This is a preprint of: “Bifurcations of the Riccati quadratic polynomial differential system”, Jaume Llibre, Bruno D. Lopes, Paulo Ricardo da Silva, Internat. J. Bifur. Chaos Appl. Sci. Engrg., vol. 31(6), 2150094:1–13, 2021. DOI: [10.1142/S0218127421500942] PHASE PORTRAITS OF THE RICCATI QUADRATIC POLYNOMIAL DIFFERENTIAL SYSTEMS JAUME LLIBRE1 , BRUNO D. LOPES 2 AND PAULO R. DA SILVA3 Abstract. In this paper we characterize the phase portrait of the Ric- cati quadratic polynomial differential systems ẋ = α2 (x), ẏ = ky 2 + β1 (x)y + γ2 (x), 2 with (x, y) ∈ R , γ2 (x) non-zero (otherwise the system is a Bernoulli differential system), k 6= 0 (otherwise the system is a Lienard differen- tial system), β1 (x) a polynomial of degree at most 1, α2 (x) and γ2 (x) polynomials of degree at most 2, and the maximum of the degrees of α2 (x) and ky 2 + β1 (x)y + γ2 (x) is 2. We give the complete description of their phase portraits in the Poincaré disk (i.e. in the compactification of R2 adding the circle S1 of the infinity) modulo topological equivalence. 1. Introduction and statement of the main results Numerous problems of applied mathematics are modeled by quadratic polynomial differential systems, see for instance [?]. Excluding linear sys- tems, such systems are the ones with the lowest degree of complexity, and the large bibliography on the subject proves its relevance. We refer for example to the books of Ye Yanqian et al. [?], Reyn [?], and Artes, Llibre, Schlomiuk, Vulpe [?], and the surveys of Coppel [?], and Chicone and Jinghuang [?] are excellent introductory readings to the quadratic polynomial differential sys- tems. In this paper we characterize the phase portraits of the Riccati quadratic differential systems (1) ẋ = α2 (x), ẏ = ky 2 + β1 (x)y + γ2 (x), 2010 Mathematics Subject Classification. 37G15, 37D45. Key words and phrases. Riccati system, Poincaré compactification, dynamics at infinity. ∗ The first author is partially supported by the Ministerio de Economı́a, Industria y Competitividad, Agencia Estatal de Investigación grant MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. The second author is sup- ported by PNPD/CAPES-IMECC/UNICAMP. The third author was partially supported by CAPES, CNPq and FAPESP. All the authors are supported by FP7-PEOPLE-2012- IRSES-316338. 1 2 J. LLIBRE, B.D. LOPES AND P.R. DA SILVA with (x, y) ∈ R2 , γ2 (x) non-zero (otherwise the system is a Bernoulli dif- ferential system), k 6= 0 (otherwise the system is a Lienard differential system), β1 (x) a polynomial of degree at most 1, α2 (x) and γ2 (x) poly- nomials of degree at most 2, and the maximum of the degrees of α2 (x) and ky 2 + β1 (x)y + γ2 (x) is 2. In (??) the dot denotes derivative with respect to the time. Proposition 1. A Riccati quadratic differential system (??) is topologically equivalent to one of the following systems: (i) ẋ = x(x + 1), ẏ = y 2 + (ax + b)y + cx2 + dx + e; (ii) ẋ = x2 , ẏ = y 2 + (ax + b)y + cx2 + dx + e; (iii) ẋ = x, ẏ = y 2 + (ax + b)y + cx2 + dx + e; (iv) ẋ = 1, ẏ = y 2 + (ax + b)y + cx2 + dx + e; (v) ẋ = x2 + 1, ẏ = y 2 + (ax + b)y + cx2 + dx + e. with c2 + d2 + e2 6= 0 in all these systems. We note that the Riccati systems have no periodic orbits. In fact, the equilibrium points of systems (i), (ii) and (iii) are on invariant straight lines and systems (iv) and (v) do not have equilibrium points, and consequently they do not have limit cycles, because it is well known that a periodic orbit in the plane must surrounds at least one equilibrium point. The objective of this work is to classify the phase portraits of the Riccati quadratic polynomial differential systems (??) in the Poincaré disk modulo topological equivalence. As any polynomial differential system, system (??) can be extended to an analytic system on a closed disk of radius one, whose interior is diffeomorphic to R2 and its boundary, the circle S1 , plays the role of the infinity. This closed disk is denoted by D2 and called the Poincaré disk, because the technique for doing such an extension is precisely the Poincaré compactification for a polynomial differential system in R2 , which is described in details in chapter 5 of [?]