest nilpotent centers must appear in cubic differential equations [Nusse & Yorke , 1995]. The phase portraits of some cubic Hamiltonian vector fields with a nilpotent center at the origin were classified in [Colak et al., 2014]. Li el at. [Li et al., 2018] study the bi-center conditions for a class of Z2 -equivariant cubic differential equations with two nilpotent singular points. In recent years an increasing interest appears for studying the qualitative theory of the non-smooth differential equations, mainly due to the fact that these equations can analyze some real phenomena more accurately, see for instance [Andronov et al., 1966; Banerjee & Verghese, 2001; Filippov, 1988; Lv et al., 2019]. In this paper we will study the following piecewise smooth systems, ( ) (ẋ, ẏ) = F ± (x, y), G± (x, y) , ±s(x, y) ≥ 0, (2) where (F + (x, y), G+ (x, y)) and (F − (x, y), G− (x, y)) are smooth vector fields, and they are separated by the discontinuity curve Γ = s−1 (0). The center-focus problem in the piecewise smooth systems becomes much more complicated than that in the smooth systems. Gasull and Torregrosa [Gasull & Torregrosa, 2003] developed an efficient method for computing the Lyapunov constants of piecewise smooth systems (2), which can be used to solve the center-focus problem of piecewise smooth systems with linear type singular points. By computing the Lyapunov constants, the authors of [Chen & Zhang, 2012; Tian et al., 2015] gave a complete classification on the linear type center conditions of the origin in several classes of Bautin switching systems. There are some results for determining the linear type center conditions in cubic piecewise smooth systems, see [Coll et al., 1999; Guo et al., 2019]. As far as we known there is extremely rare to find works for studying the center-focus problem in piecewise smooth systems with non-elementary singular points, see [Chen et al., 2021, 2018]. This work is more challenging compared to the smooth systems. In this paper using the Poincaré compactification we will study the global dynamics of a class of piecewise quadratic Hamiltonian systems with a nilpotent type center at (0, 0) which comes from the combination of two cusps of both Hamiltonian systems, described by ( 2 2 )   y + a20 x − 2b 02 xy + a 02 y ( )    , for x ≥ 0, ẋ − x2 − 2a20 xy + b02 y 2 = ( ) (3) ẏ   y + A x 2 − 2b xy + a y 2   20 02 02 , for x ≤ 0,  x2 − 2A20 xy + b02 y 2 where x = 0 is the discontinuity line, a20 , a02 , b02 and A20 are real parameters. Note that systems (3) are continuous on x = 0. The Hamiltonian functions of the right system and the left one of systems (3) are 1 1 a02 3 H(x, y)+ = x3 + y 2 + a20 x2 y − b02 xy 2 + y (4) 3 2 3 and 1 1 a02 3 H(x, y)− = − x3 + y 2 + A20 x2 y − b02 xy 2 + y , (5) 3 2 3 respectively. 3 1.1 S=5, R=1 1.2 S=11, R=3 1.3 S=17, R=5 1.4 S=18, R=6 1.5 S=17, R=5 1.6 S=18, R=6 1.7 S=19, R=5 1.8 S=20, R=6 1.9 S=27, R=9 1.10 S=26, R=8 1.11 S=26, R=8 1.12 S=25, R=7 1.13 S=14, R=5 1.14 S=13, R=4 1.15 S=12, R=4 1.16 S=14, R=4 1.17 S=21, R=7 1.18 S=20, R=6 1.19 S=21, R=7 1.20 S=20, R=6 1.21 S=9, R=3 1.22 S=13, R=3 1.23 S=16, R=5 1.24 S=15, R=4 Fig. 1. Topological phase portraits of cases 1.1-1.24 of Proposition 1. 4 1.25 S=23, R=8 1.26 S=23, R=8 1.27 S=22, R=7 1.28 S=22, R=7 1.29 S=22, R=7 1.30 S=18, R=6 1.31 S=15, R=5 1.32 S=25, R=9 1.33 S=24, R=8 1.34 S=25, R=9 1.35 S=25, R=9 1.36 S=24, R=8 1.37 S=24, R=8 1.38 S=24, R=8 Fig. 2. Topological phase portraits of cases 1.25-1.38 of Proposition 1. We denote by S and R the number of separatrices and of canonical regions of a phase portrait in the Poincaré disc D2 , respectively. It is known that a separatrix of a polynomial differential system in the Poincaré disc is an orbit such that the orbits in one side of this orbit have different α- or ω-limit than the orbits in the other side of this orbit in any small neighborhood of it. And the separatrices include all the infinite orbits, all the finite singular points, the two orbits at the boundary of the hyperbolic sectors of the finite and infinite singular points, the limit cycles. If Σ denotes the closed set of all separatrices in the Poincaré disc D2 , and the components of D2 \ Σ are called the canonical regions. For more details on separatrices and canonical regions see Chapter 1 of [Dumortier et al., 2006]. Our main result is the following one. Theorem 1. In the Poincaré disc the phase portraits of systems (3) with a nilpotent center at (0, 0) are topologically equivalent to the following 38 phase portraits of Figures 1 and 2. In Section 2 we provide a brief introduction to the Poincaré compactification and some basic results that we shall need for proving Proposition 1. In Section 3 we show how to obtain the continuous piecewise quadratic Hamiltonian systems (3). In Section 4 we characterize the global phase portraits of systems (3) in the Poincaré disc. 5 2. Preliminaries 2.1. Poincaré compactification In order to study the phase portrait of the infinite singular points of systems (3) we use the Poincaré compactification, see more details in Chapter 5 of [Dumortier et al., 2006]. In fact this compactification identifies the plane R2 with the interior of the closed unit disc D2 centered at the origin of coordinates of the plane, and extends analytically the differential systems to its boundary, usually called the circle of the infinity. The singular points in the interior of the disc D2 are called finite singular points, and the ones on the boundary of D2 are called infinite singular points. In the planar R2 we describe the Poincaré compactification for the piecewise polynomial differential systems ( ) (x˙1 , x˙2 ) = X ± (x1 , x2 ), Y ± (x1 , x2 ) , ±x1 ≥ 0, (6) whose associated polynomial vector fields are P + = (X + (x1 , x2 ), Y + (x1 , x2 )) and P − = (X − (x1 , x2 ), Y − (x1 , x2 )), respectively. The degree of P + is the maximum of the degrees of the polyno- mials X + and Y + , in a similar way is defined the degree of P − . The degrees of P + and P − are defined as d+ and d− , respectively. For studying the neighborhood of the boundary of the disc D2 , i.e the neighborhood of the infinity of R , we use four local charts Ui = {(x1 , x2 ) ∈ D2 : xi > 0} and Vi = {(x1 , x2 ) ∈ D2 : xi < 0}, for i = 1, 2, 2 with the corresponding diffeomorphisms ϕi : U i → R 2 , ψi : Vi → R2 , (7) defined by ϕi (x1 , x2 ) = (x2 /x1 , 1/x1 ) = (u, v) and ψi (x1 , x2 ) = (x1 /x2 , 1/x2 ) = (u, v). Thus the coordinates (u, v) play different roles in the distinct local charts. The expressions for the corresponding vector fields (6) in the local charts U1 and V1 are given by ( [ ( ) ( )] ( )) d+ + 1 u + 1 u d+ +1 + 1 u (u̇, v̇) = v − uX , +Y , , −v X , (8) v v v v v v and ( [ ( ) ( )] ( )) d− −1 d− − 1 u − 1 u d− d− +1 − 1 u (u̇, v̇) = (−1) v − uX , +Y , , (−1) v X , (9) v v v v v v respectively. And the expressions of vector fields (6) in the local chart U2 with u ≥ 0 and u ≤ 0 are given by ( [ ( ) ( )] ( )) d+ + u 1 + u 1 d+ +1 + u 1 (u̇, v̇) = v X , − uY , , −v Y , , (10) v v v v v v and ( [ ( ) ( )] ( )) d− − u 1 − u 1 d− +1 − u 1 (u̇, v̇) = v X , − uY , , −v Y , , (11) v v v v v v respectively. The expressions of the vector fields (6) in the local chart V2 with u ≥ 0 and u ≤ 0 are + − equal to the expressions in the local chart U2 multiplied by (−1)d −1 and (−1)d −1 , respectively. In fact it is sufficient to study the infinite singular points in U1 |v=0 and V1 |v=0 , and at the origin of U2 and V2 for studying all the infinite singular points of systems (6). Note that when the origin of U2 is an infinite singular point it comes from the combination of the two origins of systems (10) and (11). 