This is a preprint of: “On a class of polynomial differential systems of degree 4: phase portraits and limit cycles”, Jaume Llibre, Tayeb Salhi, Topol. Methods Nonlinear Anal., vol. 57(1), 441–463, 2021. DOI: [10.12775/TMNA.2020.042] ON A CLASS OF POLYNOMIAL DIFFERENTIAL SYSTEMS OF DEGREE 4: PHASE PORTRAITS AND LIMIT CYCLES JAUME LLIBRE1 AND TAYEB SALHI2 Abstract. In this paper we characterize the phase portraits in the Poincaré disc of the class of polynomial differential systems of the form ẋ = −y, ẏ = x + ax4 + bx2 y 2 + cy 4 , with a2 + b2 + c2 6= 0, which are symmetric with respect to the x-axis. Such systems have a center at the origin of coordinates. Moreover using the averag- ing theory of five order we study the number of limit cycles which can bifurcate from the period annulus of this center when it is perturbed inside the class of all polynomial differential systems of degree 4. 1. Introduction and statement of the main results By definition a polynomial differential system in R2 is a differential system of the form dx dy (1) = ẋ = P (x, y), = ẏ = Q(x, y), dt dt where the dependent variables are x and y, and the independent one (the time) is denoted by t, and P (x, y) and Q(x, y) are polynomials in the variables x and y with real coefficients. We denote by m = max{deg P, deg Q} the degree of the polynomial differential system. A singular point p of a polynomial differential system (1) is a center if there is a neighborhood U of p such that U \ {p} is filled by periodic orbits. A limit cycle of a polynomial differential system (1) is a periodic orbit isolated in the set of all periodic orbits of system (1). The phase portrait of a differential system is the decomposition of its domain of definition as union of all its oriented orbits. The phase portraits of the polynomial differential systems are drawn in the Poincaré disc which, roughly speaking, is the closed disc centered at the origin of coordinates with radius one, the interior of this disc is diffeomorphic to R2 and its boundary S1 corresponds to the infinity of R2 , each point of S1 provides a direction for going or coming from infinity. For more details see Chapter 5 of [4], we shall use the conditions and notations introduced in that chapter. 2010 Mathematics Subject Classification. Primary 34C05, 34A34. Key words and phrases. Polynomial differential systems, polynomial vector fields, phase por- traits, centers, limit cycles. 1 2 J. LLIBRE AND T. SALHI In the qualitative theory of polynomial differential systems in R2 two of the main problems are: first the characterization of their centers together with their phase portraits in the Poincaré disc, and second the analysis of the limit cycles that bifurcate from the periodic orbits surrounding those centers when they are perturb inside some class of polynomial differential systems. The polynomial differential systems of the form (2) ẋ = −y, ẏ = x + Qn (x, y) where Qn is a homogeneous polynomial of degree n started to be studied by Volok- itin and Ivanov in [19] trying to characterize their centers. Later on this character- ization was continued by Giné [5], and by Giné, Llibre and Valls in [6, 7]. Polynomial differential systems (2) with n = 1 are linear, and of course they can have centers, but linear centers perturbed inside the class of all linear differential systems cannot produce limit cycles, because any linear differential systems has no isolated periodic solutions. Polynomial differential systems (2) with n = 2 is a particular class of quadratic polynomial differential systems, and the centers of the quadratic polynomial differ- ential systems as their phase portraits in the Poincaré disc have been classified by Bautin [1], Kapteyn [9, 10], Schlomiuk [18], Vulpe [20], Żola̧dek [21], ... Again polynomial differential systems (2) with n = 3 are a particular class of cu- bic polynomial differential systems without quadratic terms. Volokitin and Ivanov in [19] proved that the centers of this class have their vector fields symmetric with respect to one of the coordinate axes, i.e. are reversible centers. These reversible centers as its phase portraits in the Poincaré disc have been classified by Żola̧dek [22, 23], and Buzzi, Llibre and Medrado [3]. Previously where characterized the centers without their phase portraits of all the cubic polynomial differential sys- tems without quadratic terms by Malkin [13] and Vulpe and Sibirskii [17]. When systems (2) with n = 3 are perturb inside the class of all cubic polynomial differen- tial systems the limit cycles that bifurcate from the periodic orbits of those centers were studied in [8, 11]. Giné in [5] proved that the polynomial differential systems (2) with n = 4 have a center at the origin if and only if its