This is a preprint of: “Phase Portraits of the complex Abel polynomial differential systems”, Jaume Llibre, Clàudia Valls, Chaos Solitons Fractals, vol. 148, 111050:1–5, 2021. DOI: [10.1016/j.chaos.2021.111050] PHASE PORTRAITS OF THE COMPLEX ABEL POLYNOMIAL DIFFERENTIAL SYSTEMS JAUME LLIBRE1 AND CLAUDIA VALLS2 Abstract. In this paper we characterize the phase portraits of the complex Abel polynomial differential equations ż = (z − a)(z − b)(z − c), with z ∈ C, a, b, c ∈ C. We give the complete description of their phase portraits in the Poincaré disc (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, or in physics, chemistry, economics, ... are modeled by polynomial differential systems. Excluding linear systems the quadratic polynomial differential systems are the ones with the lowest degree of complexity, and the large bibliography on them proves their relevance, see the books [1, 12, 13] and the surveys [3, 4]. After the quadratic polynomial differential systems come the cubic ones, which also have many applications. Among the cubic polynomial differential systems we emphasize the Abel systems, see for instance the papers [2, 6, 8] where the Abel systems are applied to modelize problems from Ecology, control theory for electrical circuits and cosmology, respectively. In this paper we characterize the phase portraits of the complex Abel differential equations (1) ż = (z − a)(z − b)(z − c), with z ∈ C, a, b, c ∈ C and the dot means derivative with respect to the time t ∈ R. We write z = x + iy, a = a1 + ia2 , b = b1 + ib2 , c = c1 + ic2 , with x, y ∈ R and ai , bi , ci ∈ R for i = 1, 2. The complex differential equation (1) becomes the real differential system ẋ = −a1 b1 c1 + a2 b2 c1 + a2 b1 c2 + a1 b2 c2 + (a1 b1 − a2 b2 + a1 c1 + b1 c1 − a2 c2 − b2 c2 )x − (a2 b1 + a1 b2 + a2 c1 + b2 c1 + a1 c2 + b1 c2 )y − (a1 + b1 + c1 )x2 + 2(a2 + b2 + c2 )xy + (a1 + b1 + c1 )y 2 + x3 − 3xy 2 , (2) ẏ = −a2 b1 c1 − a1 b2 c1 − a1 b1 c2 + a2 b2 c2 + (a2 b1 + a1 b2 + a2 c1 + b2 c1 + a1 c2 + b1 c2 )x + (a1 b1 − a2 b2 + a1 c1 + b1 c1 − a2 c2 − b2 c2 )y − (a2 + b2 + c2 )x2 − 2(a1 + b1 + c1 )xy + (a2 + b2 + c2 )y 2 + 3x2 y − y 3 . 2010 Mathematics Subject Classification. Primary 34A05. Secondary 34C05, 37C10. Key words and phrases. complex Abel system, Poincaré compactification, dynamics at infinity. 1 2 J. LLIBRE AND C. VALLS The objective of this work is to classify the phase portraits of the Abel polynomial differential systems (2) in the Poincaré disc modulo topological equivalence. As any polynomial differential system, system (2) can be extended to an analytic system on a closed disc D of radius one, whose interior is diffeomorphic to R2 and its boundary, the circle S1 , plays the role of the infinity. This closed disc is denoted by D2 and called the Poincaré disc, because the technique for doing such an extension is the Poincaré compactification for a polynomial differential system in R2 , which is described in details in Chapter 5 of [5], see also subsection 2.1. In this paper we shall use the notation of that chapter. By using this compactification technique the dynamics of system (2) in a neighborhood of the infinity can be studied. See subsection 2.2 for the definition of equivalent topological phase portraits, and for seeing that it is sufficient to draw the separatrices in the Poincaré disc and an orbit in every canonical region for characterizing a phase portrait in the Poincaré disc. Theorem 1. There are only six different topologically phase portraits of the Abel differential system (2) in the Poincaré disc, see Figure 1. In Figure 1 the phase portrait (a) is realized when a1 = a2 = −1, b1 = b2 = 0, and c1 = c2 = −1/2; (b) is realized when a1 = a2 = 0, b1 = −3/2, b2 = −3, c1 = −2, c2 = −1; (c) is realized when a1 = a2 = b1 = 0, b2 = 1, c1 = 0, c2 = 2; (d) is realized when a1 = a2 = 0, b1 = 2, b2 = −1, c1 = −2, c2 = −1; (e) is realized when a1 = a2 = 0, b1 = −8, b2 = −6, c1 = −3, c2 = −1; (f) is realized when a1 = −1, a2 = 0, b1 = −1, b2 = 0, c1 = 0, c2 = 1; (g) is realized when a1 = a2 = b1 = b2 = 0, c1 = −1, c2 = −1/2; (h) is realized when a1 = a2 = b1 = b2 = c1 = 0, c2 = −1; and (i) is realized when a1 = a2 = b1 = b2 = c1 = c2 = 0. 