. In this paper we shall use the notation of that chapter. By using this compactification technique the dynamics of system (??) in a neighborhood of the infinity can be studied and we have the following result. Theorem 2. The phase portraits of the Riccati system (??) in the Poincaré disk are topologically equivalent to one of the 74 phase portraits presented in Figures 1, 2 and 3. The phase portraits of the systems of Proposition 1 are provided in Tables 1, 2, 3, 4 and 5 where ∆F1 = b2 − 4e, ∆F2 = (b − a)2 − 4(c − d + e), (2) ∆I1 = (a − 1) − 4c, ∆I2 = a2 − 4c. 2 Three papers on generalizations of Riccati differential equations can be found in [?, ?, ?]. This paper is organized as follows. In section ?? we prove Proposition ??, and study the finite equilibria. In section ?? we study the infinite equilibria. Finally in section ?? we prove Theorem ??. RICCATI QUADRATIC POLYNOMIAL DIFFERENTIAL SYSTEM 3 Phase Portraits of systems (i) conditions P 1, P 2, P 3, P 4, P 5 ∆I1 > 0, ∆F1 > 0, ∆F2 > 0 P 6, P 7, P 8, P 9 ∆I1 > 0, ∆F1 > 0, ∆F2 = 0 P 10 ∆I1 > 0, ∆F1 > 0, ∆F2 < 0 P 11, P 12, P 13, P 14 ∆I1 > 0, ∆F1 = 0, ∆F2 > 0 P 15, P 16 ∆I1 > 0, ∆F1 = 0, ∆F2 = 0 P 17 ∆I1 > 0, ∆F1 = 0, ∆F2 < 0 P 18 ∆I1 > 0, ∆F1 < 0, ∆F2 > 0 P 19 ∆I1 > 0, ∆F1 < 0, ∆F2 = 0 P 20 ∆I1 > 0, ∆F1 < 0, ∆F2 < 0 P 21, P 22, P 23 ∆I1 = 0, ∆F1 > 0, ∆F2 > 0 P 24, P 25 ∆I1 = 0, ∆F1 > 0, ∆F2 = 0 P 26 ∆I1 = 0, ∆F1 > 0, ∆F2 < 0 P 27 ∆I1 = 0, ∆F1 = 0, ∆F2 > 0 P 28 ∆I1 = 0, ∆F1 = 0, ∆F2 = 0 P 29 ∆I1 = 0, ∆F1 = 0, ∆F2 < 0 P 30 ∆I1 = 0, ∆F1 < 0, ∆F2 > 0 P 31 ∆I1 = 0, ∆F1 < 0, ∆F2 = 0 P 32 ∆I1 = 0, ∆F1 < 0, ∆F2 < 0 P 33 ∆I1 < 0, ∆F1 > 0, ∆F2 > 0 P 34 ∆I1 < 0, ∆F1 > 0, ∆F2 = 0 P 35 ∆I1 < 0, ∆F1 > 0, ∆F2 < 0 P 36 ∆I1 < 0, ∆F1 = 0, ∆F2 > 0 P 37 ∆I1 < 0, ∆F1 = 0, ∆F2 = 0 P 38 ∆I1 < 0, ∆F2 = 0, ∆I1 < 0 P 39 ∆I1 < 0, ∆F1 < 0, ∆F2 > 0 P 40 ∆I1 < 0, ∆F1 < 0, ∆F2 = 0 P 41 ∆I1 < 0, ∆F1 < 0, ∆F2 < 0 Table 1. The phase portraits of systems (i). 4 J. LLIBRE, B.D. LOPES AND P.R. DA SILVA Phase Portraits of systems (ii) conditions P 42, P 43, P 44 ∆I1 > 0, ∆F1 > 0 P 45, P 46, P 47 ∆I1 > 0, ∆F1 = 0 P 48 ∆I1 > 0, ∆F1 < 0 P 49, P 50, P 51 ∆I1 = 0, ∆F1 > 0 P 52, P 53, P 54 ∆I1 = 0, ∆F1 = 0 P 55 ∆I1 = 0, ∆F1 < 0 P 56 ∆I1 < 0, ∆F1 > 0 P 57, P 58 ∆I1 < 0, ∆F1 = 0 P 41 ∆I1 < 0, ∆F1 < 0 Table 2. The phase portraits of systems (ii). Phase Portraits of systems (iii) conditions P 59, P 60, P 61 ∆I2 > 0, ∆F1 > 0 P 62, P 63, P 64 ∆I2 > 0, ∆F1 = 0 P 65 ∆I2 > 0, ∆F1 < 0 P 66, P 67 ∆I2 = 0, ∆F1 > 0 P 68, P 69 ∆I2 = 0, ∆F1 = 0 P 32 ∆I2 = 0, ∆F1 < 0 P 35 ∆I2 < 0, ∆F1 > 0 P 38 ∆I2 < 0, ∆F1 = 0 P 41 ∆I2 < 0, ∆F1 < 0 Table 3. The phase portraits of systems (iii). Phase Portrait of systems (iv) conditions P 70, P 71 ∆I2 > 0 P 72, P 73, P 74 ∆I2 = 0 P 41 ∆I2 < 0 Table 4. The phase portraits of systems (iv). Phase Portraits of systems (v) conditions P 70, P 71 ∆I1 > 0 P 72, P 73, P 74 ∆I1 = 0 P 41 ∆I1 < 0 Table 5. The phase portraits of systems (v). 2. Finite equilibrium points We start this section with the proof of Proposition ??. RICCATI QUADRATIC POLYNOMIAL DIFFERENTIAL SYSTEM 5 P1_Ricatti1.pdf P2_Ricatti.pdfP3_Ricatti.pdfP4_Ricatti.pdf P5_Ricatti.pdfP6_Ricatti.pdfP7_Ricatti.pdfP8_Ricatti.pdf P9_Ricatti.pdfP10_Ricatti.pdf P11_Ricatti.pdf P12_Ricatti.pdf P13_Ricatti.pdf P14_Ricatti.pdf P15_Ricatti.pdf P16_Ricatti.pdf P17_Ricatti.pdf P18_Ricatti.pdf P19_Ricatti.pdf P20_Ricatti.pdf P21_Ricatti.pdf P22_Ricatti.pdf P23_Ricatti.pdf P24_Ricatti.pdf P25_Ricatti.pdf P26_Ricatti.pdf P27_Ricatti.pdf P28_Ricatti.pdf Figure 1. Phase portraits of systems (1) the Poicaré disk. Proof of Proposition ??. Since α2 (x) is a polynomial of degree at most 2, we have, using a rescaling of the time if necessary, ẋ = (x − r)(x − s) with r 6= s, ẋ = (x − r)2 , ẋ = (x − r), ẋ = 1, ẋ = (x − r)2 + s2 with s 6= 0. If ẋ = (x − r)(x − s), r 6= s, considering the change of coordinates x−r x1 = , y1 = cy and T = (r − s)t, r−s we get a system (i). If ẋ = (x − r)n , n = 1, 2, considering the change of coordinates x1 = x − r, y1 = cy, we get systems (ii) for n = 2 and systems (iii) for n = 1. If ẋ = 1, considering the change of coordinates x1 = x and y1 = cy, we get a systems (iv). If ẋ = (x − r)2 + s2 , considering the change of coordinates x1 = (x − r)/s, y1 = cy and T = st, we get a system (v). Proposition 3. The finite equilibrium points of the Riccati quadratic poly- nomial differential system (??) are described below. (a) Systems (i) have at most 4 equilibria which can be either a saddle, or a stable or an unstable node, or a saddle–node. (b) Systems (ii) have at most 2 equilibria which can be either a saddle– node either semi–hyperbolic or nilpotent. (c) Systems (iii) have at most 2 equilibria which can be either a saddle or an unstable node, or a saddle-node. (d) Systems (iv) and (v) have no finite equilibria. Proof. Systems (i): Consider the Riccati quadratic polynomial differential systems (3) ẋ = x(x + 1), ẏ = y 2 + (ax + b)y + cx2 + dx + e. 6 J. LLIBRE, B.D. LOPES AND P.R. DA SILVA P29_Ricatti.pdf P30_Ricatti.pdf P31_Ricatti.pdf P32_Ricatti.pdf P33_Ricatti.pdf P34_Ricatti.pdf P35_Ricatti.pdf P36_Ricatti.pdf P37 P38 P39_Ricatti.pdf P40_Ricatti.pdf P41 P42_Ricatti.pdf P43_Ricatti.pdf P44_Ricatti.pdf P45_Ricatti.pdf P46_Ricatti.pdf P47_Ricatti.pdf P48 P49 P50_Ricatti.pdf P51_Ricatti.pdf P52_Ricatti.pdf P53_Ricatti.pdf P54 P55_Ricatti.pdf P56_Ricatti.pdf Figure 2. Continuation of phase portraits of systems (1) in the Poicaré disk. The equilibrium points of system (??) are p ! p ! b+ ∆F1 b− ∆F1 (x1 , y1 ) = 0, − , (x2 , y2 ) = 0, − , 2 2 RICCATI QUADRATIC POLYNOMIAL DIFFERENTIAL SYSTEM 7 P57_Ricatti.pdf P58_Ricatti.pdf P59_Ricatti.pdf P60_Ricatti.pdf P61_Ricatti.pdf P62_Ricatti.pdf P63_Ricatti.pdf P64_Ricatti.pdf P65_Ricatti.pdf P66_Ricatti.pdf P67_Ricatti.pdf P68 P69_Ricatti.pdf P70_Ricatti.pdf P71_Ricatti.pdf P72_Ricatti.pdf P73_Ricatti.pdf P74_Ricatti.pdf Figure 3. Continuation of the phase portraits of systems (1) in the Poicaré disk. p ! p ! −a + b + ∆F2 −a + b − ∆F2 (x3 , y3 ) = −1, − , (x4 , y4 ) = −1, − , 2 2 where ∆F1 and ∆F2 are given by (??). The eigenvalues p of the Jacobian matrix of system p (??) evaluated at (xi , yi ) are (1, (−1)i ∆F1 ) for i = 1, 2, and (−1, (−1)i ∆F2 ) for i = 3, 4, when they exist. From the classification of the hyperbolic and semi-hyperbolic equilibrium points (see for instance Theorems 2.18 and 2.19 of [?]), we have the fol- lowing (when the equilibrium point is not hyperbolic we mention this fact explicitly). (i) If ∆F1 > 0 and ∆F2 > 0, system (??) has two saddles, a stable node and an unstable node. (ii) If ∆F1 > 0 and ∆F2 = 0, system (??) has a saddle, a stable node and a semi–hyperbolic saddle–node. (iii) If ∆F1 > 0 and ∆F2 < 0, system (??) has a saddle and a stable node. (iv) If ∆F1 = 0 and ∆F2 > 0, system (??) has a saddle, an unstable node and a semi–hyperbolic saddle–node. (v) If ∆F1 = 0 and ∆F2 = 0, system (??) has two semi–hyperbolic saddle–nodes. (vi) If ∆F1 = 0 and ∆F2 < 0 system (??) has one semi–hyperbolic saddle– node. (vii) If ∆F1 < 0 and ∆F2 > 0, system (??) has a saddle and an unstable node. (viii) If ∆F1 < 0 and ∆F2 = 0, system (??) has one semi–hyperbolic saddle–node. (ix) If ∆F1 < 0 and ∆F2 < 0, system (??) has not equilibria. 8 J. LLIBRE, B.D. LOPES AND P.R. DA SILVA Systems (ii): Consider the Riccati quadratic polynomial differential systems (4) ẋ = x2 , ẏ = y 2 + (ax + b)y + cx2 + dx + e. We have that the finite equilibrium points of system (??) are p ! p ! b + ∆F1 b − ∆F1 (5) (x1 , y1 ) = 0, − , (x2 , y2 ) = 0, − , 2 2 where ∆F1 is given by (??). The eigenvalues of the Jacobian p matrix of system (??) evaluated at (xi , yi ) for all i = 1, 2 are 0 and (−1)i ∆F1 . Then we have (i) If ∆F1 > 0 systems (??) have two semi–hyperbolic saddle–nodes. (ii) If ∆F1 = 0 then systems (??) have one nilpotent saddle–node equi- librium point. (iii) If ∆F1 < 0 system (??) has not equilibrium points. Systems (iii): Consider the Riccati quadratic polynomial differential sys- tems (6) ẋ = x, ẏ = y 2 + (ax + b)y + cx2 + dx + e. The equilibrium points of systems (??) are given by (??). Then system (??) has 0, 1 or 2 equilibrium points if ∆F1 is negative, zero or positive, respec- tively. The eigenvalues of the Jacobian p matrix of system (??) evaluated at i (xi , yi ) for i = 1, 2 are 1 and (−1) ∆F1 . Thus we have: (i) If ∆F1 > 0 systems (??) have a saddle and an unstable node. (ii) If ∆F1 = 0 systems (??) have a semi–hyperbolic saddle–node. (iii) If ∆F1 < 0 system (??) has no equilibria. Systems (iv) and (v): These systems are chordal quadratic systems, or qua- dratic system without finite singularities.  3. Infinite equilibrium points For a complete description of the Poincaré compactification method we refer to chapter 5 of [?]