vector field is symmetric about one of the coordinate axes. From these centers the ones that are symmetric with respect to the y-axis where classified recently by Benterki and Llibre in [2], together with their phase portraits. So for the polynomial differential systems (2) with n = 4 only remains to classify the centers and their phase portraits symmetric with respect to the x-axis. These is the objective of this paper. More precisely, the first objective of this paper is to study the phase portraits in the Poincaré disc of the centers of systems (2) with n = 4 which are symmetric with respect to the x–axis, i.e. the phase portraits of the systems (3) ẋ = −y, ẏ = x + ax4 + bx2 y 2 + cy 4 . If c 6= 0 then without loss of generality we can assume that c = 1 doing the change 1 1 1 of variables x = X/c 3 , y = Y /c 3 . If c = 0 we can consider b = 1 doing x = X/b 3 , 1 1 1 y = Y /b 3 , and if c = b = 0 then we can assume a = 1 doing x = X/a 3 , y = Y /a 3 . Therefore we only need to study the phase portraits of system 3 in the cases: A CLASS OF POLYNOMIAL DIFFERENTIAL SYSTEMS OF DEGREE 4 3      a > 0;     b > 0, a = 0;      a < 0;           a > 0;       a > 0;  b = 1, a = 0;  c = 1, b = 0, a = 0; And c = 0, a < 0;        a < 0;         b = 0, a = 1.      a > 0;     b < 0, a = 0;   a < 0. The second objective is to study the limit cycles that bifurcate from the periodic orbits of the centers of systems (3) when they are perturbed inside the class of all polynomial differential systems of degree 4. Our first result is the following Theorem 1. The phase portraits in the Poincaré disc of systems (3) with a2 + b2 + c2 6= 0 are topologically equivalent to one of the fourteen phase portraits given in Figure 1. We perturb the centers of system (3) as follows 5 X ẋ = −y + εi Pi (x, y), i=1 (4) 5 X ẏ = x + ax4 + bx2 y 2 + cy 4 + εi Qi (x, y), i=1 where Pi (x, y) are polynomials of degree 4. For the definition of the averaging theory of order k = 1, . . . , 5, see section 2. In what follows we state our second main result. Theorem 2. For ε 6= 0 sufficiently small the number of limit cycles of the differ- ential system (4) obtained using averaging theory up to order five is two. We prove Theorem 2 in section 5. 2. The singular points and their local phase portraits The singular points finite or infinite are hyperbolic if its two eigenvalues have non-zero real part, are semi-hyperbolic if one of its eigenvalues is zero and the other is non-zero, are linearly zero if its linear part is identically zero. For instance the local phase portraits of hyperbolic and semi-hyperbolic singular points can be obtained from Theorems 2.15 and 2.19 of [4], respectively. The local phase portraits of linearly zero singular points can be studied doing the changes of variables called blow-ups, see Chapters 2 and 3 of [4]. The unique singular points with eigenvalues non-zero but which are not hyperbolic have eigenvalues purely imaginary, and they only can be foci or centers, see [4]. Proposition 3. The polynomial differential systems (3) have 4 J. LLIBRE AND T. SALHI Figure 1. All topologically different phase portraits in the Poincaré disc of systems (3). (a) two finite singular points if a 6= 0, a center at (0, 0), and a saddle at (−a−1/3 , 0); (b) one finite singular point if a = 0, a center at (0, 0). Proof. Since the eigenvalues √ of the linear part at the origin are ±i and at the point (−a−1/3 , 0) are ± 3, and the system is reversible, i.e. it is invariant by A CLASS OF POLYNOMIAL DIFFERENTIAL SYSTEMS OF DEGREE 4 5 (x, y, t) 7−→ (x, −y, −t), it follows that the origin is a center and the other is a hyperbolic saddle.  For studying the infinite singular points in the Poincaré disc, we use the defini- tions and notations given in chapter 5 of [4], then we have the following result. Proposition 4. Let ∆ = b2 − 4a. The polynomial differential systems (3) have in the local chart U1 : (a) no infinite points in the following cases: I: ∆ < 0; II: ∆ = 0 and b > 0;√ III: ∆ > 0 and b − ∆ > 0; V: c = 0, b = 1 and a > 0; VI: c = 0, b = 0 and a = 1; (b) one singular point, the origin (0, 0) which is a lineary zero and its local phase portrait is a topological saddle in the following cases: I: c = 1, b > 0 and a = 0; II: c = 1, b = 0 and a = 0; III: c = 0, b = 1 and a = 0; (c) two singular points in the following cases:   p I: two linearly zero topological saddles at ± −b/2, 0 if ∆ = 0, c = 1 and b < 0;  q  √ II: two semi-hyperbolic saddle-nodes at ± −(b − ∆)/2, 0 if c = 1, √ √ ∆ > 0, b + ∆ > 0, and b − ∆ < 0, moreover these two saddle- nodes have the two hyperbolic sectors in v > 0 (resp. v < 0), and the parabolic one in v < 0 (resp. v > 0); √  III: two semi-hyperbolic saddle-nodes at ± −a, 0 if c = 0, b = 1 and a < 0, moreover these two saddle-nodes have the two hyperbolic sectors in v > 0 (resp. v < 0), and the parabolic one in v < 0 (resp. v > 0); (d) three singular points, a linearly zero√ topological  saddle at (0, 0) and two semi-hyperbolic saddle-nodes at ± −b, 0 if c = 1, b < 0 and a = 0, moreover these two saddle-nodes have the two hyperbolic sectors in v > 0 (resp. v < 0), and the parabolic one in v < 0 (resp. v > 0); w w v u Figure 2. Blow-up at the origin of U1 in all the cases of statement (b) of Proposition 4. 