2.2. Topological skeleton The union of the set of all separatrices with an orbit of each canonical region form the separatix skeleton. We say that the separatrix skeletons S1 and S2 of the flows (D2 , ϕ1 ) and (D2 , ϕ2 ) are topologically equivalent if there exists a homeomorphism h : S1 → S2 which sends orbits to orbits reversing or preserving the direction of all orbits. In a similar way it is defined the topological equivalence between the flows (D2 , ϕ1 ) 6 and (D2 , ϕ2 ). From [Markus, 1954; Neumann, 1975; Peixoto, 1973] we have the following theorem which 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 [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 topological equivalent. 2.3. Topological index Then we present two important theorems on the topological index of a singular point in differential systems on surfaces, which are extremely useful tools to analyze the type of the singular points, for more details see Chapter 6 of [Dumortier et al., 2006]. Theorem 3. We denote by p an isolated singular 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 singular point p equals 1 + (e − h)/2. Corollary 2.1. The topological indices of a center, a cusp, a saddle and a node equal 1, 0, −1 and 1, respectively. Theorem 4. For any continuous vector field with finitely many singular points, the sum of their topological indices is 2. 3. Obtaining systems (3) Without loss of generality the piecewise quadratic Hamiltonian systems (3) are obtained from the more general quadratic differential systems having a nilpotent singular point at the origin of coordinates, i.e. ( 2 2 + )   y + a 20 x + a 11 xy + a02 y = F (x, y) ( )    , for x ≥ 0, ẋ b20 x2 + b11 xy + b02 y 2 = G+ (x, y) = ( ) (12) ẏ   y + A20 x2 + A11 xy + A02 y 2 = F − (x, y)   , for x ≤ 0.  B20 x2 + B11 xy + B02 y 2 = G− (x, y) Let H + (x, y) be the cubic Hamiltonian of the right system of systems (12). Integrating G+ (x, y) of (12) with respect to x and we have ∫ + H1 (x, y) =h1 (y) − G+ (x, y)dx (13) b20 3 b11 2 2 =h1 (y) − x − x y − b02 xy , 3 2 for some real polynomials h1 . And integrating F + (x, y) of (12) with respect to y and we have ∫ H2+ (x, y) =h2 (x) + F + (x, y)dy (14) 1 a11 2 a02 3 =h2 (x) + y 2 + a20 x2 y + xy + y , 2 2 3 for some real polynomials h2 . We equate H1+ (x, y) to H2+ (x, y) and have a11 = −2b02 , b11 = −2a20 , (15) h1 (y) = y 2 /2 + a02 y 3 /3 and h2 (x) = −b20 x3 /3. Similar to the right system of (12), we have A11 = −2B02 , B11 = −2A20 , (16) from the left one of (12). 7 Then systems (12) become the piecewise quadratic Hamiltonian systems ( )   y + a20 x2 − 2b02 xy + a02 y 2 = y + Φ+ (x, y) ( )    b20 x2 − 2a20 xy + b02 y 2 = Ψ+ (x, y) , for x ≥ 0, ẋ = ( ) (17) ẏ   y + A20 x2 − 2B02 xy + A02 y 2 = y + Φ− (x, y)   , for x ≤ 0.  B20 x2 − 2A20 xy + B02 y 2 = Ψ− (x, y) For analyzing the origin of the right and the left systems of (17), we define ∞ ∑ ± f (x) = c± kx k k=2 as the unique solution of y + Φ± (x, y) = 0 in a neighborhood of the origin, respectively. And assuming that ∞ ∑ Ψ± (x, f ± (x)) = αk± xk . (18) k=2 It follows from Theorem 3.5 of [Dumortier et al., 2006] that when αm ̸= 0 and  {  m = 2k + 1 αm < 0, then the origin is a center or a f ocus,  αm > 0, then the origin is a saddle, (19)   m = 2k, then the origin is a cusp. We note that a20 and A20 cannot be zero, otherwise the positive y-axis or the negative one are invariant and consequently the origin of the piecewise differential systems cannot be center. From the above results we have the following for the right system and the left one of the piecewise Hamiltonian systems (17): α2+ = b20 , α3+ = 2a220 , (20) α2− = B20 , α3− = 2A220 . In short, if α2± ̸= 0 then the corresponding Hamiltonian system of the continuous piecewise differential systems (17) has a cusp at the origin. But if α2± = 0, since α3± > 0, then the corresponding Hamiltonian system has a saddle, and then the continuous piecewise differential systems (17) cannot have a center at the origin. So we must assume that α2± ̸= 0. In summary, the piecewise differential systems (17) only can have a center at the origin if both Hamil- tonian systems have a cusp at the origin. Therefore, assume that b20 ̸= 0 and B20 ̸= 0 and consequently α2+ ̸= 0 and α2− ̸= 0. If we have a periodic orbit around the origin of the piecewise differential systems (17) which intersects the straight line x = 0 in the points (0, y1 ) and (0, y2 ) with y1 < y2 and both different from zero, then y1 and y2 must be solutions of H + (0, y1 ) − H + (0, y2 ) = (y1 − y2 )e1 /6 = 0 and H − (0, y1 ) − H − (0, y2 ) = (y1 − y2 )e2 /6 = 0, where e1 = 3y1 + 2a02 y12 + 3y2 + 2a02 y1 y2 + 2a02 y22 , e2 = 3y1 + 2A02 y12 + 3y2 + 2A02 y1 y2 + 2A02 y22 . We calculate the Gröbner basis for the two polynomials e1 and e2 , and obtain three polynomials (a02 − A02 )y22 , (a02 − A02 )(y1 + y2 ) and 3y1 + 2A02 y12 + 3y2 + 2A02 y1 y2 + 2A02 y22 . Note that if A02 = a02 then we have a continuum of solutions for y1 and y2 , and consequently we can have a continuum of periodic orbits, which allow the existence of centers surrounding the origin. Therefore in what follows A02 = a02 . Note that if A02 ̸= a02 , then y2 = 0 and we cannot have periodic orbits, because they cannot pass through the singular point of the piecewise differential system localized at the origin of coordinates. In order that the piecewise differential systems (17) be continuous on x = 0, i.e., Ψ+ (x, y)|x=0 = − Ψ (x, y)|x=0 we obtain B02 = b02 . Otherwise there will be more complex bifurcations. Hence with the two conditions A02 = a02 and B02 = b02 we have that the continuous piecewise quadratic Hamiltonian systems (17) depends on six parameters. This number of parameters is very big in order to study all the global 8 phase portraits of these systems. So in this paper we restrict our attention to the subfamily of continuous piecewise quadratic Hamiltonian systems (3) which have b20 = −1 and B20 = 1. 4. Global phase portraits of systems (3) Since the right system of the continuous piecewise quadratic Hamiltonian systems (3) becomes the left system of systems (3) throngh the transformation (x, y, t, a20 , a02 , b02 ) → (−x, −y, t, −A20 , −a02 , −b02 ), the right and left systems have the same topological phase portraits. So we only present the analysis of the right system of (3). Furthermore we can assume a02 ≥ 0 because the right system of Hamiltonian systems (3) is invariant under the transformation (x, y, t, a20 , a02 , b02 ) → (x, −y, −t, −a20 , −a02 , b02 ). Thus we study these phase portraits in two different cases a02 = 0 and a02 > 0. 4.1. The phase portraits when a02 = 0 The continuous piecewise quadratic Hamiltonian systems (3) become ( )   y + a20 x2 − 2b02 xy ( )    , for x ≥ 0, ẋ − x2 − 2a20 xy + b02 y 2 = ( ) (21) ẏ   y + A20 x2 − 2b02 xy   , for x ≤ 0.  