. In what follows we remember some formulas. Consider a polynomial differential system in R2 with degree 2. (7) ẋ = P (x, y), ẏ = Q(x, y) or equivalently its associated polynomial vector field X = (P, Q). As we said before, any polynomial differential system can be extended to an analytic differential system on a closed disk of radius one centered at their origin of coordinates, whose interior is diffeomorphic to R2 and its boundary, the circle S1 , plays the role of the infinity. We consider 4 open charts covering the disk D: RICCATI QUADRATIC POLYNOMIAL DIFFERENTIAL SYSTEM 9 φ1 : R2 −→ U1 , φ1 (x, y) = (1/v, u/v), φ2 : R2 −→ U2 , φ1 (x, y) = (u/v, 1/v) and ψk : R2 −→ Vk , ψk (x, y) = −φk (x, y), k = 1, 2 with U1 = {(u, v) ∈ D : u2 + v 2 ≤ 1 and u > 0}, 2 2 U2 = {(u, v) ∈ D : u + v ≤ 1 and v > 0}, 2 2 V1 = {(u, v) ∈ D : u + v ≤ 1 and u < 0}, V2 = {(u, v) ∈ D : u2 + v 2 ≤ 1 and v < 0}. The Poincaré compactification is denoted by p(X). The expression of p(X) in the chart U1 is (8) u̇ = v 2 (−uP + Q), v̇ = −v 3 P, where P and Q are evaluated at (1/v, u/v). The expression of p(X) in the chart U2 is (9) u̇ = v 2 (P − uQ), v̇ = −v 3 Q, where P and Q are evaluated at (u/v, 1/v). Moreover in all these local charts the points (u, v) of the infinity have its coordinate v = 0. The expression for the extend differential system in the local chart Vi , i = 1, 2 is the same as in Ui multiplied by −1. Proposition 4. On the circle of the infinity, for any systems of Proposition ?? the origin of U2 , denoted by n, is an attracting node and the origin of V2 , denoted by s, is a repelling node of the Riccati quadratic polynomial differential system (??). Moreover, the remaining infinite equilibrium points are described below. (a) For systems (i), (ii) and (v) three situations can occur. – 4 equilibrium points being 2 saddles, 1 attracting node and 1 repelling node; – 2 equilibrium points being 2 saddle-nodes; – The only equilibria are n and s. (b) For systems (iii) three situations can occur. – 4 equilibrium points being 4 nilpotent saddle-nodes; – 2 equilibrium points being 2 semi–hyperbolic saddle-nodes; – The only equilibria are n and s. (c) For systems (iv) three situations can occur. – 4 equilibrium points being 2 semi-hyperbolic saddles, 1 semi- hyperbolic attracting node and 1 semi-hyperbolic repelling node; – 2 equilibrium points being 2 semi-hyperbolic saddle-nodes; – The only equilibria are n and s. 10 J. LLIBRE, B.D. LOPES AND P.R. DA SILVA Proof. Systems (i): First we analyze the phase portrait in the local chart U1 . The expression of the system in this chart is (10) u̇ = v((b − 1)u + d) + ev 2 + p(u), v̇ = −(v + v 2 ), where p(u) = u2 + (a − 1)u + c. Note that (u0 , 0) is an infinite equilibrium point of (??) if, and only if, p(u0 ) = 0. System (??) has 0, 1 or 2 two infinite equilibrium points: p ! 1 − a + (−1)i ∆I1 (11) Si = ,0 , 2 for i = 1, 2, where ∆I1 is given (??). p The eigenvalues of the Jacobian matrix of system (??) are −1 and (−1)i ∆I1 . Thus we have: (i) If ∆I1 > 0 systems (??) have a saddle and a stable node. (ii) If ∆I1 = 0 systems (??) have a semi–hiperbolic saddle–node. (iii) If ∆I1 < 0 systems (??) have no equilibrium points. Now we analyze the phase portrait in the local chart U2 , we need to the study the origin of U2 , the others infinite singularity ahead, have been studied in the local chart U1 . The expression of the system in this chart is u̇ = v(vue − u(du + b− 1)) + q(u), (12) v̇ = −v 1 + au + cu2 − v 2 (b + du) − ev 3 , where q(u) = −u(1 + (a − 1)u + cu2 ). The eigenvalues of the Jacobian matrix at the origin of U2 of system (??) are −1 and −1. Therefore system (??) has a stable node at (0, 0). Thus, the equilibrium points of system (??), system (i), on the circle S1 are classified as follows. n v2 q1 u1 p1 p2 q2 u2 v1 s Figure 4. Finite and infinite equilibrium of system (??), systems (i). (a) If ∆I1 > 0 system (??) has 6 equilibrium points. – 2 saddles: u1 and v1 diametrically opposed to u1 ; – 2 attracting nodes: u2 and n the origin of U2 ); RICCATI QUADRATIC POLYNOMIAL DIFFERENTIAL SYSTEM 11 – 2 repelling nodes: v2 diametrically opposed to u2 and s the origin of V2 diametrically opposed to n. (b) If ∆I1 = 0 system (??) has 4 equilibrium points. – 