6 J. LLIBRE AND T. SALHI  q  √ (e) four semi-hyperbolic saddle-nodes at ± −(b + ∆)/2, 0 and  q  √ √ ± −(b − ∆)/2, 0 if c = 1, ∆ > 0, and b + ∆ < 0, moreover these four saddle-nodes have the two hyperbolic sectors in v > 0 (resp. v < 0), and the parabolic one in v < 0 (resp. v > 0). The polynomial differential systems (3) have at the origin of the local chart U2 (f) a hyperbolic stable node if c = 1, and a linearly zero singularity if c = 0, which is a topological stable node if b = 1, and if b = 0 we have a topological stable node if a = 1. A summary of the results of Proposition 4 is given in Table 1.   ∆ < 0; c=1 ∆ = 0, and b > 0; No infinite singular points in U1 if   ∆ > 0, b > 0 and a > 0; b = 1 and a > 0; c=0  = 0 and a = 1; b b > 0 and a = 0, which is saddle; c=1 One infinite singular points in U1 if b = 0 and a = 0, which is saddle; c = 0,b = 1 and a = 0, is saddle;  ∆ = 0 and b√< 0, which are saddles; √ c=1 ∆ > 0, b + ∆ > 0 and b − ∆ < 0, Two infinite singular points in U1 if  which are semi-hyperbolic saddles-nodes; c = 0, b = 1 and a < 0, which are semi- hyperbolic saddles-nodes; Three infinite singular points in U1 if c = 1, b < 0 and a = 0, which √ are saddles; Four infinite singular points in U1 if c = 1, ∆ > 0, and b + ∆ < 0, which are semi-hyperbolic saddles-nodes. Table 1. The conditions for the existence of the infinite singular points in the local chart U1 in function of the parameters a, b and c. Here ∆ = b2 − 4a. w w v u u u Figure 3. Blow-up at the origin of U2 in the case c = 0 and b = 1. A CLASS OF POLYNOMIAL DIFFERENTIAL SYSTEMS OF DEGREE 4 7 w v w l l u Figure 4. Blow-up at the origin of U2 in the case c = 0, b = 0 and a = 1. Using the Poincaré compactification on the local chart U1 and V1 . The system (3) in the local chart U1 becomes . (5) u̇ = cu4 + u2 v 3 + bu2 + v 3 + a, v = uv 4 . For each real root u∗ of the polynomial f (u) = u̇ |v=0 = cu4 + bu2 + a, the point (u∗ , 0) is an infinite singular point in U1 . Since f (u) = c(u + r1 )(u − r1 )(u + r2 )(u − r2 ), with s √ s √ −(b + ∆) −(b − ∆) r1 = , r2 = , ∆ = b2 − 4a ≥ 0, when c = 1. 2 2 √ If c = 0 and b = 1 we have u∗ = ± −a, it follows that in the infinity of the local chart U1 there can be 0, 1, 2, 3 or 4 singularities. The eigenvalues λ1 and λ2 for these infinite singular points are: √ • for (u, v) = (±r1 , 0) we have λ1 = ∓2√∆r1 and λ2 = 0; • for (u, v) = (±r√2 , 0) we have λ1 = ±2 ∆r √ 2 and λ2 = 0; • for (u, v) = (± −a, 0) we have λ1 = ±2 −a and λ2 = 0. Proof of statement (a) of Proposition 4. The proof follows directly from the fact that the polynomial f (u) has no real roots under the five assumptions of statement (a).  Proof of statement (b) of Proposition 4. If c = 1, b > 0 and a = 0 the origin of local chart U1 is a linearly zero singularity. In the direction u we do the blow up (u, v) = (l3 , l2 w), then system (5) becomes 1 1 l˙ = l4 (l6 w3 + cl6 + lw3 + b), ẇ = l3 (l6 w4 − 2cl6 w − 2w4 − 2bw). 3 3 Rescaling the independent variable we eliminate the common factor l3 /3 and we get the new system l˙ = l7 w3 + cl7 + lw3 + bl, ẇ = l6 w4 − 2cl6 w − 2w4 − 2bw. This new differential system √ on l = 0 has two singular points, the origin which is a saddle, and the point (0, 3 −b) is semi-hyperbolic with eigenvalue 0 and b. In order to obtain the local phase portrait at this semi-hyperbolic infinite singular point we use Theorem 2.19 of [4], and we see that this point is a topological saddle. Going 8 J. LLIBRE AND T. SALHI back through the change of variables, we get the local phase portrait at the origin of the chart U1 shown in Figure 2, i.e. we have at the origin of U1 a topological saddle. The proof of the case a = 0, b = 1, and c = 0, is the same as the proof of the previous case. If a = 0, b = 0 and c = 1 system (3) in the local chart U1 becomes . u̇ = u4 + u2 v 3 + v 3 , v = uv 4 . In the direction u we consider the blow up (u, v) = (l3 , l4 w), and in the new variables the system writes 1  1 l˙ = l10 l6 w3 + w3 + 1 , ẇ = − l9 (l6 w4 + 4w4 + 4w). 