x2 − 2A20 xy + b02 y 2 Then we consider the infinite singular points of the right system of systems (21) using the Poincaré compactification. Since the infinite singular points at U1 |v=0 are complicated, we first consider the origin in the local chart U2 . In U2 the right system of systems (21) writes u̇ = −3b02 u + v + 3a20 u2 + u3 , v̇ = −v(b02 − 2a20 u − u2 ). (22) The origin O1 of system (22) is a singular point, whose linear part has two eigenvalues −b02 and −3b02 . Hence, the origin is an attracting node when b02 > 0, or a repelling node when b02 < 0, or a nilpotent singularity point when b02 = 0. Furthermore applying Theorem 3.5 of [Dumortier et al., 2006] we obtain that the nilpotent point at the origin is a repelling node when a20 = b02 = 0, or of type E-H (i.e. the local phase portrait of this singular point is formed by one elliptic sector and one parabolic sector) when b02 = 0 and a20 ̸= 0. Since the degree of systems (21) is 2, the flow in a neighborhood of the origin O2 of V2 has the opposite sense with respect to the flow one in U2 . Now we consider the infinite singular points in U1 |v=0 . In the local chart U1 the right system of systems (21) has the form u̇ = −1 − 3a20 u + 3b02 u2 − u2 v, v̇ = −v(a20 − 2b02 u + uv). (23) 4.1.1. Subcase b02 = 0. System (23) has no infinite singular points when a20 = 0, and it has one infinite singular point (A1 , 0) = (−1/(3a20 ), 0), which is an attracting node or a repelling node, when a20 > 0 or a20 < 0, respectively. We consider the finite singular points of the right system of (21). If a20 = 0 the right system of (21) has only one finite singular point at the origin. We obtain that the phase portrait with all singular points in the right half Poincaré disc is shown in Figure 3(a). If a20 ̸= 0 the right system of (21) has two finite singular points (0, 0) and p1 = (1/(2a220 ), −1/(4a320 )), where p1 is a saddle. Then we obtain that the right half phase portraits with all singular points are shown in Figure 3(b) and Figure 3(c) when a20 > 0 and a20 < 0, respectively. 9 O O O O 1 1 1 1 q1 A1 q1 A1 A1 O O O O 2 2 2 2 (a) (b) (c) (d) O O 1 1 q1 A1 A1 q1 q2 A 2 A2 O O 2 2 (e) (f) Fig. 3. The right half phase portraits with all finite and infinite singular points of systems (21). 4.1.2. Subcase b02 ̸= 0 We separate the study of the infinite singular points at U1 |v=0 in three cases: i) 3a220 + 4b02 < 0, ii) 3a220 + 4b02 = 0, and iii) 3a220 + 4b02 > 0. i) If 3a220 + 4b02 < 0 system (23) has no infinite singular points. We analyze the finite singular points of the right system of (21). i.1) When a220 + b02 < 0 the right system of systems (21) has only one finite singular point at origin. i.2) When a220 + b02 = 0 the right system of systems (21) has two possible finite singular points (0, 0) and p1 = (−1/a220 , 1/a320 ). Since −1/a220 < 0, the singular point p1 is a virtual singular point. i.3) When a220 + b02 > 0 the right system of systems (21) has three possible finite singular points (0, 0), p1 = (x1 , y1 ) and p2 = (x2 , y2 ) where 1 a20 x1,2 = , y1,2 = − , a220 + 2b02 ± M1 2a420 + 3a220 b02 ± 2a220 M1 ± 2b02 M! √ M1 = a220 (a220 + b02 ). Since x1,2 < 0 in this subcase, these two points p1,2 are virtual singular points. In the above subcases the right half phase portrait of systems (21) is topological equivalently to the one of Figure 3(a). ii) If 3a220 + 4b02 = 0 system (23) has one infinite singular point (A1 , 0) = (−2/(3a20 ), 0), which is of type E-H. The right system of (21) has two singular points (0, 0) and p1 = (−1/a220 , 2/a320 ), but p1 is a virtual singular point. Then we obtain that the right half phase portrait of system (21) is shown in Figure 3(d). iii) If 3a220 + 4b02 > 0 system (23) has two infinite singular points (A1 , 0) and (A2 , 0), where √ √ 3a20 + 3(3a220 + 4b02 ) 3a20 − 3(3a220 + 4b02 ) A1 = , A2 = , 6b02 6b02 which are two nodes. And the right system of systems (21) has three possible singular points (0, 0), p1 = (x1 , y1 ) and 10 p2 = (x2 , y2 ). The Jacobian matrix of the right system of systems (21) at a finite singular point (x, y) is ( ) 2(a20 x − b02 y) 1 − 2b02 x . (24) −2(x + a20 y) −2(a20 x − b02 y) Furthermore we have a220 + b02 > 0 when 3a220 + 4b02 > 0. We claim that there are no finite singular points for the right system of systems (21) whose linear part is identically zero. Then we compute the Gröbner basis for the polynomials ẋ, ẏ, a20 x − b02 y and 1 − 2b02 x is 1. We again calculate the Gröbner basis for four polynomials ẋ, ẏ, a20 x − b02 y and x + a20 y, then we obtain four polynomials (a220 + b02 )y, −y(a20 − b202 y), y(1 + a20 b02 y) and x + a20 y. From a220 + b02 > 0 we have that the right system of systems (21) has no nilpotent singular points different from (0, 0), because these four polynomials must be zero. Thus all the remaining finite singular points are hyperbolic, or semi-hyperbolic, or centers and by Theorems 2.15 and 2.19 of [Dumortier et al., 2006], then the remaining finite singular points p1,2 must be saddles or centers because this system is Hamiltonian. By Theorem 4, on the Poincaré sphere the known singular points of the right system of (21) have total index 4. Then the sum of indices of the singular points p1,2 must be −2. Thus p1,2 are two saddles. Assume that b02 > 0 then we have x1 > 0 and x2 > 0. Furthermore we obtain the right half phase portraits of Figure 3(e). In particular, when a20 = 0 we have x1 = x2 > 0. When −3a220 /4 < b02 < 0 we have x1 > 0 and x2 < 0. Hence the singular point p2 is a virtual singular point. We have the right half phase portrait of Figure 3(f). From the above analysis we have the following result. Proposition 1. For the right system of systems (21) the following statements hold. (I) If either b02 = 0 and a20 = 0, or 3a220 + 4b02 < 0 the right system of (21) has two infinite singular points O1,2 , which are a repelling node and an attracting node, respectively, and one finite singular point the (0, 0), see Figure 3(a). (II) If b02 = 0 and a20 ̸= 0 the right system of (21) has three infinite singular point (A1 , 0) = (−1/(3a20 ), 0), which is an attracting node for a20 > 0 or a repelling node for a20 < 0, and O1,2 , which are of type E-H, and two finite singular points (0, 0) and p1 = (1/(2a220 ), −1/(4a320 )), where p1 is a saddle, see Figures 3(b) and 3(c). (III) If b02 < 0 and 3a220 + 4b02 = 0 the right system of (21) has three infinite singular points (A1 , 0) = (−2/(3a20 ), 0), which is of type E-H, and O1,2 , which are a repelling node and an attracting node, respectively, and one finite singular point the (0, 0), see Figure 3(d). (IV) If b02 ̸= 0 and 3a220 + 4b02 > 0 the right system of (21) has four infinite singular points (A1,2 , 0) and O1,2 , which are nodes. (IV.1) If b02 > 0 the right system of (21) has three finite singular points (0, 0), p1 = (x1 , y1 ) and p2 = (x2 , y2 ), where p1,2 are two saddles, see Figure 3(e). (IV.2) If b02 < 0 the right system of (21) has two finite singular points (0, 0) and p1 = (x1 , y1 ), where p1 is a saddle, see Figure 3(f ). As in the above proposition we obtain the left half phase portrait of systems (21). Proposition 2. For the left system of systems (21) the following statements hold. (I) If either b02 = 0 and A20 = 0, or 3A220 −4b02 < 0 the left system of (21) has two infinite singular points O1,2 , which are an attracting node and a repelling node, respectively, and one finite singular point the (0, 0), whose phase portraits are topologically equivalent to the ones of Figure 3(a). (II) If b02 = 0 and A20 ̸= 0 the left system of (21) has three infinite singular points (B1 , 0) = (1/(3A20 ), 0), which is a repelling node for A20 > 0 or an attracting node for A20 < 0, and O1,2 , which are of type E-H, and two finite singular points the (0, 0) and p∗1 = (−1/(2A220 ), −1/(4A320 )), where p∗1 is a saddle, whose phase portraits are topologically equivalent to the ones of Figures 3(b) or 3(c). (III) If b02 > 0 and 3A220 − 4b02 = 0 the left system of (21) has three infinite singular points (B1 , 0) = (2/(3A20 ), 0), which is of type E-H, and O1,2 , which are an attracting node and a repelling node, 11 respectively, and one finite singular point the (0, 0), whose phase portraits are topologically equivalent to the ones of Figure 3(d). (IV) If b02 ̸= 0 and 3A220 − 4b02 > 0 the left system of (21) has four infinite singular points (B1,2 , 0) and O1,2 , which are four nodes. (IV.1) If b02 < 0 the left system of (21) has three finite singular points (0, 0), p∗1 = (x∗1 , y1∗ ) and p∗2 = (x∗2 , y2∗ ), where p∗1,2 are two saddles, whose phase portraits are topologically equivalent to the ones of Figure 3(e). (IV.2) If b02 > 0 the left system of (21) has two finite singular points (0, 0) and p∗1 = (x∗1 , y1∗ ), where p∗1 is a