2 saddle-node: u12 and v12 (diametrically opposed to u12 ; – 1 attracting node: n; – 1 repelling node: s. (c) If ∆I1 < 0 system (??) has 2 equilibrium points. – 1 attracting node: n; – 1 repelling node: s. Systems (ii): The expression of the system in the local chart U1 is (13) u̇ = v((d + bu) + ev) + p(u), v̇ = −v, where p(u) = u2 + (a − 1)u + c, and in the local chart U2 is u̇ = −v(u(d + bu) + vue)  + q(u), (14) v̇ = −v c + du + cu2 − v 2 (b + du) − ev 3 , where q(u) = −u(1 + (a − 1)u + cu2 ). The equilibrium point at infinity and their classification are exactly the same of system (i). Systems (iii): The expression of this system in the local chart U1 is u̇ = v(u(b − 1) + d + ev) + p(u), (15) v̇ = −v 2 , where p(u) = u2 + au + c. System (??) has 0, 1 or 2 equilibrium points. p ! −a + (−1)i ∆I2 Si = ,0 2 for i = 1, 2, where ∆I2 is given by (??). p The eigenvalues of the Jacobian i matrix of system (??) are 0 and (−1) ∆I2 . Thus we have: (i) If ∆I2 > 0 systems (??) have two nilpotent saddle–nodes. (ii) If ∆I2 = 0 systems (??) have a saddle–node with both eigenvalues being zero. (iii) If ∆I2 < 0 systems (??) have no equilibrium points. The expression of the system in the local chart U2 is u̇ = v(−v(eu) − u(−1+ b + du)) + q(u), (16) v̇ = −v 1 + au + cu2 − v 2 (b + du) − ev 3 , where q(u) = −u(1 + (a − 1)u + cu2 ). The equilibrium points at infinity and their classification are exactly the same of systems (i). In summay, the equilibrium points of system (??), system (iii), on the circle S1 are classified as follows. (a) If ∆I2 > 0 system (??) has 6 equilibrium points. – 4 saddle-nodes: u1 , v1 diametrically opposed to u1 , u2 and v2 diametrically opposed to u2 ; 12 J. LLIBRE, B.D. LOPES AND P.R. DA SILVA – 1 attracting node: n ; – 1 repelling node: s diametrically opposed to n. (b) If ∆I2 = 0 system (??) has 4 equilibrium points. – 2 saddle-node: u12 and v12 diametrically opposed to u12 ; – 1 attracting node: n; – 1 repelling node: s. (c) If ∆I2 < 0 system (??) has 2 equilibrium points. – 1 attracting node: n; – 1 repelling node: s. Systems (iv): The expression of the system in the local chart U1 is (17) u̇ = v(d + bu + (e − u)v) + p(u), v̇ = −v 3 , where p(u) = u2 + au + c. System (??) has 0, 1 or 2 equilibrium points. p ! −a + (−1)i ∆I2 Si = ,0 2 for i = 1, 2, where ∆I2 is given by (??). p The eigenvalues of the Jacobian matrix of system (??) are 0 and (−1)i ∆I2 . Thus we have: (i) If ∆I2 > 0 systems (??) have a semi-hyperbolic stable node and a semi-hyperbolic saddle. (ii) If ∆I2 = 0 systems (??) have a semi-hyperbolic saddle-node. (iii) If ∆I2 < 0 systems (??) have no equilibrium points. The expression of the system in the local chart U2 is u̇ = v(v(1 − eu) − u(b+ du)) + q(u), (18) v̇ = −v 1 + au + cu2 − v 2 (b + du) − ev 3 , where q(u) = −u(1 + (a − 1)u + cu2 ). The equilibrium points at infinity and their classification are exactly the same of systems (i). In short, the equilibrium points of system (??), systems (iv), on the circle S1 are classified as follows. (a) If ∆I2 > 0 system (??) has 6 equilibrium points. – 2 semi-hyperbolic saddles: u1 and v1 diametrically opposed to u1 ; – 1 semi-hyperbolic attracting node: u2 ; – 1 attracting node: n ; – 1 semi-hyperbolic repelling node: v2 diametrically opposed to u2 ; – 1 repelling node: s diametrically opposed to n. (b) If ∆I2 = 0 system (??) has 4 equilibrium points. – 2 semi-hyperbolic saddle-nodes: u12 and v12 diametrically op- posed to u12 ; – 1 attracting node: n ; – 1repelling node: s diametrically opposed to n. (c) If ∆I2 < 0 system (??) has 2 equilibrium points. RICCATI QUADRATIC POLYNOMIAL DIFFERENTIAL SYSTEM 13 – 1 attracting node: n; – 1 repelling node: s. Systems (v): The expression of the system in the local chart U1 is u̇ = v((d + bu) + v(e − u)) + p(u) (19) v̇ = −(v + v 3 ). where p(u) = u2 + (a − 1)u + c. The equilibrium points at infinity and their classification are exactly the same than of systems (i). The expression of the system in the local chart U2 is u̇ = v(v(1 − eu) − u(b+ du)) + q(u), (20) v̇ = −v 1 + au + cu2 − v 2 (b + du) − ev 3 , where q(u) = −u(1 + (a − 1)u + cu2 − u). The origin and its classification is exactly the same than of systems (i).  