3 3 Doing a rescaling of the independent variable we eliminate the common factor l9 /3 and we get the system l˙ = l(l6 w3 + w3 + 1), ẇ = −(l6 w4 + 4w4 + 4w). This differential system on l = 0 has two singular points, the origin with eigenvalues 1 and −4 which is a saddle, and a semi-hyperbolic point at (0, −1) with eigenvalues 0 and 12. Applying Theorem 2.19 of [4] we conclude that this semi-hyperbolic point is a topological saddle. Going back through the changes of variables, we get the local phase portrait at this point, i.e. the origin of U1 is a topological saddle. This completes the proof of statement (b).  Proof of statement (c) of Proposition 4. If ∆ = 0, c = 1 and b < 0 the singular p points ± −b/2, 0 of system (3) in the chart U1 are linearly zero. p  The singular point −b/2, 0 , by the change of variables (u, v) 7−→ (l, v) with p u = l+ −b/2, goes to the origin of U1 , after that we do the blow-up (l, v) 7−→ (l, w) where w= v/l. Now from the obtained differential system we eliminate the common factor l doing a rescaling of the independent variable we get that the differential system (5) becomes q √ l˙ = w3 l4 + − 2b c 1 w3 l3 + 2c (2c − b) w3 l2 + cl3 + 2 −2bcl2 − 2bl, q √ . b 1 w = − 2c w4 l2 + 2c (b − 2c) w4 l − cwl2 − 2 −2bcwl + 2bw. This system has a unique singular point on l = 0 the origin which is a saddle with eigenvalues ±2b . Going back through all the changes p  of variables, we get the local phase portrait at the singular points −b/2, 0 . The same study can be made at  p  the singular point − −b/2, 0 . This proves the case I of the statement (c). The case II can be proved in a similar way. √ If c = 0, b = 1 and a < 0√the singular points (± −a, 0) of system (3) in the chart U1 with eigenvalues ±2 −a and 0 are semi-hyperbolic. Form Theorem 2.19 of [4], they are saddle-nodes. This proves the case III of statement (c). Hence the proofs of statement (c) is done.  A CLASS OF POLYNOMIAL DIFFERENTIAL SYSTEMS OF DEGREE 4 9 Proof of statement (d) of Proposition 4. When c = 1, b < 0 and a = 0 system (3) has three singular points in the local chart U1 . The origin of this chart is a linearly zero topological saddle, the√proof is similar to the proof of the case I of statement (b). The singular points ± −b, 0 are semi-hyperbolic sadlle-nodes using Theorem 2.19 of [4]. Hence the proof of statement (d) is done.  Proof of statement (e) of Proposition 4. This proof is similar to the proof of the statement (c).  Proof of statement (f ) of Proposition 4. The unique possible singular points at in- finity which are not covered by the local charts U1 and V1 are the origins of co- ordinates of the local charts U2 and V2 . So we only need to study if the origin of the chart U2 is a singular point or not, and if it is we must study its local phase portrait. System (3) in the local chart U2 becomes u̇ = −au5 − bu3 − u2 v 3 − u − v 3 , v̇ = −au4 v − bu2 v − uv 4 − v. If c = 1 the eigenvalues at the origin are −1 of multiplicity 2, so the origin is a hyperbolic stable node. If c = 0 and b = 1 in the direction u we do the blow up (u, v) −→ (u, uw), and we obtain the system u̇ = −u5 w3 − au5 − u3 w3 − u3 , ẇ = u2 w4 . Rescaling the independent variable of the differential system we eliminate the com- mon factor u2 between u̇ and ẇ. The new differential system becomes u̇ = −u3 w3 − au3 − uw3 − u, ẇ = w4 . Applying Theorem 2.19 of [4] to the origin of this system, we get the origin is a semi-hyperbolic saddle-node. Going back through the changes of variables, we obtain the local phase portrait at the origin of U2 , which is the one shown in Figure 3. If c = 0, b = 0 and a = 1 system (3) in the local chart U2 becomes u̇ = −u5 − u2 v 3 − v 3 , v̇ = −u4 v − uv 4 . In the direction u we do the blow-up (u, v) 7−→ (l3 , l5 w), and we obtain the system 1  1 12 6 4 l˙ = − l13 l6 w3 + w3 + 1 , ẇ = l (2l w + 5w4 + 2w). 3 3 Doing a rescaling of the independent variable we eliminate the common factor l12 , and the differential system becomes 1  1 l˙ = − l l6 w3 + w3 + 1 , ẇ = (2l6 w4 + 5w4 + 2w). 3 3 On l = 0 this differential system has two singular points, the originp with eigen- values −1/3 and 2/3, so it is a hyperbolic saddle, and the point (0, 3 −2/5) with eigenvalues −1/5 and −2, which is a stable hyperbolic node. Going back through the change of variables we get that the local phase portrait at origin of U2 is the one shown in Figure 4. So statement (f) is proved.  10 J. LLIBRE AND T. SALHI 3. Phase portraits in the Poincaré disc We study the phase portraits of system (3) in the Pioncaré disc, first for the case c = 0, b = 0, a = 1, after for the case c = 0, b = 1; and finally for the case c = 1. Figure 5. Phase portrait for c = 0, b = 0, and a = 1. The global phase portrait has nine separatrices (S = 9) and three canonical regions (R = 3), see the definitions of separatrix and of a canonical region in section 1.9 of [4]. 