saddle, whose phase portraits are topologically equivalent to the ones of Figure 3(f ). Here √ √ 3A20 + 3(3A220 − 4b02 ) 3A20 − 3(3A220 − 4b02 ) B1 = , B2 = , 6b02 6b02 1 A20 x∗1,2 = − 2 , ∗ y1,2 =− 4 , A20 − 2b02 ± M2 2A20 − 3A220 b02 ± 2A220 M2 ∓ 2b02 M2 √ M2 = A220 (A220 − b02 ). Proposition 3. The phase portraits of systems (3) with a02 = 0 are topologically equivalent to the 12 phase portraits showed in Figure 1, in Table 1 we provide the values of the parameters which realize the different phase portraits. Right system Left system Phase portraits A20 = 0 1.1 a20 = 0 A20 ̸= 0 1.2 A20 = 0 1.2 b02 = 0 A20 = −a20 1.3 a20 ̸= 0 ̸ −a20 a20 A20 < 0, A20 = 1.4 A20 = a20 1.5 a20 A20 > 0, A20 ≠ a20 1.6 A20 = 0 1.3 3a220 + 4b02 < 0 A20 ̸= 0 1.4 A20 = 0 1.7 b02 < 0 3a220 + 4b02 = 0 A20 ̸= 0 1.8 A20 = 0 1.10-1.12 3a220 + 4b02 > 0 A20 ̸= 0 1.9-1.11 3A220 − 4b02 < 0 1.3 a20 = 0 3A220 − 4b02 = 0 1.7 b02 > 0 3A220 − 4b02 > 0 1.10-1.12 3A220 − 4b02 < 0 1.4 a20 ̸= 0 3A220 − 4b02 = 0 1.8 3A220 − 4b02 > 0 1.9-1.11 Table 1. The conditions for the phase portraits of systems (3) with a02 = 0. Proof. We just prove subcases 1.1,1.9-1.12, because the proof of the other phase portraits follow from Propositions 1 and 2. (a) For the case a20 = b02 = A20 = 0 systems (21) have one finite singular point the (0, 0), which is a center, and two infinite singular points O1,2 . The infinity singular points O1,2 in the right system of systems (21) are a repelling node and an attracting node, respectively. And these two singular points in the left system of systems (21) are an attracting node and a repelling node, respectively. Then the infinity singular points O1,2 in systems (21) are both formed by one parabolic sector, which comes from the combination of two nodes. Hence we obtain the phase portrait 1.1 of Figure 1. (b) For the case b02 > 0, a20 ̸= 0 and 3A220 − 4b02 > 0 systems (21) have six nodes (A1 , 0), (A2 , 0), (B1 , 0), (B2 , 0), O1 and O2 at infinity, and four finite singular points, i.e. the center (0, 0), three saddles 12 O1 B2 q1 A1 B1 q1* q2 A2 O2 Fig. 4. The local phase portraits with all finite and infinite singular points of systems (21) with b02 > 0 and 3A220 − 4b02 > 0. (a) (b) (c) (d) (e) (f) Fig. 5. The possible phase portraits of systems (21). p1,2 = (x1,2 , y1,2 ) and p∗1 = (x∗1 , y1∗ ), see Figure 4. Since the finite singular points of the piecewise differential systems (21) are saddle, center or cusp there must be a saddle or a cusp in the boundary of the period annulus of center. (b.1) We assume that p2 is in the boundary of the period annulus of the origin, creating a center-loop. If the separatrices of p1 connect with the infinite singular points O1 , (A1 , 0), (A2 , 0) and (B1 , 0), respectively, then the separatrices of p∗1 must connect with the infinite singular points O2 , (A2 , 0), (B1 , 0) and (B2 , 0), respectively. Thus we obtain the phase portrait 1.9 of Figure 1, which can be realized when a20 = 1, b02 = 2 and A20 = 2. If the separatrices of p∗1 connect with the infinite singular points O1 , O2 , (B1 , 0) and (B2 , 0), respectively. Then the separatrices of p1 connect with the infinite singular points O1 , O2 , (A1 , 0) and (A2 , 0), respectively. Then we have the phase portrait of Figure 5(a), which can be realized when a20 = 0.1, b02 = 1 and A20 = 1.3. Since the separatrix skeleton of this phase portrait is equivalent with the one of phase portrait 1.9 of Figure 1, by Theorem 2, they are topologically equivalent. From these two phase portraits it follows by the continuity of the phase portraits with respect to the parameters must exist one phase portrait such that one separatrix of the saddle p1 connects with one separatrix of the saddle p∗1 , providing the phase portrait 1.10 of Figure 1. (b.2) Similarly assume that p∗1 is in the boundary of the period annulus of the origin, creating a center- loop, then we have the phase portraits of Figures 5(b)-5(d), which are topologically equivalent to 1.9-1.10 13 Fig. 6. A tri-heteroclinic loop. of Figure 1, respectively. From the phase portrait 1.9 and the phase portrait of Figure 5(b), it follows by the continuity of the phase portraits with respect to the parameters that two separatrices of the saddle p2 will connect with two separatrices of the saddle p∗1 producing a heteroclinic loop. Thus we obtain the phase portrait 1.11 of Figure 1. (b.3) Assume that p1 is in the boundary of the period annulus of the origin we also have the phase portraits, which are topologically equivalent to the phase portraits 1.9-1.11 of Figure 1. (c) For the case b02 > 0, a20 = 0 and 3A220 − 4b02 > 0 systems (21) also have the above six infinite singular points (A√1,2 , 0), (B1,2 , 0) and O1,2 , and four √ finite singular points (0, 0), p1,2 and p∗1 , where p1 = (1/(2b02 ), 1/(2b02 b02 )), p2 = (1/(2b02 ), −1/(2b02 b02 )), see Figure 4. Since H + (x, y)|p1 = H + (x, y)|p2 = 1/(24b302 ) we have that the saddles p1 and p2 are on the same energy level. If p∗1 is in the boundary of the period annulus of the origin, we have the phase portrait of Figure 5(e), which is realized when a20 = 0, b02 = 1 and A20 = 2. In this subcase the phase portrait is topologically equivalent to the phase portraits 1.10 of Figure 1. If the saddles p1 and p2 are in the boundary of the period annulus of the origin, producing a heteroclinic loop, we have the phase portrait of Figure 5(f), which is realized when a20 = 0, b02 = 2 and A20 = 2. This phase portrait is topologically equivalent to 1.11 of Figure 1. From the phase portraits in Figures 5(e) and 5(f) it follows by the continuity of the phase portraits with respect to the parameters b02 and A20 that the three saddles p1,2 and p∗1 are in the boundary of the period annulus of the origin, which produces a tri-heteroclinic loop, see Figure 6. Then we have the phase portrait 1.12 of Figure 1.  4.2. The phase portraits when a02 > 0. We first consider the origin in the local chart U2 of the right system of systems (3), the system becomes u̇ = a02 − 3b02 u + v + 3a20 u2 + u3 , v̇ = −v(b02 − 2a20 u − u2 ). (25) The origin of system (25) is not a singular point. Note that u̇|v=0 = a02 + O(u) showing that the flow is increasing in the direction u. In the local chart U1 the right system of systems (3) has the form u̇ = −1 − 3a20 u + 3b02 u2 − a02 u3 − u2 v, v̇ = −v(a20 − 2b02 u + a02 u2 + uv). (26) Similarly we consider the following two subcases b02 = 0 and b02 ̸= 0. 4.2.1. Subcase b02 = 0. System (26) has at most three infinite singular points. And the Jacobian matrix of system (26) at (u, 0) is given by ( ) −3(a20 + a02 u2 ) −u2 J= , (27) 0 −(a20 + a02 u2 ) 14 A3 A2 q2 A2 q2 A1 A1 A1 A1 q1 q1 q1 q1 (a) (b) (c) (d) Fig. 7. The right half phase portraits with all finite and infinite singular points of systems (3) with a02 > 0 and b02 = 0. which has a pair of eigenvalues with same sign. For analyze the infinite singular points we compute the Gröbner basis for u′ |v=0 = −1 − 3a20 u − a02 u3 and a20 + a02 u2 , then we obtain three polynomials a02 + 4a320 , −2a220 + a02 u, 1 + 2a20 u. (28) Hence system (26) has one nilpotent infinite singular point when a02 = −4a320 . From u′ |v=0 = 0 we obtain that if a02 + 4a320 > 0 system (26) has one infinite singular point √ ( 2a a − 3 2(a2 + √a3 (a + 4a3 ))2 ) 02 20 02 02 02 20 (A1 , 0) = √ √ ,0 , a02 3 4(a202 + a302 (a02 + 4a320 )) which is an attracting node. Assume that a20 > 0 then the right system of systems (3) has three finite singular points (0, 0), p1 = (0, −1/a02 ) and p2 = (2a20 /(a02 + 4a320 ), −1/(a02 + 4a320 )), where p1 is a center and p2 is a saddle. We have that the right half phase portrait with all singular points of systems (3) in the Poincaré disc is topologically equivalent to Figure 7(a). When a20 = 0 the singular