4. Proof of Theorem ?? We start this section considering the Tables 1, ..., 5, one for each of the possible Riccati systems. In each table, we list the conditions about the parameters and indicate the possible phase portraits. 4.1. Proof of Theorem ??. We remember the notation introduced in pre- vious sections ∆F1 = b2 − 4e, ∆F2 = (b − a)2 − 4(c − d + e), ∆I1 = (a − 1)2 − 4c and ∆I2 = a2 − 4c. 4.1.1. Proof of Theorem ?? – System (i). We begin the proof considering the assumptions of the first row of Table ??. These systems have 4 finite equi- librium p1 , p2 , q1 , q2 and 6 infinite equilibrium n, s, u1 , u2 , v1 , v2 , according to sections ?? and ??, see Figure (??). Let r1 be the straight line joining v1 , p1 and u1 , and r2 be the straight line joining v1 , q2 and u1 : r1 = y − u1 x − k1 = 0, r2 = y − u1 x − k2 = 0, where 1 p p k1 = (1 − b + (a − 1)2 − 4c + (a − b)2 − 4(c − d + e)) 2 and p 1 k2 = (−b − b2 − 4c). 2 We analyze the position of q1 with respect to r1 and the position of p2 with respect to r2 . We have four possibilities. Assume the first possibility. By Lemma ?? (see Appendix) the vector field X(x, y) = (α2 (x), ky 2 + β1 (x)y + γ2 (x)) has only the equilibrium p1 as a contact point with r1 , and the equilibrium q2 as a contact point with r2 . Thus p1 divides r1 into two semi-straight lines and we have the direction of 14 J. LLIBRE, B.D. LOPES AND P.R. DA SILVA Figure 5. Straight lines r1 and r2 and the directions of the vector field X(x, y) = (α2 (x), ky 2 + β1 (x)y + γ2 (x)). the field downward between v2 and p1 and upward between p1 and u1 . In fact this is due to the fact that the repelling node is below the line r1 , and there is a trajectory with α-limit q1 and ω-limit n. Similarly we concluded that q2 divides r2 into two semi-straight lines and we have the direction of the vector field downward between v2 and q2 and upward between q2 and u1 . Thus the only way to complete the phase portrait is shown in figure P 1. In the second case and in an analogous way, we conclude that the phase portrait is shown in figure P 2. The third case does not occur, because the conditions r1 (q1 ) > 0 and r2 (p2 ) < 0 will never be satisfied at the same time. In the fourth case we concluded that p1 divides r1 into two semi-straight lines and the direction of the field is downward between v2 and p1 and upward between p1 and u1 . Moreover q2 divides r2 into two semi-straight lines and the direction of the field is upward between v2 and q2 and downward between q2 and u1 . There are three possibilities to complete the phase portrait. To analyze this case we consider the straight line S : y = mx + n joining p1 and q2 . The coeficients are π2 (q2 ) − π2 (p1 ) p p p m= = (−a− ∆F1 − ∆F2 )/2 and n = (−b− ∆I1 )/2 π1 (q2 ) − π1 (p1 ) where π1 (x, y) = x and π2 (x, p y) = y. p We analyze how p the straight line S reaches the infinite. If −a− ∆F1 − ∆F2 < 1−a− ∆I1 , then u2 is above S, and the only p possibility p to complete p the phase portrait is shown in figure P 3. If −a− ∆F1 − ∆F2 > 1−a− ∆I1 then u2 is below S, and the phase p p p portrait is shown in figure P 5. Finally, if −a− ∆F1 − ∆F2 = 1−a− ∆I1 then u2 belong to S, the phase portrait is shown in figure P 4. Now we explicit the parameter values for each phase portrait. • P1 : (a, b, c, d, e) = (0, 0, 0, 3.75, −0.25). • P2 : (a, b, c, d, e) = (0, 0, 0, −3.75, −4). • P3 : (a, b, c, d, e) = (0, 0, −0, 75, −0.75, −0.25). • P4 : (a, b, c, d, e) = (0, 0, −2, −2, −0.25). • P5 : (a, b, c, d, e) = (0, 0, −3.75, −3.75, −0.25). Assume the conditions in the second row of Table ??. systems (i) have 3 finite equilbria p1,2 , q1 , q2 and 6 infinite equilibria n, s, u1 , u2 , v1 , v2 . Note RICCATI QUADRATIC POLYNOMIAL DIFFERENTIAL SYSTEM 15 that p1,2 comes from the collision of p1 and p2 (these equilibria exist when we assume the conditions of the first row of Table ??) when ∆F2 → 0. Con- sequently systems (i) have at most five phase portraits which are obtained from the 5 possible phase portraits of row 1 of Table ??. Applying Lemma 7, we can see that effectively only the 4 phase portraits listed in row 2 of Table ?? occur. Next we explicit the parameter values for each phase portrait. • P6 : (a, b, c, d, e) = (0, 0, 0, −3, −3). • P7 : (a, b, c, d, e) = (0, 0, −1, −2, −1). • P9 : (a, b, c, d, e) = (0, 0, −29, −30, −1). • P8 : We cannot explicit a choice of (a, b, c, d, e). However its existence follows from continuity when we pass from the phase portraits P7 to P9 . The analysis of the phase portraits for the conditions listed in the other rows of Table ?? is analogous to the one that we did above. We will only give an example for each phase portrait. • P10 : (a, b, c, d, e) = (0, 0, 0, −1, −0.25). • P11 : (a, b, c, d, e) = (0, 0, −1, 10, 0). • P12 : (a, b, c, d, e) = (2, 0, −1, −1, 0). • P13 : (a, b, c, d, e) = (1, 0, −1, −1, 0). • P14 : (a, b, c, d, e) = (−2, 0, −1, −1, 0). • P15 : (a, b, c, d, e) = (4, 0, 2, −2, 0). • P16 : (a, b, c, d, e) = (1, 0, −1, −1.25, 0). • P17 : (a, b, c, d, e) = (0, 0, −1, −10, 0). • P18 : (a, b, c, d, e) = (0, 1, 0, 1, 0.75). • P19 : (a, b, c, d, e) = (0, 0, −0.75, 0.25, 1). • P20 : (a, b, c, d, e) = (1, 0, −1, −1.25, 1). • P21 : (a, b, c, d, e) = (1, 2, 0, 4.75, 0). • P22 : (a, b, c, d, e) = (2, 0, 0.25, −9.75, −10). • P23 : (a, b, c, d, e) = (2, 0, 0.25, −0.75, −1). • P24 : (a, b, c, d, e) = (0, 0, 0.25, −0.75, −1). • P25 : (a, b, c, d, e) = (1, 1, 0, 0.2, 0.2). • P26 : (a, b, c, d, e) = (0, 0, 0.25, −1.75, −1). • P27 : (a, b, c, d, e) = (0, 0, 0.25, 1.25, 0). • P28 : (a, b, c, d, e) = (1, 1, 0, 0.25, 0.25). • P29 : (a, b, c, d, e) = (1, 0, 0, −1, 0). • P30 : (a, b, c, d, e) = (0, 0, 0.25, 2.25, 1). • P31 : (a, b, c, d, e) = (0, 0, 0.25, 1.25, 1). • P32 : (a, b, c, d, e) = (0, 0, 0.25, 0.25, 1). • P33 : (a, b, c, d, e) = (0, 0, 1.25, 1.25, −1). • P34 : (a, b, c, d, e) = (1, 1, 1, 1.2, 0.2). • P35 : (a, b, c, d, e) = (1, 1, 2, 0, 0.2). • P36 : (a, b, c, d, e) = (1, 2, 1, 2, 1). • P37 : (a, b, c, d, e) = (2, 0, 2, 1, 0). • P38 : (a, b, c, d, e) = (1, 0, 1, −1, 0). 16 J. LLIBRE, B.D. LOPES AND P.R. DA SILVA • P39 : (a, b, c, d, e) = (1, 0, 1, 2, 1). • P40 : (a, b, c, d, e) = (1, 0, 1, 1.75, 1). • P41 : (a, b, c, d, e) = (0, 0, 1.25, 1.25, 1). 4.1.2. Proof of Theorem ?? –System (ii). The phase portraits listed in row 1 of Table ?? are obtained from row 1 of Table ??. Note that system (ii) has only x = 0 as an invariant vertical line, which comes when the two straight lines x = 0, x = −1 of system (i) collide at x = 0. Thus we consider the phase portraits represented in the figures P 1, P 2, P 3, P 4 and P 5, excluding what occurs in the strip −1 ≤ x ≤ 0. This reduces the possible phase por- traits to P 42, P 43 and P 44 obtained from P 1, P 2 and P 3 respectively. Note that no new configurations can be obtained from P 4 and P 5 because the phase portraits are equal in the complement of the strip −1 ≤ x ≤ 0. The possibilities listed in the other rows of Table ?? are obtained in a similar way. Below we list values of the parameters that realize each one of the possible phase portraits. • P42 : (a, b, c, d, e) = (1, 1, −1, 4, −1). • P43 : (a, b, c, d, e) = (1, 1, −1, −4, −1). • P44 : (a, b, c, d, e) = (1, 1, −1, 0, −1). • P45 : (a, b, c, d, e) = (1, 1, −1, 4, 0.25). • P46 : (a, b, c, d, e) = (1, 1, −1, 0, 0.25). • P47 : (a, b, c, d, e) = (2, 1, −1, 1, 0.25). • P48 : (a, b, c, d, e) = (1, 2, −1, 0, 2). • P49 : (a, b, c, d, e) = (1, 1, 0, 4, −1). • P50 : (a, b, c, d, e) = (1, 1, 0, −2, −1). • P51 : (a, b, c, d, e) = (1, 1, 0, 0, −1). • P52 : (a, b, c, d, e) = (1, 1, 0, 4, 0.25). • P53 : (a, b, c, d, e) = (2, 1, 0.25, 1, 0.25). • P54 : (a, b, c, d, e) = (1, 1, 0, −2, 0.25). • P55 : (a, b, c, d, e) = (1, 1, 0, 0, 1). • P56 : (a, b, c, d, e) = (1, 1, 1, 0, 0.2). • P57 : (a, b, c, d, e) = (1, 1, 1, 0, 0.25). • P58 : (a, b, c, d, e) = (2, 1, 0.3, 1, 0.25). 4.1.3. Proof of Theorem ?? –System (iii). If ∆I2 > 0 and ∆FI > 0, corre- sponding to the case considered in the first row of Table ??, systems (iii) have 2 finite equilibria q1 , q2 and 6 infinite equilibria n, s, u1 , u2 , v1 , v2 , ac- cording to sections ?? and ??. We consider the straight line r joining v2 , q1 and u2 . Applying Lemma (??) we can prove that the following configurations cannot occur: (a) both unstable separatrix of q2 have ω-limit n; (b) the left hand side of unstable separatrix of q2 has ω-limit n and the right hand side separatrix of q2 has ω-limit u1 ; RICCATI QUADRATIC POLYNOMIAL DIFFERENTIAL SYSTEM 17 (c) the left hand side of unstable separatrix of q2 has ω-limit v2 and the right hand side separatrix of q2 has ω-limit n; (d) the left hand side of unstable separatrix