2. Preliminaries 2.1. Poincaré compactification. For a complete description of the Poincaré com- pactification method we refer to chapter 5 of [5]. In what follows we remember some formulas. Consider a polynomial differential system in R2 with degree 3. ẋ = P (x, y), ẏ = Q(x, y) or equivalently its associated polynomial vector field X = (P, Q). As we said be- fore, any polynomial differential system can be extended to an analytic differential system on a closed disc 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. In order to study the compactified vector field we consider four open charts covering the disc D: φ1 : R2 −→ U1 , φ1 (x, y) = (1/v, u/v), 2 φ2 : R −→ U2 , φ1 (x, y) = (u/v, 1/v) COMPLEX ABEL POLYNOMIAL DIFFERENTIAL SYSTEM 3 (a) 3 centers (b) 1 center 2 foci (c) 3 nodes (d) 1 node 2 foci (e) 3 foci (f) 1 doble 1 center (g) 1 doble 1 focus (h) 1 doble 1 node (i) 1 triple Figure 1. The nine topological phase portraits in the Poincaré disc of system (2) if we distinguish between nodes and foci, without this dis- tintion the phase portraits (c), (d) and (e) are topologically equivalent, and also the phase portraits (g) and (h) are topologically equivalent. So there are only six different topological phase portraits. 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}, U2 = {(u, v) ∈ D : u2 + v 2 ≤ 1 and v > 0}, V1 = {(u, v) ∈ D : u2 + v 2 ≤ 1 and u < 0}, V2 = {(u, v) ∈ D : u2 + v 2 ≤ 1 and v < 0}. 4 J. LLIBRE AND C. VALLS The Poincaré compactification is denoted by p(X). The expression of p(X) in the chart U1 is u̇ = v 3 (−uP + Q), v̇ = −v 4 P, where P and Q are evaluated at (1/v, u/v). The expression of p(X) in the chart U2 is u̇ = v 3 (P − uQ), v̇ = −v 4 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 . 2.2. Topological equivalence of two polynomial vector fields. Let X1 and X2 be two polynomial vector fields on R2 . We say that they are topologically equivalent if there exists a homeomorphism on the Poincaré disc D which preserves the infinity S1 and sends the orbits of π(p(X1 )) to orbits of π(p(X2 )), preserving or reversing the orientation of all the orbits. A separatrix of the Poincaré compactification π(p(X)) is one of following orbits: all the orbits at the infinity S1 , the finite singular points, the limit cycles, and the two orbits at the boundary of a hyperbolic sector at a finite or an infinite singular point, see for more details on the separatrices [9, 10]. The set of all separatrices of π(p(X)), which we denote by ΣX , is a closed set (see [10]). A canonical region of π(p(X)) is an open connected component of D \ ΣX . The union of the set ΣX with an orbit of each canonical region form the separatrix con- figuration of π(p(X)) and is denoted by Σ0X . We denote the number of separatrices of a phase portrait in the Poincaré disc by S, and its number of canonical regions by R. Two separatrix configurations Σ0X1 and Σ0X2 are topologically equivalent if there is a homeomorphism h : D −→ D such that h(Σ0X1 ) = Σ0X2 . According to the following theorem which was proved by Markus [9], Neumann [10] and Peixoto [11], it is sufficient to investigate the separatrix configuration of a polynomial differential system, for determining its global phase portrait. Theorem 2. Two Poincaré compactified polynomial vector fields π(p(X1 )) and π(p(X2 )) with finitely many separatrices are topologically equivalent if and only if their separatrix configurations Σ0X1 and Σ0X2 are topologically equivalent. 