points p1 and p2 collide providing a cusp. We obtain the right half phase portrait of Figure 7(b). Assuming a20 < 0 the singular point p1 is a saddle and p2 is a virtual singular point. This right half phase portrait is topological equivalent to phase portrait in Figure 7(b). If a02 + 4a320 = 0 system (26) has two infinite singular points (A1 , 0) = (1/a20 , 0) and (A2 , 0) = (−1/(2a20 ), 0), where one is an attracting node and the other is a nilpotent point of type E-H, respectively. And the right system of systems (3) has two finite singular points (0, 0) and p1 = (0, −1/(4a320 )), where p1 is a saddle. We have that the right half phase portrait of systems (3) is topologically equivalent to Figure 7(c). If a02 + 4a320 < 0 system (26) has three infinite singular points (A1,2,3 , 0), which are two attracting nodes and a repelling node. The explicit expressions of A2,3 in terms of parameter a20 , a02 and b02 are big and we do not provide them here. And the right system of systems (3) has three finite singular points (0, 0) and p1 = (0, −1/a02 ) and p2 = (2a20 /(a02 + 4a320 ), −1/(a02 + 4a320 )), where p1,2 are two saddles. Then we obtain right half phase portrait of Figure 7(d). 4.2.2. Subcase b02 ̸= 0. From u̇|v=0 = 1 + 3a20 u + 3b02 u2 − a02 u3 = 0 we obtain that system (26) has one infinite singular point when either a02 = a320 and b02 = −a220 , or D1 = a02 + 4a02 a320 + 6a02 a20 b02 − 3a220 b202 − 4b302 > 0. And system (26) has two and three infinite singular points when D1 = 0 and D1 < 0, respectively. The Jacobian matrix of system (26) at (u, 0) is ( ) −3(a20 − 2b02 u + a02 u2 ) −u2 J= , (29) 0 −(a20 − 2b02 u + a02 u2 ) 15 q1 q1 q1 q2 A1 A1 A1 A1 (a) (b) (c) (d) A2 A2 q1 A1 q1 A1 A1 A2 (e) (f) (g) Fig. 8. The right half phase portraits with all finite and infinite singular points of systems (3) with a02 > 0 and b02 ̸= 0. which has a pair of eigenvalues with the same sign. And we compute the Gröbner basis for u̇|v=0 and a20 − 2b02 u + a02 u2 , and obtain six polynomials D1 , −4a02 a220 − 5a02 b02 − a20 b202 + 2(a202 + b302 )u, a02 + a20 b02 + 2(a20 a02 − b202 )u, a20 − 2b02 u + a02 u2 , (30) − 2a220 − 2b02 + (a02 + a20 b02 )u, −1 − 2a20 u + b02 u2 . Thus one infinite singular point of system (26) is nilpotent when D1 = 0. We consider the finite singular points of the right system of systems (3). We compute the Gröbner basis for ẋ and ẏ with respect to x and y, and obtain nine polynomials, and the following two a02 x2 + a20 b02 x2 + b02 y + 2a20 a02 xy − 2b202 xy, (31) − x2 {b02 + [2a20 a02 − 2b02 (a220 + 2b02 )]x − D1 x2 }, are enough for the analysis. Then we obtain that the right system of systems (3) has at most two finite singular points other than (0, 0). The Jacobian matrix of the right system of systems (3) at a finite singular point (x, y) is ( ) 2(a20 x − b02 y) 1 − 2b02 x + 2a02 y . (32) −2(x + a20 y) −2(a20 x − b02 y) We claim that there are no finite singular points for the right system of systems (3) whose linear part is identically zero. Indeed, we obtain that a20 x−b02 y and 1−2b02 x+2a02 y have no common solutions, because the Gröbner basis for the polynomials ẋ, ẏ, a20 x − b02 y and 1 − 2b02 x + 2a02 y is 1. We again calculate the Gröbner basis for four polynomials ẋ, ẏ, a20 x − b02 y and x + a20 y, then we obtain five polynomials (a220 + b02 )y, y(a02 − a20 b02 + a202 y + b302 y), y(a20 + a02 a20 y − b202 y), y(1 + a02 y + a20 b02 y) and x + a20 y. It means that when a220 + b02 = 0 it has a nilpotent singular point different from (0, 0), because these five polynomials must be zero. Otherwise all the remaining finite singular points must be saddles or centers. i) If a02 = a320 and b02 = −a220 the right system of systems (3) has a nilpotent infinite singular point (A1 , 0) = (−1/a20 , 0), which is a node, and a finite singular point at the origin, whose right half phase portrait is topologically equivalent to Figure 8(a). 16 ii) If D1 > 0 system (26) has an infinite singular point ( [ √ √ √ ] ) 1 2 3 2(a02 a20 − b202 ) 3 2 (A1 , 0) = 2b02 + √ √ − 4(M3 + a02 D1 ) , 0 , 2a02 3 M3 + a202 D1 where M3 = a202 + 3a02 a20 b02 − 2b302 , and this infinite singular √ point is √ an attracting node. 2 ii.1) Assume that a20 + b02 < 0, i.e. b02 < 0 and − −b02 < a20 < −b02 , the right system of systems (3) has only one finite singular point the (0, 0). We have that the right half phase portrait in the Poincaré disc is topologically equivalent to Figure 8(a). ii.2) Assuming a220 + b02 = 0 then the right system of systems (3) has two possible finite singular points (0, 0) and p1 = (a20 /(a02 − a320 ), −1/(a02 − a320 )). √ 3 3 √ ii.2.1)√ When a20 /(a02 − a20 ) < 0, i.e either b02 < 0, a02 < −b02 and a20 = −b02 , or b02 < 0 and a20 = − −b02 the singular point p1 is a virtual singular point. The right half phase portrait with all singular points in the Poincaré disc is topologically equivalent √ to Figure 8(a). √ ii.2.2) When a20 /(a02 − a320 ) > 0, i.e b02 < 0, a02 > −b302 and a20 = −b02 the singular point p1 is a cusp. Hence we have the right half phase portrait of Figure 8(b). ii.3) Assuming a220 + b02 > 0 the right system of systems (3) has two possible singular points p1,2 = (x1,2 , y1,2 ) other than (0, 0), where √ a02 a20 − a220 b02 − 2b202 ∓ (a02 + a20 b02 ) a220 + b02 x1,2 = , D1 √ (33) a02 + 2a320 + 3a20 b02 ± 2(a220 + b02 ) a220 + b02 y1,2 = − . D1 By Theorem 4 the sum of indices of p1,2 must be 0, thus p1,2 are a saddle and a center. 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. Then we have the following subcases. √ ii.3.1) √ When x√ 1 < 0 and x2 < 0, i.e. either b02 < 0 and R[D1 , 1] < a20 < − −b02 , or b02 < 0, a02 < −b302 and −b02 < a20 < R[D1 , 2], the points p1,2 are two virtual singular points. The right half phase portrait in this subcase is topologically equivalent to the ones of Figure 8(a). ii.3.2) When x1 < 0 and x2 > 0, i.e. b02 > 0 and a20 > R[D1 , 1], the point p2 is a virtual singular point. The right half phase portrait is topological equivalent to Figure 8(c). √ ii.3.3) √ When x 1 > 0 and x 2 > 0, i.e. either b 02 < 0, a02 < −b302 and a20 > R[D1 , 3], or b02 < 0, √ a02 ≥ −b302 and a20 > −b02 , we have the right half phase portrait of Figure 8(d). iii) If D1 = 0 system (26) has two infinite singular points (A1 , 0) and (A2 , 0), which are an attracting node and a nilpotent point of type E-H. And the right system of systems (3) has one possible finite singular point ( ) b02 a02 + a20 b02 p1 = − ,− 2a02 a20 − 2b02 (a220 + 2b02 ) 4(a220 + b02 )(−a02 a20 + a220 b02 + 2b202 ) other than the origin, where p1 is a saddle. 2 2 √ When b02 /(a02 a20 − a20 b02 − 2b02 ) > 0, i.e. either b02 < 0 and a20 = R[D1 , 1], or b02 < 0, iii.1) a02 < −b302 and a20 = R[D1 , 2], the point p1 is a virtual singular point. We obtain the right half phase portrait of Figure 8(e). √ iii.2) When b02 /(a02 a20 − a220 b02 − 2b202 ) < 0, i.e. either b02 < 0, a02 < −b302 and a20 = R[D1 , 3], or b02 > 0 and a20 = R[D1 , 1], we have that the right half phase portrait in the Poincaré disc is topologically equivalent to Figure 8(f) or 8(g). iv) If D1 < 0 system (26) has three infinite singular points (A1,2,3 , 0), which are two attracting nodes and one repelling node. And the right system of systems (3) has two possible singular points p1,2 = (x1,2 , y1,2 ) different from the origin. By Theorem 4 on the Poincaré sphere the sum of indices of p1,2 must be −2, thus, p1,2 are two saddles. Then we have only the following two subcases. 