3.1. Phase portrait for c = 0, b = 0 and a = 1. In this case the infinity has no singular points in the chart U1 , and the origins of the charts U2 and V2 are singular points. The system has two finite singular points a center and a saddle. From Propositions 3 and 4 the local phase portraits at the singularities of system (2) are given in the left picture of Figure 5. Then from these local phase portraits, there is a unique possible global phase portrait in the Poincaré disc given in the right picture of Figure 5. 3.2. Phase portrait for c = 0, b = 1. Then we draw its diagram of bifurcation on the straight lien in Figure 6, and its singular points in Table 2. l4 p l5 a Figure 6. Bifurcation diagram in the case c=0 and b=1. Regions infinite singular points finite singular points U1 U2 l4 : a < 0 two origin center + saddle p: a=0 one origin center l5 : a > 0 ∅ origin center + saddle Table 2. The singular points of system (2) in the cases c = 0 and b = 1. A CLASS OF POLYNOMIAL DIFFERENTIAL SYSTEMS OF DEGREE 4 11 3.2.1. Phase portrait in l5 (c = 0, b = 1, a > 0), see Figure 6. In this case the local phase portraits of the singularities of system (2) coincide with case of c = 0, b = 0, a = 1. So we get the global phase portrait of Figure 5. 3.2.2. Phase portrait in p (c = 0, b = 1, a = 0),see Figure 6. From Propositions 3 and 4 the local phase portraits at the singularities of system (2) are given in the left picture of Figure 7. Then from these local phase portraits there is a unique possible global phase portrait in the Poincaré disc given in the right picture of Figure 7. Figure 7. Phase portrait in p (c = 0, b = 1, a = 0). The global phase portrait has ten separatrices (S = 10) and two canonical regions (R = 2). Figure 8. Phase portrait in l4 (c = 0, b = 1, a < 0). The global phase portrait has nineteen separatrices (S = 19) and five canonical regions (R = 5). 3.2.3. Phase portrait in l4 (c = 0, b = 1, a < 0), see Figure 6. From Propositions 3 and 4 the local phase portraits at the singularities of system (2) are given in the left picture of Figure 8. Then from these local phase portraits there is a unique possible global phase portrait in the Poincaré disc given in the right picture of Figure 8. 12 J. LLIBRE AND T. SALHI b l1 R3 R1 a b l3 l2 R2 b2-4a Figure 9. Bifurcation diagram in the case c=1. 3.3. Phase portrait for c = 1. According with Table 1, we draw in Figure 9 the diagram of bifurcation on the plane (a, b) the case c = 1. We illustrate the singular points corresponding to the bifurcation diagram of Figure 9 in Table 3 : Regions infinite singular points finite singular points U1 U2 √ R1 : (b > −2 a) ∧ (a > 0) ∅ origin center + saddle √ l2 : (b = −2 a) ∧ (a > 0) two origin center + saddle √ R2 : (b < −2 a) ∧ (a > 0) four origin center + saddle l3 : (b < 0) ∧ (a = 0) three origin center R3 : a < 0 two origin center + saddle l1 : (b ≥ 0) ∧ (a = 0) one origin center Table 3. The singular points of system (2) in the case c = 1. √ 3.3.1. Phase portrait in R1 (b > −2 a, a > 0), see Figure 9. In this case the local phase portraits of the singularities of system (2) coincide with case of c = 0, b = 0, a = 1. So we get the global phase portrait of Figure 5. 3.3.2. Phase portrait in l1 (b > 0, a = 0), see Figure 9. In this case the local phase portraits of the singularities of system (2) coincide with case of c = 0, b = 1, a = 0. So we get the global phase portrait of Figure 7. √ 3.3.3. Phase portrait in l2 (b = −2 a, a > 0), see Figure 9. From Propositions 3 and 4 the local phase portraits at the singularities of system (2) are given in the left picture of Figure 10. Then from these local phase portraits and using the fact that A CLASS OF POLYNOMIAL DIFFERENTIAL SYSTEMS OF DEGREE 4 13 the phase portrait is symmetric with respect to the x-axis there is a unique possible global phase portrait in the Poincaré disc given in the right picture of Figure 10. √ Figure 10. Phase portrait in l2 (b = −2 a, a > 0). The global phase portrait has eighteen separatrices (S = 18) and four canonical regions (R = 4). 3.3.4. Phase portrait in R3 (a < 0), see Figure 9. From Propositions 3 and 4 the local phase portraits at the singularities of system (2) are given in Figure 11. Then from these local phase portraits and using the fact that the phase portrait is symmetric with respect to the x-axis there are three possible global phase portraits in the Poincaré disc given in Figure 12. Figure 11. Local phase portrait in R3 (a < 0). 3.3.5. Phase portrait in l3 (c = 1, b < 0, a = 0), see Figure 9. From Propositions 3 and 4 the local phase portraits at the singularities of system (2) are given in Figure 13. Then from these local phase portraits and using the fact that the phase portrait is symmetric with respect to the x-axis there are three possible global phase portraits in the Poincaré disc given in Figure 14. 