of q2 has ω-limit v2 and the right hand side separatrix of q2 has ω-limit u1 ; (e) the left hand side of unstable separatrix of q2 has ω-limit u1 and the right hand side separatrix of q2 has ω-limit v2 . Taking into account this previous informative the only possible phase por- traits are P59 , P60 and P60 remain. The other lines of Table ?? are similarly analyzed. Below we list the parameter values that realize each one of the possible phase portraits. • P59 : (a, b, c, d, e) = (1, 2, 0.2, 1, 0.2). • P60 : (a, b, c, d, e) = (1, 1, 0.2, 2, 0.2). • P61 : (a, b, c, d, e) = (1, 1, 0.2, ∗∗, 0.2). • P62 : (a, b, c, d, e) = (2, 2, 0.2, 1, 1). • P63 : (a, b, c, d, e) = (1, 2, 0.2, 1, 1). • P64 : (a, b, c, d, e) = (∗∗, 2, 0.2, 1, 1. • P65 : (a, b, c, d, e) = (8, 2, 2, 1, 5). • P66 : (a, b, c, d, e) = (2, 2, 1, 1, 0.2). • P67 : (a, b, c, d, e) = (2, 1, 1, 1, 0.2). • P68 : (a, b, c, d, e) = (2, 2, 1, 1, 1). • P69 : (a, b, c, d, e) = (1, 2, 0.25, 1, 1). 4.1.4. Proof of Theorem ?? –Systems (iv) and (v). The classification given in Tables 4 and 5 follows directly from the analysis of singularities at infinity. We list a parameter value that realize each phase portrait. • P70 : (a, b, c, d, e) = (1, 1, 0, 0, 0). • P71 : (a, b, c, d, e) = (1, 1, 0, 0, 1). • P72 : (a, b, c, d, e) = (1, 1, 0, 0, 0). • P73 : (a, b, c, d, e) = (1, 1, 0, 0, 1). • P74 : (a, b, c, d, e) = (1, 1, 0, 0, −1). 5. Appendix: Semi-hyperbolic equilibrium points The following two lemmas are very useful in the proofs and they proved in Chapter 11 of [?]. Lemma 5. If the straight line passing through two singular points S1 and S2 of a quadratic system is not an integral line, then it must be formed by three open line segments without contact points ∞S1 , S1 S2 and S2 ∞. Moreover the trajectories cross ∞S1 and S2 ∞ in one direction, and cross S1 S2 in the opposite direction. Lemma 6. The straight line connecting one finite singular point and a pair of infinite singular points in a quadratic system is either formed by trajectories or it is a line with exactly one contact point. This contact point 18 J. LLIBRE, B.D. LOPES AND P.R. DA SILVA is the finite singular point. For the latter case the flow goes in different directions on each half–line. References [1] J.C. Artés, J. Llibre, D. Schlomiuk and N. Vulpe, Geometric configurations of singularities of planar polynomial differential systems. A global classification in the quadratic case, to appear in Springer. [2] C. Chicone and T. Jinghuang, On General Properties of Quadratic Systems, The American Mathematical Monthly, 89, No. 3 (1982), 167–178. [3] W.A. Coppel, A Survey of Quadratic Systems, J. Differential Equations 2 (1966), 293–304. [4] F. Dumortier, J. Llibre and J.C. Artés, Qualitative Theory of Planar Differential Systems, Springer, 2006. [5] P. Figueroa and M. Pinto, Riccati equations and nonoscillatory solutions of third order differential equations, Dynam. Systems Appl. 17 (2008), no. 3-4, 459–475. [6] A. Gasull and J. Llibre, On the nonsingular quadratic differential equations in the plane, Proc. Amer. Math. Soc. 104 (1988), 793-794. [7] A. Gasull, Sheng Li-Ren, J. Llibre, Chordal quadratic systems, Rocky Mountain J. of Math. 16 (1986),751-782. [8] J. Llibre, R. Oliveira, C. Valls, Phase portraits for Z2 -symmetric cubic Riccati polynomial differential equations, Topology and its Applications 234 (2018), 220–237. [9] S.D. Maharaj, A.K. Tiwari, R. Mohanlal, R. Narain , Riccati equations for bounded radiating systems, J. Math. Phys. 57-9 (2016), 092501, 12p. [10] J.W. Reyn, Phase portraits of planar quadratic systems, Mathematics and its Appli- cations, 583, Springer, 2007. [11] A.G. Wu, F.J. Sun and Y. Zhang, A novel iterative algorithm for solving coupled Riccati equations, Appl. Math. Comput. 364 (2020), 124645, 14 pp. [12] Ye Yanqian el al., Theory of limit cycles, Trans. of Mathematical Monographs 66, Amer. Math. Soc., Providence, RI, 2 edition, 1984. 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain. 2 IMECC–UNICAMP, CEP 13081–970, Campinas, São Paulo, Brazil. 3 Departamento de Matemática – IBILCE–UNESP, Rua C. Colombo, 2265, CEP 15054–000 S. J. Rio Preto, São Paulo, Brazil. Email address: jllibre@mat.uab.cat Email address: brunodomicianolopes@gmail.com Email address: paulo.r.silva@unesp.br