17 A1 A1 q1 A2 q2 A 2 q1 A3 A3 (a) (b) Fig. 9. The right half phase portraits with all finite and infinite singular points of systems (3) with a02 > 0 and b02 = 0. √ iv.1) When x1 x2 < 0, i.e. either b02 < 0, a02 < −b302 and R[D1 , 2] < a20 < R[D1 , 3], or b02 < 0 and a20 < R[D1 , 1], one of the points p1,2 is virtual. Then we have that the right half phase portrait in the Poincaré disc is topologically equivalent to the ones of Figure 9(a). iv.2) When x1 > 0 and x2 > 0, i.e. b02 > 0 and a20 < R[D1 , 1], we have the right half phase portrait of Figure 9(b). Then we have the following results. Proposition 4. For the right system of systems (3) the following statements hold. (I) If b02 = 0 and a02 + 4a320 > 0 the right system of systems (3) has one infinite singular point (A1 , 0), which is an attracting node. (I.1) If a20 > 0 the right system of systems (3) has three finite singular points (0, 0), p1 = (0, −1/a02 ) and p2 = (2a20 /(a02 + 4a320 ), −1/(a02 + 4a320 )), where p1 is a center and p2 is a saddle, see Figure 7(a). (I.2) If a20 ≤ 0 the right system of systems (3) has two finite singular points (0, 0) and p1 = (0, −1/a02 ), where p1 is a saddle, see Figure 7(b). (II) If b02 = 0 and a02 + 4a320 = 0 the right system of systems (3) has two infinite singular points (A1 , 0) = (1/a20 , 0), (A2 , 0) = (−1/(2a20 ), 0), which are an attracting node and a point of type E- H, respectively, and two finite singular point (0, 0) and p1 = (0, −1/a02 ), where p1 is a saddle see Figure 7(c). (III) If b02 = 0 and a02 +4a320 < 0 the right system of systems (3) has three infinite singular points (A1,2,3 , 0), which are a repelling node and two attracting nodes, and three finite singular points (0, 0) and p1,2 , where p1,2 are two saddles, see Figure 7(d). (IV) If either a02 = a320 and b02 = −a220 , or D1 > 0 the right system of systems (3) has an infinite singular point (A1 , 0), which is an attracting node. √ √ √ (IV.1) If either b02 < 0 and R[D1 , 1] < a20 < −b02 , or b02 < 0, a02 ≤ −b302 and −b02 ≤ a20 < R[D1 , 2], the right system√ of systems (3) has one finite singular point (0, 0), see Figure 8(a). √ (IV.2) If b02 < 0, a02 > −b302 and a20 = −b02 the right system of systems (3) has two finite singular points (0, 0) and p1 = (a20 /(a02 − a320 ), −1/(a02 − a320 )), where p1 is a cusp, see Figure 8(b). (IV.3) If b02 > 0 and a20 > R[D1 , 1] the right system of systems (3) has two finite singular points (0, 0) and p1 = (x1 , y1 ), where p√ 1 is a saddle, see Figure 8(c). √ √ (IV.4) If either b02 < 0, a02 < −b302 and a20 > R[D1 , 3], or b02 < 0, a02 ≥ −b302 and a20 > −b02 the right system of systems (3) has three finite singular points (0, 0) and p1,2 = (x1,2 , y1,2 ), where p1 is a saddle and p2 is a center, see Figure 8(d). (V) If D1 = 0 the right system of systems (3) has two infinite singular points (A1,2 , 0), which are an at- tracting node and a point of type E-H. √ (V.1) If either b02 < 0 and a20 = R[D1 , 1], or b02 < 0, a02 < −b302 and a20 = R[D1 , 2], the right 18 system of systems (3) has one finite √ singular point (0, 0), see Figure 8(e). (V.2) If either b02 < 0, a02 < −b302 and a20 = R[D1 , 3], or b02 > 0 and a20 = R[D1 , 1], the right system of systems (3) has two finite singular points (0, 0) and p1 , where p1 is a saddle, see Figures 8(f ) and 8(g). (VI) If D1 < 0 the right system of systems (3) has three infinite nodes (A1,2,3 , 0), which are two attracting nodes and one repelling node. √ (VI.1) If either b02 < 0, a02 < −b302 and R[D1 , 2] < a20 < R[D1 , 3], or b02 < 0 and a20 < R[D1 , 1], the right system of systems (3) has two finite singular points (0, 0) and p1 = (x1 , y1 ), where p1 is a saddle, see Figure 9(a). (VI.2) If b02 > 0 and a20 < R[D1 , 1] the right system of systems (3) has three finite singular points (0, 0) and p1,2 = (x1,2 , y1,2 ), where p1,2 are two saddles, see Figure 9(b). Proposition 5. For the left system of systems (3) the following statements hold. (I) If b02 = 0 and a02 + 4A320 > 0 the left system of systems (3) has one infinite singular point (B1 , 0), which is a repelling node. (I.1) If A20 > 0 the left system of systems (3) has three finite singular points (0, 0), p∗1 = (0, −1/a02 ) and p∗2 = −(2A20 /(a02 + 4A320 ), −1/(a02 + 4A320 )), where p∗1 is a center and p∗2 is a saddle, whose phase portraits are topologically equivalent to the ones of Figure 7(a). (I.2) If A20 ≤ 0 the left system of system (3) has two finite singular points (0, 0) and p∗1 , where p∗1 is a saddle, whose phase portraits are topologically equivalent to the ones of Figure 7(b). (II) If b02 = 0 and a02 + 4A320 = 0 the left system of systems (3) has two infinite singular points (B1 , 0) = (−1/A20 , 0), (B2 , 0) = (1/(2A20 ), 0), which are a repelling node and a point type of E-H, respectively, and two finite singular points (0, 0) and p∗1 = (0, −1/a02 ), where p∗1 is a saddle, whose phase portraits are topologically equivalent to the ones of Figure 7(c). (III) If b02 = 0 and a02 + 4A320 < 0 the left system of systems (3) has three infinite singular points (B1,2,3 , 0), which are two repelling nodes and an attracting node, and three finite singular points (0, 0) and p∗1,2 , where p∗1,2 are two saddles, whose phase portraits are topologically equivalent to the ones of Figure 7(d). (IV) If either a02 = A320 and b02 = A220 , or D2 > 0 the left system of systems (3) has an infinite singular point (B1 , 0), which is a repelling node. √ √ (IV.1) If either b02 > 0 and R[D2 , 1] < A20 < b02 , or b02 > 0 and b02 ≤ A20 < R[D2 , 2], the left system of systems (3) has one finite singular point (0, 0), whose phase portraits are topologically equivalent to the ones of Figure 8(a). √ √ √ √ (IV.2) If either b02 > 0, a02 < − b302 and A20 = − b02 , or b02 > 0, a02 > b302 and A20 = b02 , the left system of systems (3) has two finite singular points (0, 0) and p∗1 = (A20 /(A320 −a02 ), 1/(A320 −a02 )), where p∗1 is a cusp, whose phase portraits are topologically equivalent to the ones of Figure 8(b). (IV.3) If b02 < 0 and A20 > R[D2 , 1] the left system of systems (3) has two finite singular points (0, 0) and p∗1 = (x∗1 , y1∗ ), where p∗1 is a saddle, whose phase portraits are topologically equivalent to the ones of Figure 8(c). √ √ √ (IV.4) If either b02 > 0, a02 ≥ b302 and A20 > b02 , or b02 > 0, a02 < b302 and A20 > R[D2 , 3], the left system of systems (3) has three finite singular points (0, 0) and p∗1,2 = (x∗1,2 , y1,2 ∗ ), where p∗ is a 1 saddle and p∗2 is a center, whose phase portraits are topologically equivalent to the ones of Figure 8(d). 19 B3 B3 B2 q2* B2 q2* q1* q1 A1 q1* q1 B1 A1 B1 q2 A2 (a) (b) Fig. √ 10. The local phase portraits with all finite and infinite singular √ points of systems (3) with (a) either b02 < 0, a02 < √ −b302 , a20 > R[D1 , 3] and A20 < R[D2 , 1], or b02 < 0, a02 > −b302 , a20 > −b02 and A20 < R[D2 , 1], (b) b02 < 0, √ a02 < −b302 , a20 = R[D1 , 3] and A20 < R[D2 , 1]. Here D2 = a202 + 4a02 A320 − 6a02 A20 b02 − 3A220 b202 + 4b302 , √ a A + A 2 b − 2b2 ± (A b − a ) A2 − b 02 20 02 20 02 02 02 x∗1,2 = − 20 02 20 , D2 √ ∗ a02 + 2A320 − 3A20 b02 ± 2 A220 − b02 y1,2 =− . D2 (V) If D2 = 0 the left system of systems (3) has two infinite singular points (B1,2 , 0), which are a repelling node and a point of type E-H point. √ (V.1) If either b02 > 0 and A20 = R[D2 , 1], or b02 > 0, a02 < b302 and A20 = R[D2 , 2], the left system of systems (3) has one finite singular point (0, 0), whose phase portraits are topologically equivalent to the ones of Figure 8(e). √ (V.2) If either b02 < 0 and A20 = R[D2 , 1], or b02 > 0, a02 < b302 and A20 = R[D2 , 3], the left system of systems (3) has two finite singular points (0, 0) and p∗1 = (−b02 /2(a02 A20 + A220 b02 − 2b202 ), (a02 − A20 b02 )/4(A220 − b02 )(a02 A20 + A220 b02 + 2b202 )), where p∗1 is a saddle, whose phase portraits are topolog- ically equivalent to the ones of Figures 8(f ) or 8(g). (VI) If D1 < 0 the left system of systems (3) has three infinite singular points (B1,2,3 , 0), which are two repelling nodes and an