14 J. LLIBRE AND T. SALHI Figure 12. The global phase portraits in R3 (c = 1, a < 0). Going from the left to the right these phase portraits have 19, 18, 19 separa- trices and 5, 4, 5 canonical regions, respectively. The phase portrait on the left can be obtained by taking a = −1 and b = 2, the one on the right can be obtained with a = −1 and b = −5. The phase portrait in the middle exists by continuity on the parameter b. Figure 13. The local Phase portrait in l3 (c = 1, b < 0, a = 0). Figure 14. The global phase portraits in l3 (c = 1, b < 0, a = 0). Going from the left to the right these phase portraits have 20, 19, 20 separatrices, and 4, 3, 4 canonical regions, respectively. The phase por- trait on the left can be obtained by taking b = −1 and the one on the right can be obtained with b = −5. The phase portrait in the middle exists by continuity on the parameter b. A CLASS OF POLYNOMIAL DIFFERENTIAL SYSTEMS OF DEGREE 4 15 √ 3.3.6. Phase portrait in R2 (c = 1, b < −2 a, a > 0), see Figure 9. From Propo- sitions 3 and 4 the local phase portraits at the singularities of system (2) are given in Figure 15. Then from these local phase portraits and using the fact that the phase portrait is symmetric with respect to the x-axis there are five possible global phase portraits in the Poincaré disc given in Figure 16. √ Figure 15. Local phase portrait in R2 (c = 1, b < −2 a, a > 0). √ Figure 16. The global Phase portraits in R2 (c = 1, b < −2 a, a > 0). Going from the upper left to the down right these phase portraits have 28, 27, 29, 27, 29 separatrcies, and 6, 5, 7, 6, 7 canonical regions, respectively. The phase portrait on the upper right can be obtained by 1 taking c = 1, b = −4.1, and a = 10 , and the down left phase portrait 1 can be obtained for c = 1, b = −4.2, a = 10 . The second, third and fourth global portraits exist by continuity on the parameter b. 16 J. LLIBRE AND T. SALHI 4. The averaging theory up to order 5 In this section we present the basic results from the averaging theory that we shall need for proving the main results of this paper. We consider the differential system of the form k X (6) ż(t) = εi Fi (t, z) + εk+1 R(t, z, ε), i=0 where Fi : R × D −→ R, R : R × D × (−εf , εf ) −→ R are continuous functions, T −periodic in the first variable, and D is an open subset of R, and ε a small parameter. From [12] we define the following functions yi (t, z) for k = 1, 2, 3, 4, 5 related to system (6) : Z s Z s y1 (s, z) = F1 (t, z)dt, y2 (s, z) = [2F2 (t, z) + 2∂F1 (t, z)y1 (t, z)]dt, 0 0 Z s y3 (s, z) = [6F3 (t, z)+6∂F2 (t, z)y1 (t, z)+3∂ 2 F1 (t, z)y1 (t, z)2 +3∂F1 (t, z)y2 (t, z)]dt, 0 Z s y4 (s, z) = [24F4 (t, z) + 24∂F3 (t, z)y1 (t, z) + 0 12∂ 2 F2 (t, z)y1 (t, z)2 + 12∂F2 (t, z)y2 (t, z) + 12∂ 2 F1 (t, z)y1 (t, z)y2 (t, z) + 4∂ 3 F1 (t, z)y1 (t, z)3 + 4∂F1 (t, z)y3 (t, z)]dt, Z s y5 (s, z) = [120F5 (t, z) + 120∂F4 (t, z)y1 (t, z) + 0 60∂ 2 F3 (t, z)y1 (t, z)2 + 60∂F3 (t, z)y2 (t, z) + 60∂ 2 F2 (t, z)y1 (t, z)y2 (t, z) + 20∂ 3 F2 (t, z)y1 (t, z)3 + 20∂F2 (t, z)y3 (t, z) + 20∂ 2 F (s, z)y1 (t, z)y3 (t, z) + 15∂ 2 F1 (t, z)y2 (t, z)2 + 30∂ 3 F1 (t, z)y1 (t, z)2 y2 (t, z)) + 5∂ 4 F1 (t, z)y1 (t, z)4 + 5∂F1 (t, z)y4 (t, z)]dt. Here ∂ k Fl (s, z) means the k − th partial derivative of the function Fl (s, z) with respect to the variable z. Also from [12] we have the functions Z T f1 (z) = F1 (t, z)dt, 0 Z T f2 (z) = [F2 (t, z) + ∂F1 (t, z)y1 (t, z)]dt, 0 Z T 1 1 f3 (z) = [F3 (t, z) + ∂F2 (t, z)y1 (t, z) + ∂ 2 F1 (t, z)y1 (t, z)2 + ∂F1 (t, z)y2 (t, z)]dt, 0 2 2 A CLASS OF POLYNOMIAL DIFFERENTIAL SYSTEMS OF DEGREE 4 17 Z T f4 (z) = [F4 (t, z) + ∂F3 (t, z)y1 (t, z) + 0 1 2 1 1 ∂ F2 (t, z)y1 (t, z)2 + ∂F2 (t, z)y2 (t, z) + ∂ 2 F1 (t, z)y1 (t, z)y2 (t, z) + 2 2 2 1 3 3 1 ∂ F (t, z)y1 (t, z) + ∂F1 (t, z)y3 (t, z)]dt, 6 6 Z T f5 (z) = [F5 (t, z) + F4 (t, z)y1 (t, z) + 0 1 2 1 1 ∂ F3 (t, z)y1 (t, z)2 + ∂F3 (t, z)y2 (t, z) + ∂ 2 F2 (t, z)y1 (t, z)y2 (t, z) + 2 2 2 1 3 1 1 ∂ F2 (t, z)y1 (t, z)3 + ∂F2 (t, z)y3 (t, z) + ∂ 2 F1 (s, z)y1 (t, z)y3 (t, z) + 6 6 6 1 2 2 1 3 2 ∂ F1 (t, z)y2 (t, z) + ∂ F1 (t, z)y1 (t, z) y2 (t, z) + 8 4 1 4 4 1 ∂ F1 (t, z)y1 (t, z) + ∂F1 (t, z)y4 (t, z)]dt. 24 24 For more details see [12]. 5. Proof of theorem 2 Consider system (3) we shall study which periodic solutions of its center become limit cycles when we perturb the center inside the whole class of polynomial dif- ferential systems of degree 4 as in system (3). The polynomial Pi and Qi given in system (3) are 4 X (i) Pi (x, y) = akj xk y j , 0 4 X (i) Qi (x, y) = bkj xk y j . 