This is a preprint of: “Polynomial differential systems with even degree have no global centers”, Jaume Llibre, Clàudia Valls, J. Math. Anal. Appl., vol. 503, 125281:1–5, 2021. DOI: [10.1016/j.jmaa.2021.125281] POLYNOMIAL DIFFERENTIAL SYSTEMS WITH EVEN DEGREE HAVE NO GLOBAL CENTERS JAUME LLIBRE1 AND CLAUDIA VALLS2 Abstract. Let ẋ = P (x, y), ẏ = Q(x, y) be a differential system with P and Q real polynomials, and let d = max{deg P, deg Q}. A singular point p of this differential system is a global center if R2 \ {p} is filled with periodic orbits. We prove that if d is even then the polynomial differential systems have no global centers. 1. Introduction and statement of the main results A singular point q of a vector field defined in R2 is a center if it has a neighbourhood filled of periodic orbits with the unique exception of q. The period annulus of the center q is the maximal neighbourhood U of q such that all the orbits contained in U are periodic except of course q. A center is global if its period annulus is R2 \ {q}. The notion of center goes back to Poincaré, see [6]. It is well known that any quadratic polynomial system (i.e. n = 2) has no global centers. The proof of this result is very large. It is based in classifying all the centers of the quadratic systems and then see that they are not global centers, see [2, 3, 4, 7, 8]. Let P, Q ∈ R[x, y] and d = max{degP, degQ}. We will show that the polynomial differential system (1) ẋ = P (x, y), ẏ = Q(x, y), with d even do not have global centers. This is the main aim of this paper. Our proof for all d even is shorter than the existing one for d = 2. Theorem 1. The polynomial differential system (1) with even degree has no global centers. The proof of Theorem 1 is given in section 3. In the following section we state and prove some auxiliary results that will be used during the proof. 2010 Mathematics Subject Classification. Primary 34C05. Key words and phrases. global centers, polynomial differential system, Poincaré compact- ification. 1 2 J. LLIBRE AND C. VALLS 2. Auxiliary results In the proof of Theorem 1 we will use the Poincaré compactification of a planar polynomial vector field X (x, y) = (P (x, y), Q(x, y)) of degree d. The Poincaré compactification of X , denoted by p(X ), is an induced vector field on S2 = {y = (y1 , y2 , y3 ) ∈ R3 : y12 + y22 + y32 = 1}. We call S2 the Poincaré sphere. For more details on the Poincaré compactification see [5, Chapter 5]. Here we just introduce what will be needed. Denote by Tp S2 be the tangent space to S2 at the point p. Assume that X is defined in the plane T(0,0,1) S2 = R2 . Consider the central projection f : T(0,0,1) S2 → S2 . This map defines two copies of X , one in the open northern hemisphere H+ and other in the open southern hemisphere H− . Denote by X 1 the vector field Df ◦ X defined on S2 except on its equator S1 = {y ∈ S2 : y3 = 0}. Clearly S1 is identified to the infinity of R2 . In order to extend X 1 to a vector field on S2 (including S1 ) it is necessary that X satisfies suitable conditions. In the case that X is a planar polynomial vector field of degree n then p(X ) is the only analytic extension of y3d−1 X 0 to S2 . On S2 \ S1 = H+ ∪ H− there are two symmetric copies of p(X ), one in H+ and other in H− , and knowing the behaviour of p(X ) around S1 , we know the behaviour of X at infinity. The Poincaré compactification has the property that S1 is invariant under the flow of p(X ). The singular points of X are called the finite singular points of X or of p(X ), while the singular points of p(X ) contained in S1 , i.e. at infinity, are called the infinite singular points of X or of p(X ). It is known that the infinity singular points appear in pairs diametrically opposed. To study the vector field p(X ) we use six local charts on S2 given by Uk = {y ∈ S2 : yk > 0}, Vk = {y ∈ S2 : yk < 0} for k = 1, 2, 3. The corresponding local maps φk : Uk → R2 and ψk : Vk → R2 are defined as φk (y) = −ψk (y) = (ym /yk , yn /yk ) for m < n and m, n 6= k. We denote by z = (u, v) the value of φk (y) or ψk (y) for any k, such that (u, v) will play different roles depending on the local chart we are considering. The points of the infinity S1 in any chart have v = 0. The expression for p(X) in local chart (U1 , φ1 ) is given by        d 1 u 1 u d+1 1 u (2) u̇ = v −uP , +Q , , v̇ = −v P , . v v v v v v The infinite singular points are the endpoints of the straight lines de- fined by the real linear factors of the homogeneous polynomial yPd (x, y) − xQd (x, y), being Pd and Qd the homogeneous parts of the polynomials P and Q of degree d. Let q be an infinite singular point and let h be a hyperbolic sector of q. We say that h is degenerate if its two separatrices are contained in the equator of S2 (i.e. in S1 ). It is well-known that an infinite singular point ON THE GLOBAL CENTERS OF POLYNOMIAL DIFFERENTIAL SYSTEMS 3 p formed by two degenerated hyperbolic sectors must have its linear part identically zero (see for instance Chapters 2,3 and Theorems 2.5, 2.19 and 3.5 of [5]. For proving Theorem 1 we will use the following proposition which char- acterizes when a polynomial differential system has a global center. Proposition 2. A polynomial vector field X (x, y) = (P (x, y), Q(x, y)) with- out a line of singular points at infinity, has a global center if and only if it has a unique finite singular point which is a center and all the infinite sin- gular points in the Poincaré sphere, if they exist, must be formed by two degenerated hyperbolic sectors. 3. Proof of Theorem 1 It is well known that any homogeneous polynomial of degree d factorizes as r1 Y r2 Y (ai x + bi y)li (αk x2 + βk xy + γk y 2 )jk , i=1 k=0 Plri1 ≥ 0 forPall where i = 1, . . . , r1 , jk ≥ 0 and βk2 − 4αk γk < 0 for k = 0, . . . , r2 r2 and i=1 li + k=0 2jk = d. Let d1 be the degree of P and d2 be the degree of Q. We assume that d = max{d1 , d2 }. The infinite singular points in the Poincaré disc of system (1) correspond to the linear factors of the quantity Gd (x, y) = yPd (x, y) − xQd (x, y) = 0 (it is well understood that Pd or Qd could be zero). We will separate the proof of Theorem 1 in two propositions dealing re- spectively with the cases Gd 6≡ 0 and Gd ≡ 0. We start with the case Gd 6≡ 0. Proposition 3. Any polynomial differential system (1) of degree d even and with Gd 6≡ 0 do not have global centers. Proof. Taking into account that Gd 6≡ 0, doing a rotation of the coordinate with respect to the origin we can assume that all the infinite singular points are in the local charts U1 ∪ V1 . We introduce the notation Gd−k (x, y) = yPd−k (x, y) − xQd−k (x, y) = 0, k = 0, . . . , d. In the local chart U1 system (1), using system (2), can be written as u̇ = −Gd (1, u) + vGd−1 (1, u) + . . . + v d−1 G0 (1, 0), (3) v̇ = −vPd (1, u) − v 2 Pd−1 (1, u) − . . . − v d P0 (1, u). 4 J. LLIBRE AND C. VALLS The Jacobian matrix of any singular point (ū, 0) of the local chart U1 is of the form ! ∂ Gd (1, ū) Gd−1 (1, ū) ∂u . 0 −Pd (1, ū) So the singular point (ū, 0) if it exists (that is if Gd (1, ū) = 0) must be formed by two degenerate hyperbolic sectors, and as pointed out above it must be ∂ linearly zero. Hence Gd (1, ū) = 0. This implies that Gd (1, ū) = 0 and ∂u ∂ Gd (1, ū) = 0 and so the singular point (ū, 0) must have multiplicity two ∂u as a zero of Gd (1, u). This implies that if Gd has a real linear factor then it must have at least multiplicity two and so in general must be of the form (recall that Gd has degree d + 1) r1 Y r2 Y (4) Gd = (ai x + bi y)li (αk x2 + βk xy + γk y 2 )jk , i=1 k=0 Plri1 ≥ 2 forPall where i = 1, . . . , r1 , jk ≥ 0 and βk2 − 4αk γk < 0 for k = 0, . . . , r2 r2 and i=1 li + k=0 2jk = d + 1. Note that since d + 1 is odd in (4), there exists at least i ∈ {1, . . . , r1 } and without loss of generality we can assume that it is i = 1, such that l1 ≥ 3 is odd. Then r1 Y r2 Y Gd (x, y) = (a1 x + b1 y)l1 (ai x + bi y)li (αk x2 + βk xy + γk y 2 )jk . i=2 k=0 We can assume without loss of generality that b1 6= 0, otherwise we do a rotation with respect to the origin. Note that r1 Y r2 Y l1 li Gd (1, u) = (a1 + b1 u) (ai + bi u) (αk + βk u + γk u2 )jk . i=2 k=0 Setting the new variable a1 + b1 