attracting √ node. (VI.1) If either b02 > 0, a02 < b302 and R[D2 , 2] < A20 < R[D2 , 3], or b02 > 0 and A20 < R[D2 , 1], the left system of systems (3) has two finite singular points (0, 0) and p∗1 = (x∗1 , y1∗ ), where p∗1 is a saddle, whose phase portraits are topologically equivalent to the ones of Figure 9(a). (VI.2) If b02 < 0 and A20 < R[D2 , 1] the left system of systems (3) has three finite singular points (0, 0) and p∗1,2 = (x∗1,2 , y1,2 ∗ ), where p∗ are two saddles, whose phase portraits are topologically equivalent to 1,2 the ones of Figure 9(b). Proposition 6. The phase portraits of systems (3) with a02 > 0 are topologically equivalent to 37 phase portraits of Figures 1 and 2, the corresponding conditions realizing these phase portraits are given in Tables 2 and 3. Proof. We just prove subcases 1.25-1.29 and 1.32-1.38, because the proof of the other cases follows from Propositions 4 and 5. √ (a)√For the subcases either b02 < 0, a02 < −b302 , a20 > R[D1 , 3] and A20 < R[D2 , 1], or b02 < 0, √ a02 > −b302 , a20 > −b02 and A20 < R[D2 , 1], systems (3) has four nodes (A1 , 0), (B1 , 0), (B2 , 0) and (B3 , 0) in infinity, and five finite singular points, i.e. two centers (0, 0) and p2 , three saddles p1 and p∗1,2 , see Figure 10(a). 20 Right system Left system Phase portraits A20 > 0, A20 ̸= a20 1.13 A20 = a20 1.14 a20 > 0 a02 + 4A320 > 0, A20 ≤0 1.15 a02 + 4A320 = 0 1.16 a02 + 4A320 < 0 1.17-1.20 A20 > 0 1.15 a02 + 4A320 > 0, A20 ≤0 1.21 a20 ≤ 0, a02 + 4a320 > 0 a02 + 4A320 = 0 1.2 b02 = 0 a02 + 4A320 < 0 1.3,1.4 A20 > 0 1.16 a02 + 4A320 > 0, A20 ≤0 1.2 a02 + 4a320 = 0 a02 + 4A320 = 0 1.22 a02 + 4A320 < 0 1.7,1.8 A20 > 0 1.17-1.20 a02 + 4A320 > 0, A20 ≤0 1.4 a02 + 4a320 < 0 a02 + 4A320 = 0 1.7,1.8 √ a02 + 4A320 < 0 1.9-1.12 R[D1 , 1] < √a20 < −b02 A20 > R[D2 , 1] 1.21 or a02 ≤ −b302 , A20 = R[D2 , 1] 1.2 √ −b02 ≤ a20 < R[D1 , 2] A20 < R[D2 , 1] 1.3,1.4 √ A20 > R[D2 , 1] 1.15 √ a02 > −b302 , a20 = −b02 A20 = R[D2 , 1] 1.16 √ A20 < R[D2 , 1] 1.17-1.20 a02 < −b302 , a20 > R[D1 , 3] A20 > R[D2 , 1] 1.13,1.14 √ √ b02 < 0 or a02 ≥ −b302 , a20 > −b02 A20 = R[D2 , 1] 1.23,1.24 A20 < R[D2 , 1] 1.25-1.29 a20 = R[D √1 , 1] A20 > R[D2 , 1] 1.2 or a02 < −b302 , a20 = R[D1 , 2] A20 = R[D2 , 1] 1.22 A20 < R[D2 , 1] 1.7,1.8 √ A20 > R[D2 , 1] 1.30,1.31 a02 < −b302 , a20 = R[D1 , 3] A20 = R[D2 , 1] 1.5,1.6 √ A20 < R[D2 , 1] 1.32-1.38 a02 < −b302 , A20 > R[D2 , 1] 1.4 R[D1 , 2] < a20 < R[D1 , 3] A20 = R[D2 , 1] 1.8 A20 < R[D2 , 1] 1.9-1.11 Table 2. The conditions for the phase portraits of systems (3) with a02 > 0. (a.1) We assume that the saddle p∗1 is in the boundary of the period annulus of center p2 , creating a center-loop. (a.1.1) If the saddle p1 is in the boundary of the period annulus of the origin, creating the second center-loop, then the separatrices of p∗2 connect with the infinite singular points (A1 , 0), (B1 , 0), (B2 , 0) and (B3 , 0), respectively. Thus we obtain the phase portraits 1.25 and 1.26 of Figure 1, which can be realized by a20 = 2, a02 = 1, b02 = −1.5, A20 = 0.5, and a20 = 2, a02 = 1, b02 = −1.5, A20 = −0.5, respectively. From the phase portrait 1.25 and 1.26, it follows by the continuity of the phase portraits with respect to the parameters that there must exist one phase portrait that one separatrix of saddle p∗1 connects with one separatrix of saddle p∗2 providing the phase portrait 1.27 of Figure 1. (a.1.2) Similarly, if the saddle p∗2 is in the boundary of the period annulus of the origin, then we have the phase portrait of Figure 12(a), which is realized when a20 = 1, a02 = 1, b02 = −0.5, A20 = −1.5. This phase portrait is topological equivalent to the phase portrait 1.26 of Figure 1. From the phase portrait of Figure 12(a) and the phase portrait 1.26, by the continuity of the phase portraits with respect to the parameters that there must exist one phase portrait that p1 and p∗2 are in the boundary of the period annulus of the origin producing a heteroclinic loop, which corresponding to the phase portrait 1.28 of Figure 1. 21 Right system Left system Phase portraits √ R[D√2 , 1] < A20 < b02 , 1.21 or b02 √≤ A20 < R[D √ 2 , 2] 3 a02 > b02 , A20 = b02 1.15 √ √ a02 ≥ b302 , A20 > b02 a20 > R[D1 , 1] √ 1.13,1.14 or a02 < b302 , A20 > R[D2 , 3] A20 = R[D √2 , 1] 1.2 or a02 < b302 , A20 = R[D2 , 2] √ a02 < b302 , A20 = R[D2 , 3] 1.5,1.6 R[D2 , 2] √ < A20 < R[D2 , 3] 1.3,1.4 a02 < b302 , or A20 < R[D2 , 1] √ R[D√2 , 1] < A20 < b02 , 1.2 b02 > 0 or b02 √≤ A20 < R[D √ 2 , 2] 3 a02 > b02 , A20 = b02 1.16 √ √ a02 ≥ b302 , A20 > b02 a20 = R[D1 , 1] √ 1.23,1.24 or a02 < b302 , A20 > R[D2 , 3] A20 = R[D √2 , 1] 1.22 or a02 < b302 , A20 = R[D2 , 2] √ a02 < b302 , A20 = R[D2 , 3] 1.6 √ a02 < b302 , R[D2 , 2] < A20 < R[D2 , 3] 1.7,1.8 or A20 < R[D2 , 1] √ R[D√2 , 1] < A20 < b02 , 1.7,1.8 or b02 √≤ A20 < R[D √ 2 , 2] 3 a02 > b02 , A20 = b02 1.17-1.20 √ √ a02 ≥ b302 , A20 > b02 a20 < R[D1 , 1] √ 1.25-1.29 or a02 < b302 , A20 > R[D2 , 3] A20 = R[D √2 , 1] 1.7,1.8 or a02 < b302 , A20 = R[D2 , 2] √ a02 < b302 , A20 = R[D2 , 3] 1.32-1.38 √ a02 < b302 , R[D2 , 2] < A20 < R[D2 , 3] 1.9-1.11 or A20 < R[D2 , 1] Table 3. The conditions for the phase portraits of systems (3) with a02 > 0. (a.2) Assume that the saddle p∗1 is in the boundary of the period annulus of the origin. Then the saddle p1 is in the boundary of the period annulus of center p2 . Thus we have the phase portraits Figure of 12(b) and 12(c), which are realized when a20 = 1.3, a02 = 1, b02 = −1.2, A20 = −0.1 and a20 = 1.3, a02 = 1, b02 = −1.2, A20 = 0.1, respectively. These two phase portraits are topological equivalent to the phase portraits 1.26 and 1.25 of Figure 1, respectively. From the phase portrait 1.25 to the phase portrait in Figure 12(c), it follows by the continuity of the phase portraits with respect to the parameters that there must exist one phase portrait that the saddles p1 and p∗1 are in the boundary of the period annulus of the origin and the center p2 producing a twin-heteroclinic loop, see Figure 11. Then we obtain the phase portrait 1.29 of Figure 1. √ (b) For the subcases b02 < 0, a02 < −b302 , a20 = R[D1 , 3] and A20 < R[D2 , 1] systems (3) has four nodes (A1 , 0), (B1 , 0), (B2 , 0) and (B3 , 0), and a point of type E-H (A2 , 0) in infinity, and four finite singular points, i.e. the center (0, 0), three saddles p1 and p∗1,2 , see Figure 10(b). (b.1) Assuming that the saddle p∗1 is in the boundary of the period annulus of the origin, which can create a center-loop. (b.1.1) If the separatrices of p∗2 connect with the infinite singular points (A1 , 0), (B1 , 0), (B2 , 0) and 22 Fig. 11. A twin-heteroclinic loop. (a) (b) (c) (d) (e) (f) (g) (h) Fig. 12. The possible phase portraits of systems (3) with a02 > 0 and b02 ̸= 0. (B3 , 0), respectively, then we have the phase portrait 1.32 of Figure 1. This phase portrait can be realized when a20 ≈ 1.27501, a02 = 1, b02 = −1.2, A20 = 0.3. (b.1.2) If the separatrices of p∗2 connect with the infinite singular points (A2 , 0), (B1 , 0), (B2 , 0) and (B3 , 0), respectively, the global phase portrait is topological equivalent to the phase portrait 1.34 of Figure 1. This phase portrait can be realized by taking a20 ≈ 1.27501, a02 = 1, b02 = −1.2, A20 = 0.1. From the phase portrait 1.32 to 1.34, it follows by the continuity of the phase portraits that there must exist one phase portrait such that one separatrix of the saddle p∗2 connects with one separatrix of the saddle p1 , which corresponding to the ones of phase portrait 1.33 of Figure 1. (b.2) Similarly assume that saddle p∗2 is in the boundary of the period annulus of the origin. (b.2.1) If the separatrices of p∗1 connect with the infinite singular points (A2 , 0), (B1 , 0), (B2 , 0) and (B3 , 0), respectively, then we have the phase