3. Infinite singular points The following lemma summarizes the information at the infinite singular points. Lemma 3. There are two pairs of infinite singular points which are saddles located at the origins of the local chats U1 , U2 , V1 and V2 . COMPLEX ABEL POLYNOMIAL DIFFERENTIAL SYSTEM 5 Proof. First we determine the local phase portrait of the infinite singular points in the local chart U1 . The expression of system (2) in this chart is u̇ = 2u − (a2 + b2 + c2 )v − (a1 + b1 + c1 )uv + (a2 b1 + a1 b2 + a2 c1 + b2 c1 + a1 c2 + b1 c2 )v 2 + 2u3 − (a2 + b2 + c2 )u2 v − (a2 b1 c1 + a1 b2 c1 + a1 b1 c2 − a2 b2 c2 )v 3 − (a1 + b1 + c1 )u3 v + (a2 b1 + a1 b2 + a2 c1 + b2 c1 + a1 c2 + b1 c2 )u2 v 2 + (a1 b1 c1 − a2 b2 c1 − a2 b1 c2 − a1 b2 c2 )uv 3 , v̇ = −v + (a1 + b1 + c1 )v 2 + 3u2 v − 2(a2 + b2 + c2 )uv 2 − (a1 b1 − a2 b2 + a1 c1 + b1 c1 − a2 c2 − b2 c2 )v 3 − (a1 + b1 + c1 )u2 v 2 + (a2 b1 + a1 b2 + a2 c1 + b2 c1 + a1 c2 + b1 c2 )uv 3 + (a1 b1 c1 − a2 b2 c1 − a2 b1 c2 − a1 b2 c2 )v 4 . The singular points (u, 0) in the local chart U1 satisfy 2u(1+u2 ) = 0. So the unique singular point in the local chart U1 is the origin. Computing the eigenvalues of the Jacobian matrix at the origin we obtain that they are 2 and −1. So the origin is a saddle. Now we analyze the phase portrait in the local chart U2 , we need to the study the origin of U2 , the other infinite singular points have been studied in the local chart U1 . The expression of the system in this chart is u̇ = −2u + (a1 + b1 + c1 )v + (a2 + b2 + c2 )uv − (a2 b1 + a1 b2 + a2 c1 + b2 c1 + a1 c2 + b1 c2 )v 2 − 2u3 + (a1 + b1 + c1 )u2 v − (a1 b1 c1 − a2 b2 c1 − a2 b1 c2 − a1 b2 c2 )v 3 + (a2 + b2 + c2 )u3 v − (a2 b1 + a1 b2 + a2 c1 + b2 c1 + a1 c2 + b1 c2 )u2 v 2 + (a2 b1 c1 + a1 b2 c1 + a1 b1 c2 − a2 b2 c2 )uv 3 v̇ = v − (a2 + b2 + c2 )v 2 − 3u2 v + 2(a1 + b1 + c1 )uv 2 − (a1 b1 − a2 b2 + a1 c1 + b1 c1 − a2 c2 − b2 c2 )v 3 + (a2 + b2 + c2 )u2 v 2 − (a2 b1 + a1 b2 + a2 c1 + b2 c1 + a1 c2 + b1 c2 )uv 3 + (a2 b1 c1 + a1 b2 c1 + a1 b1 c2 − a2 b2 c2 )v 4 . Note that the origin of the local chart U2 is a singular point whose eigenvalues of the Jacobian matrix at this point are −2 and 1. So the origin is a saddle. This completes the proof of the lemma.  4. Finite singular points The finite singular points of system (2) are the real solutions of ẋ = ẏ = 0. Computing such solutions we obtain (x, y) = (a1 , a2 ), (x, y) = (b1 , b2 ), (x, y) = (c1 , c2 ). Now we study the local phase portrait of each of the finite singular points. We will use the following three propositions. Proposition 4. Let p be a singular point of system (2) such that their eigenvalues are purely imaginary. Then p is an isochronous center. The proof of Proposition 4 is given in [7, Corollary 2.7 (a)]. 6 J. LLIBRE AND C. VALLS Proposition 5. Let p be a singular point of system (2) of multiplicity k ∈ {2, 3}. Then the local phase portrait of p is formed by 2(k − 1) elliptic sectors separated by parabolic sectors. The proof of Proposition 5 is given in [7, Corollary 2.7 (b)]. Proposition 6. System (2) has no limit cycles. The proof of Proposition 6 is given in [7, Theorem 1.3 (a)]. The finite singular point (a1 , a2 ) The eigenvalues associated to the finite singular point (a1 , a2 ) are λ1 ± iλ2 where λ1 = a21 − a22 − a1 b1 + a2 b2 − a1 c1 + b1 c1 + a2 c2 − b2 c2 λ2 = b2 c1 − a2 (b1 + c1 ) + b1 c2 − a1 (b2 + c2 − 2a2 ). We have different possibilities. If λ2 6= 0 then we have a focus, hyperbolic if λ1 6= 0 and by Proposition 4 is a center if λ1 = 0. If λ2 = 0 and λ1 6= 0 then we have a node. If λ1 = λ2 = 0 then the singular point is linearly zero. In this case we have that either b = a and c 6= a, or b = c = a. In view of Proposition 5 in the first case the singular point (a1 , a2 ) is formed by two elliptic sectors separated by parabolic sectors and in the second case the singular point (a1 , a2 ) is formed by four elliptic sectors separated by parabolic sectors. The finite singular point (b1 , b2 ) The eigenvalues associated to the finite singular point (b1 , b2 ) are λ3 ± iλ4 where λ3 = −a1 b1 + b21 + a2 b2 − b22 + a1 c1 − b1 c1 − a2 c2 + b2 c2 λ4 = a2 b1 + a1 b2 − 2b1 b2 − a2 c1 + b2 c1 − a1 c2 + b1 c2 . We have different possibilities. If λ4 6= 0 then we have a focus, hyperbolic if λ3 6= 0 and by Proposition 4 is a center if λ3 = 