0 This study will be done by applying the averaging theory described in section 4, we introduce a small parameter ε doing the scaling x = εX, y = εY. Thus we obtain a differential system (Ẋ, Ẏ ). After that we pass the differential system to polar coordinates (ṙ, θ̇). Then we take as independent variable the angle θ, and the system (ṙ, θ̇) becomes the differential equation dr/dθ. By doing a Taylor expansion truncated at 5-th order in ε we obtain an expression for dr/dθ similar to the one of the differential system (6). In short we have written our differential system (4) in the normal form (6) for applying the averaging theory. We give only the expression of functions F1 (r, θ). The explicit expressions of Fi (r, θ) for i = 2, ..., 5 are quite large so we omit them. 18 J. LLIBRE AND T. SALHI The experssion of F1 (r, θ) is (A + BC)/D where (1) (1) (1) (1) (1) A = sin θ cos θ a01 + r(a02 + b11 ) sin θ + r2 (a03 + b12 ) sin2 θ + (1) (1) (1)  r3 (a04 + b13 ) sin3 θ + b10 + (1) (1) (1) (1) (1) cos2 θ a10 + r(a11 + b20 ) sin θ + r2 (a12 + b21 ) sin2 θ + (1) (1)  r3 (a13 + b22 ) sin3 θ + (1) (1) (1) (1) (1)  r cos3 θ a20 + r(a21 + b30 ) sin θ + r2 (a22 + b31 ) sin2 θ + (1) (1) (1) (1) r2 cos4 θ(a30 + r(a31 + b40 ) sin θ) + a40 r3 cos5 θ + (1) (1) (1) (1)  sin2 θ b01 + b02 r sin θ + b03 r2 sin2 θ + b04 r3 sin3 θ , (1) (1) (1) (1)  B = − sin2 θ a01 + a02 r sin(θ) + a03 r2 sin2 θ + a04 r3 sin3 θ + (1) (1) (1) (1) (1) sin θ cos θ − a10 − r(a11 − b02 ) sin θ − r2 (a12 − b03 ) sin2 θ + (1) (1) (1)  r3 (−(a13 − b04 )) sin3 θ + b01 + (1) (1) (1) (1) cos2 θ − r(a20 − b11 ) sin θ − r2 (a21 − b12 ) sin2 θ + (1) (1) (1)  r3 (−(a22 − b13 )) sin3 θ + b10 + (1) (1) (1) (1) (1)  r cos3 θ − r(a30 − b21 ) sin θ + r2 (−(a31 − b22 )) sin2 θ + b20 − (1) (1) (1) (1) r2 cos4 θ(r(a40 − b31 ) sin θ − b30 ) + b40 r3 cos5 θ,  C = r4 sin(θ) a cos4 (θ) + b sin2 (θ) cos2 (θ) + c sin4 (θ) ; 2 D = ar3 cos5 (θ) + br3 sin2 (θ) cos3 (θ) + cr3 sin4 (θ) cos(θ) + sin2 (θ) + cos2 (θ) . The differential equation (4) is the normal form (6) for applying the averaging theory up to five order in ε, where in (6) we have now z = r, t = θ and Fk (θ, r) is the coefficient of εk in dr/dθ for k = 1, 2, 3, 4, 5 we do not write their huge expressions, easy to compute and manipulate with an algebraic manipulator as mathematica or maple. We compute the function f1 (r) defined in Section 4, and we get 1 (1) (1) f1 (r) = (a + b01 )r. 2 10 Clearly the polynomial f1 (r) cannot have positive roots. For applying the averaging (1) (1) theory of second order we need that f1 (r) ≡ 0. So we take a10 = −b01 . Computing the function f20 (r) defined in Section 4, we obtain 1 (2) (2) f2 (r) = (a + b01 )r. 2 10 Clearly the polynomial f2 (r) cannot have positive roots. For applying the averaging (2) (2) theory of third order we need that f2 (r) ≡ 0. So we take a10 = −b01 . Computing the function f3 (r) defined in Section 4, we obtain 1 (3) (2) (1) (1) (2) (3) (2) (1) (2) (1) 1 (1) (1) (1) (1) f3 (r) = (a +a00 a11 −2a20 b00 +b01 +2a00 b02 −b00 b11 )r+ (a12 +3a30 +3b03 +b21 )r3 . 2 10 8 A CLASS OF POLYNOMIAL DIFFERENTIAL SYSTEMS OF DEGREE 4 19 So the polynomial f3 (r) can have at most one positive real root. In order to apply the averaging theory of fourth order we need that f3 (r) ≡ 0. So we take (3) (3) (2) (1) (1) (2) (2) (1) (2) (1) = −a10 − a00 a11 + 2a20 b00 − 2a00 b02 + b00 b11 , b01 (1) 1 (1) (1) (1) b03 = (−a12 − 3a30 − b21 ). 3 Computing the function f4 (r) defined in Section 4, we obtain 1 (4) (3) (1) (2) (1) (1) (2) (2) (2) (2) (1) (3) f4 (r) = (a + a00 a11 + a00 a01 a11 + a00 a11 − 2a20 b00 − 2a20 b00 − 2 10 (2) (1) (1) (1) (2) (1) (4) (3) (1) (2) (1) (1) (2) (1) (1) 2a00 a20 b01 + a11 b00 b01 + b01 + 2a00 b02 + 2a00 a01 b02 + 2b00 b01 b02 + (2) (2) (1) (2) (1) (3) (1) (2) (1) (1) (2) (1) (1) (2) (2) 2a00 b02 + 2a20 b00 b10 − b00 b11 − a00 b01 b11 + b00 b10 b11 − b00 b11 )r + 1 (1) (1) (2) (1) (1) (1) (1) (2) (2) (1) (1) (a a + a12 + a11 a20 − 3a01 a30 + 3a30 + 2a00 b − 2a21 b01 + 8 02 11 (1) (1) (2) (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) 2a02 b02 + 3b03 − 3a30 b10 − b02 b11 − 2b01 b12 − 2a20 b20 − b11 b20 − (1) (1) (1) (1) (2) (2) a01 b21 − b10 b21 + b21 + 12a00 c)r3 . So the polynomial f4 (r) can have at most one positive real root. In order to apply the averaging theory of fifth order we need that f4 (r) ≡ 0. So we take (4) (4) (3) (1) (2) (1) (1) (2) (2) (2) (2) b01 = −a10 − a00 a11 − a00 a01 a11 − a00 a11 + 2a20 b00 + (1) (3) (2) (1) (1) (1) (2) (1) (3) (1) 2a20 b00 + 2a00 a20 b01 − a11 b00 b01 − 2a00 b02 − (2) (1) (1) (2) (1) (1) (2) (2) (1) (2) (1) 2a00 a01 b02 − 2b00 b01 b02 − 2a00 b02 − 2a20 b00 b10 + (3) (1) (2) (1) (1) (2) (1) (1) (2) (2) b00 b11 + a00 b01 b11 − b00 b10 b11 + b 00 b11 , (2) 1 (1) (1) (2) (1) (1) (1) (1) (2) b03 = (−a02 a11 − a12 − a11 a20 + 3a01 a30 − 3a30 − 3 (2) (1) (1) (1) (1) (1) (1) (1) (1) 2a00 b + 2a21 b01 − 2a02 b02 + 3a30 b10 + b02 b11 + (1) (1) (1) (1) (1) (1) (1) (1) (1) (1) 2b01 b12 + 2a20 b20 + b11 b20 + a01 b21 + b10 b21 − (2) (2) b21 − 12a00 c). Computing the function f5 (r) defined in Section 4 we obtain that it is of the form f5 (r) = A1 r + A2 r3 + A3 r5 . We do not give the big expressions of the independent coefficients Ai for i = 1, 2, 3. Hence the polynomial f5 (r) can have at most two positive real roots. This completes the proof of the Teorem 2. Acknowledgments The first author is partially supported by by the Ministerio de Ciencia, Inno- vación y Universidades, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER) and MDM-2014-0445, 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 partially supported by the Alge- rian Ministry of Higher Education and Scientific Research. 