portrait 1.35 of Figure 1. This phase portrait can be realized when a20 ≈ 1.27501, a02 = 1, b02 = −1.2, A20 = −0.4. From the phase portraits 1.34 and 1.35, by the continuity of the phase portraits with respect to the parameters, there must exist one phase portrait such that p∗1 and p∗2 are in the boundary of the period annulus of the origin producing a heteroclinic loop. Then the phase portrait is topologically equivalent to 1.36 of Figure 1. (b.2.2) If the separatrices of p1 connect with the infinite singular points (A1 , 0), (A2 , 0), (B2 , 0) and (B3 , 0), respectively, then we obtain the phase portrait of Figure 12(d), which is realized by taking a20 ≈ 1.44519, a02 = 1, b02 = −1.3, A20 = −0.5. From the phase portrait 1.35 to the phase portrait of Figure 12(d) there must exist one phase portrait such that one separatrix of the saddle p1 connects with one 23 separatrix of the saddle p∗1 , whose the phase portrait is topologically equivalent to 1.37 of Figure 1. (b.3) Further, we assume that the saddle p1 is in the boundary of the period annulus of the origin. Similar to the above subcases we can obtain the phase portraits of Figures 12(e)-12(h), which are topological equivalent to the phase portraits 1.32-1.34 and 1.36 of Figure 1, respectively. From the phase portrait 1.32 to the phase portrait of Figure 12(e), there must exist one phase portrait such that p∗1 and p1 are in the boundary of the period annulus of the origin producing a heteroclinic loop, which corresponding to the phase portrait 1.38 of Figure 1.  Acknowledgments The first author is partially supported by National Natural Science Foundation of China (Nos. 12001112 and 12071091), Young Innovative Talents Program in Colleges and universities of Guangdong Province (No. 2019KQNCX211) and Science and Technology Program of Guangzhou (No. 202102020443). The second author is partially by the Agencia Estatal de Investigación grant PID2019-104658GB-I00, and the H2020 European Research Council grant MSCA-RISE-2017-777911. References Andronov, A., Vitt, A. & Khaikin, S. [1966] Theory of Oscillations, Pergamon Press, Oxford. Artés, J. & Llibre, J. [1994] “Quadratic Hamiltonian vector fields,” J. Differential Equations 107, 80–95. Aziz, W., Llibre J. & Pantazi, C. [2014] “Centers of quasi-homogeneous polynomial differential equations of degree three,” Adv. Math. 254, 233–250. Banerjee, S., & Verghese, G. [2001] Nonlinear Phenomena in Power Electronics: Attractors, Bifurcations, Chaos, and Nonlinear Control, Wiley-IEEE Press, New York. Bautin, N.N. [1952] “On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type,” Mat. Sb. 30, 181–196; Amer. Math. Soc. Transl. 100, 1–19. Chen, T., Huang, L. & Yu, P. [2021] “Center condition and bifurcation of limit cycles for quadratic switching systems with a nilpotent equilibrium point,” J. Differential Equations 303, 326–368. Chen, T., Huang, L., Yu, P. & Huang, W. [2018] “Bifurcation of limit cycles at infinity in piecewise polynomial systems,” Nonlinear Anal.: Real World Appl. 41, 82–106. Chen, T., Li, S. & Llibre, J. [2020] “Z2 -equivariant linear type bi-center cubic polynomial Hamiltonian vector fields cubic system,” J. Differential Equations 269, 832-861. Chen, X. & Zhang, W. [2012] “Isochronicity of centers in switching Bautin system,” J. Differential Equa- tions 252, 2877–2899. Cima, A. & Llibre, J. [1990] “Algebraic and topological classification of the homogeneous cubic vector fields in the plane,” J. Math. Anal. Appl. 47, 420–448. Colak, I., Llibre, J. & Valls, C. [2014] “Hamiltonian linear type centers of linear plus cubic homogeneous polynomial vector fields,” J. Differential Equations 257, 1623–1661. Colak, I., Llibre, J. & Valls, C. [2014] “Hamiltonian nilpotent centers of linear plus cubic homogeneous polynomial vector fields,” Adv. Math. 259 (2014), 655–687. Coll, B., Prohens, R. & Gasull, A. [1999] “The center problem for discontinuous Liénard differential equa- tion,” Int. J. Bifurcation and Chaos 9, 1751–1761. Dulac, H. [1908] “Détermination et integration d’une certaine classe d’équations différentielle ayant par point singulier un centre,” Bull. Sci. Math. 32, 230–252. Dumortier, F., Llibre, J. & Artés, J. [2006] Qualitative Theory of Planar Differential Systems Universitext, Springer-Verlag, New York. Filippov, A.F. [1988] Differential Equation with Discontinuous Right-Hand Sides, Kluwer Academic, Netherlands. Garcia, I., Giacomini, H., Giné, J. & Llibre, J. [2016] “Analytic nilpotent centers as limits of nondegenerate centers revisited,” J. Math. Anal. Appl. 441, 893–899, Gasull, A. & Torregrosa, J. [2003] “Center-focus problem for discontinuous planar differential equations,” Int. J. Bifurcation and Chaos 13, 1755–1765. 24 Guo, L., Yu, P. & Chen, Y. [2019] “Bifurcation analysis on a class of Z2 -equivariant cubic switching systems showing eighteen limit cycles,” J. Differential Equations 266, 1221–1244. Kapteyn, W. [1911] “On the midpoints of integral curves of differential equations of the first degree,” Nederl. Akad. Wetensch. Verslag. Afd. Natuurk. Konikl. Nederland 1446–1457 (Dutch). Kapteyn, W. [1912] “New investigations on the midpoints of integrals of differential equations of the first degree,” Nederl. Akad. Wetensch. Verslag Afd. Natuurk. 20, 1354–1365; 21, 27–33 (Dutch). Li, F., Liu, Y., Liu, Y. & Yu, P. [2018] “Bi-center problem and bifurcation of limit cycles from nilpotent singular points in Z2 -equivariant cubic vector fields,” J. Differential Equations 265, 4965–4992. Liu, Y. & Li, F. [2015] “Double bifurcation of nilpotent focus,” Internat. J. Bifur. Chaos 25, 1550036 (10 pages). Liu, Y. & Li, J. [2011] “Bifurcations of limit cycles created by a multiple nilpotent critical point of planar dynamical systems,” Internat. J. Bifur. Chaos 21, 497–504. Lv, Y., Yuan, R. & Yu, P. [2019] “Dynamics in two nonsmooth predator-prey models with threshold harvesting,” Nonlinear Dyn. 74, 107–132. Malkin, K.E. [1964] “Criteria for the center for a certain differential equation,” Volz. Mat. Sb. Vyp. 2, 87–91 (in Russian). Markus, L. [1954] “Global structure of ordinary differential equations in the plane,” Trans. Amer. Math. Soc. 76, 127–148. Neumann, D.A. [1975] “Classification of continuous flows on 2–manifolds,” Proc. Amer. Math. Soc. 48, 73–81. Nusse,H.E. & Yorke, J.A. [1995] “Border-collision bifurcations for piecewise smooth one-dimensional maps,” Int. J. Bifurcation and Chaos 5, 189–207. Peixoto, M. [1973] Dynamical Systems, Proceedingsof a Symposium held at the University of Bahia, Acad. Press, New York, pp.389–420. Poincaré, H. [1881] “Mémoire sur les courbes définies par les équations différentielles, ” J. Math. 37, 375–422. Schlomiuk, D. [1993] “Algebraic particular integrals, integrability and the problem of the center,” Trans. Amer. Math. Soc. 338, 799–841. Stróżzyna, E. & Żola̧dek, H. [2012] “The analytic normal for the nilpotent singularity,” J. Differential Equations 179, 479–537. Tian, Y. & Yu, P. [2015] “Center conditions in a switching Bautin system,” J. Differential Equations 259, 1203–1226. Vulpe, N.I. & Sibirskii, K. S. [1988] “Centro-affine invariant conditions for the existence of a center of a differential system with cubic nonlinearities,” Dokl. Akad. Nauk. SSSR 301, 1297–1301 (in Russian); translation in: Soviet Math. Dokl. 38 (1989), 198–201. Żola̧dek, H. [1994] “Quadratic systems with center and their perturbations,” J. Differential Equations 109, 223–273. Żola̧dek, H. [1994] “The classification of reversible cubic systems with center,” Topol. Methods Nonlinear Anal. 4, 79–136. Żola̧dek, H. [1996] “Remarks on: “The classification of reversible cubic systems with center”, [Topol. Meth- ods Nonlinear Anal. 4 (1994), 79–136];” Topol. Methods Nonlinear Anal. 8, 335–342.