0. If λ4 = 0 and λ3 6= 0 then we have a node. Finally, if λ4 = 0 and λ3 = 0, then the singular point is linearly zero. We can consider that b 6= a because this case has been already studied. Then the condition λ4 = λ3 = 0 implies b = c. Since c 6= a then in view of Proposition 5 the singular point (b1 , b2 ) is formed by two elliptic sectors separated by parabolic sectors. The finite singular point (c1 , c2 ) The eigenvalues associated to the finite singular point (c1 , c2 ) are λ5 ± iλ6 where λ5 = a1 b1 − a2 b2 − a1 c1 − b1 c1 + c21 + a2 c2 + b2 c2 − c22 , λ6 = b2 c1 + a2 (−b1 + c1 ) + b1 c2 − 2c1 c2 + a1 (−b2 + c2 ). We have different possibilities. If λ6 6= 0 then we have a focus, hyperbolic if λ5 6= 0 and by Proposition 5 is a center if λ5 = 0. If λ6 = 0 and λ5 6= 0 then we have a node. Finally, if λ6 = λ5 = 0 then either c = a, or c = b, and these two cases have been studied before. Combining the different possibilities for the finite singular points we have in principle the following cases: three centers; two centers and a node; COMPLEX ABEL POLYNOMIAL DIFFERENTIAL SYSTEM 7 two centers and a focus; one center and two nodes; one center, one node and one focus; one center and two foci; three nodes; two nodes and a focus; one node and two foci; three foci; one point formed by two elliptic sectors separated by parabolic sectors and a center; one point formed by two elliptic sectors separated by parabolic sectors and a node; one point formed by two elliptic sectors separated by parabolic sectors and a focus; and one point formed by four parabolic sectors separated by parabolic sectors. However the cases: two centers and a node; two centers and a focus; one center and two nodes; one center, one node and one focus; and two nodes and a focus are not possible. We only show that the case two centers and a node is not possible, the other non possible cases are proved in a similar way. Indeed, the singular points a and b are centers if and only if λ1 = λ3 = 0 and c is a node if and only if λ6 = 0. Then it is easy to check that the system λ1 = λ3 = λ6 = 0 has no solutions in ai , bi and ci for i = 1, 2. In short, only nine possibilities for the finite singular points are indeed realizable, the ones of Figure 1. 5. Global phase portraits Combining the study of the local phase portraits of the finite and infinite singular points, together with Proposition 6, which states that the Abel system (2) has no limit cycles, and combining the different possibilities for the finite singular points we conclude that the possible global phase portraits are the nine described in Figure 1. Acknowledgements The first author is supported by the Ministerio de Economı́a, Industria y Com- petitividad, Agencia Estatal de Investigación grants PID2019-104658GB-I00 (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 sec- ond author is partially supported by FCT/Portugal through UID/MAT/04459/2019. References [1] J.C. Artés, J. Llibre, D. Schlomiuk and N. Vulpe, Geometric configurations of singular- ities of planar polynomial differential systems. A global classification in the quadratic case, to appear in Birkhäuser. 8 J. LLIBRE AND C. VALLS [2] D.M. Benardete, V.W. Noonburg and B. Pollina, Qualitative tools for studying periodic solutions and bifurcations as applied to the periodically harvested logistic equation, Amer. Math. Monthly 115 (2008), 202–219. [3] C. Chicone and T. 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[11] M.M. Peixoto, On the classification of flows on 2–manifolds. Academic, New York, pages 389–419, 1973. Dynamical systems (Proc. Sympos., Univ. Bahia, Salvador, 1971). [12] J.W. Reyn, Phase portraits of planar quadratic systems, Mathematics and its Applications, 583, Springer, 2007. [13] 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 Bel- laterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat 2 Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais 1049–001, Lisboa, Portugal Email address: cvalls@math.tecnico.ulisboa.pt