20 J. LLIBRE AND T. SALHI References [1] N.N. Bautin, On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or center type, Mat. Sbornik 30 (1952), 181–196, Amer. Math. Soc. Transl. Vol. 100 (1954), 1–19. [2] R. Benterki and J. Llibre, Centers and limit cycles of polynomial differential systems of degree 4 via averaging theory, J. Comput. Appl. Math. 313 (2017), 273–283. [3] C.A. Buzzi, J. Llibre and J.C. Medrado, Phase portraits of reversible linear differential systems with cubic homogeneous polynomial nonlinearities having a non-degenerate center at the Origin, Qual. Theory Dyn. Syst. 7 (2009), 369–403. [4] F. Dumortier, J. Llibre and J.C. Artés, Qualitative Theory of Planar Differential Sys- tems, UniversiText, Springer–Verlag, New York, 2006. [5] J. Giné, Conditions for the existence of a center for the Kukles homogenenous systems, Comput. Math. Appl. 43 (2002), 1261–1269. [6] J. Giné, J. LLibre and C. Valls, Centers for the Kukles homogeneous systems with odd degree, Bull. London Math. Soc. 47 (2015), 315–324. [7] J. Giné, J. LLibre and C. Valls, Centers for the Kukles homogeneous systems with even degree, J. Appl. Anal. an Comp. 7 (2017), 1534-1548. [8] J. Itikawa and J. Llibre, Limit cycles for continuous and discontinuous perturbations of uniform isochronous cubic centers, J. Comput. Appl. Math. 277 (2015), 171–191. [9] W. Kapteyn, On the midpoints of integral curves of differential equations of the first de- gree, Nederl. Akad. Wetensch. Verslag. Afd. Natuurk. Konikl. Nederland (1911), 1446–1457 (Dutch). [10] W. Kapteyn, New investigations on the midpoints of integrals of differential equations of the first degree, Nederl. Akad. Wetensch. Verslag Afd. Natuurk. 20 (1912), 1354–1365; 21, 27–33 (Dutch). [11] J. Llibre, B.D. Lopes and J.R. Moraes, Limit cycles of cubic polynomial differential systems with rational first integrals of degree 2, Appl. Math. and Comput. 250 (2015), 887– 907. [12] J. Llibre, D. D. Novaes and M. A. Teixeira, Higher order averaging theory for finding periodic solution via Brouwer degree, Nonlinearity 27 (2014), 563-583. [13] K.E. Malkin, Criteria for the center for a certain differential equation, (Russian) Volz. Mat. Sb. Vyp. 2 (1964), 87—91. [14] L. Markus, Global structure of ordinary differential equations in the plane: Trans. Amer. Math Soc. 76 (1954), 127–148. [15] D. A. Neumann, Classification of continuous flows on 2–manifolds, Proc. Amer. Math. Soc. 48 (1975), 73–81. [16] M.M. Peixoto, Dynamical Systems. Proccedings of a Symposium held at the University of Bahia, 389–420, Acad. Press, New York, 1973. [17] N.I. Vulpe and K.S. Sibirskii, Centro–affine invariant conditions for the existence of a center of a differential system with cubic nonlinearities, (Russian) Dokl. Akad. Nauk SSSR 301 (1988), 1297–1301; translation in Soviet Math. Dokl. 38 (1989), 198–201 [18] D. Schlomiuk, Algebraic particular integrals, integrability and the problem of the center, Trans. Amer. Math. Soc. 338 (1993), 799–841. [19] E.P. Volokitin and V.V. Ivanov, Isochronicity and Commutation of polynomial vector fields, Siberian Mathematical Journal 40 (1999), 22–37. [20] N. I. Vulpe, Affine–invariant conditions for the topological discrimination of quadratic sys- tems with a center, Differential Equations 19 (1983), 273–280. [21] H. Żola̧dek, Quadratic systems with center and their perturbations, J. Differential Equations 109 (1994), 223–273. [22] H. Żola̧dek, The classification of reversible cubic systems with center, Topol. Methods Non- linear Anal. 4 (1994), 79–136. [23] H. Żola̧dek, Remarks on: “The classification of reversible cubic systems with center, Topol. Methods Nonlinear Anal. 4 (1994), 79–136”, Topol. Methods Nonlinear Anal. 8 (1996), 335– 342. A CLASS OF POLYNOMIAL DIFFERENTIAL SYSTEMS OF DEGREE 4 21 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bel- laterra, Barcelona, Catalonia, Spain Email address: jllibre@mat.uab.cat 2 Département de Mathématiques, Université de Bordj Bou Arréridj, Bordj Bou Arréridj 34265, El anasser, Algeria Email address: t.salhi@univ-bba.dz