u = U (that is u = (U − a1 )/b1 we have  U −a  1 (5) Gd (1, U ) = Gd 1, =: U l1 Γ + h.o.t., b1 where Yr1   r2  j ai b1 − bi a1 li Y αk b21 − βk a1 b1 + γk a21 k Γ= 6= 0 i=2 b1 b21 k=0 (because U = 0 has exactly multiplicity l1 ) and h.o.t means the higher order terms in the variable U . Taking the new variables (U, v), it follows from (3) and (5) that the system in the local chart U1 restricted to V = 0 can be written as U̇ |v=0 = (d + 1)U l1 Γ + h.o.t., v̇|v=0 = 0. Note that the U -axis is invariant. In the positive semi-axis {U > 0, V = 0} and in a neighborhood of (U, V ) = (0, 0) the orbit travels in the opposite ON THE GLOBAL CENTERS OF POLYNOMIAL DIFFERENTIAL SYSTEMS 5 sense to the orbit in the negative semi-axis {U < 0, V = 0}, and so the local phase portrait around (U, V ) = (0, 0) cannot be formed by two degenerated hyperbolic sectors. Hence, any Hamiltonian system (1) with n even and with Hn+1 of the form (4) cannot have global centers. This concludes the proof of Proposition 3.  Proposition 4. A polynomial differential system (1) of degree d even and with Gd ≡ 0 has no global centers. Proof. Taking into account that the line at infinity is formed by singular points we must have that Gd (x, y) ≡ 0 that is yPd (x, y) ≡ xQd (x, y), which implies that there exists a polynomial Rd (x, y) of degree d − 1 odd so that (6) Pd (x, y) = xRd (x, y) and Qd (x, y) = yRd (x, y) Note if system (1) has a global center then it has a unique finite singular point which is the origin and since the period annulus of that finite singular point is R2 \ 0, then the boundary of the period annulus U of the center of p(X ) located at (0, 0, 1) is the equator of S2 or H+ . Since there are no finite singular points in H+ , except the center at (0, 0, 1), and the infinite is formed by singular points, it follows that the boundary of the period annulus U is either a finite periodic orbit γ, or it is S1 . If it is S1 then since it is formed by fixed points, then each singular point cannot be the ω-limit or de α-limit of any orbit. Now we show that it cannot be a finite periodic orbit γ. It it would be, we consider the Poincaré map π defined in a transversal section Π through γ. Since the vector field p(X ) is analytic, it follows that π is also analytic. Hence as π is the identity map in Π ∩ U it must be the identity map in Π ∩ (H+ \ U ). But then the orbits in Π ∩ (H+ \ U ) near U are also periodic, and γ is not the boundary of U , a contradiction. In the local chart U1 system (1), using system (2), can be written as u̇ = vGd−1 (1, u) + . . . + v d−1 G0 (1, u), (7) v̇ = −vRd (1, u) − v 2 Pd−1 (1, u) − . . . − v d P0 (1, u). The line at infinity v = 0 if filled by singular points. We introduce the parameterization of time ds = vdt. With this new time system (7) becomes u̇ = Gd−1 (1, u) + vGd−2 (1, u) + . . . + v d−2 G0 (1, u), (8) v̇ = −Rd (1, u) − vPd−1 (1, u) − . . . − v d−1 P0 (1, u), where now the dot means derivative in the new time s. Since Rd (1, u) 6≡ 0, there exists u so that Rd (1, u) 6= 0, and so a point (u, 0) is a regular point for system (8). Since v̇|v=0 = −Rd (1, u), and v̇|v=0,u=u 6= 0 such point which is a singular point of system (8) would be the ω-limit or 6 J. LLIBRE AND C. VALLS the α − limit of some orbit of system (7) and by the explanation above, system (1) cannot have a global center.  Proof of Theorem 1. The proof of Theorem 1 is an immediate consequence of Propositions 3 and 4.  Acknowledgements The first author is supported by the Ministerio de Economı́a, Indus- tria y Competitividad, Agencia Estatal de Investigación grant 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 second author is partially supported by FCT/Portugal through UID/MAT/04459/2019. References [1] M.J. Álvarez, A. Ferragut and X. Jarque, A survey on the blow up technique, Internat. J. Bifur. Chaos 21 (2011), 3103–3118. [2] J.C. Artés and J. Llibre, Quadratic Hamiltonian vector fields, Differential Equa- tions 107 (1994), 80–95. [3] J.C. Artés and J. Llibre, Corrigendum: Quadratic Hamiltonian vector fields, J. Differential Equations 129 (1996), 559–560. [4] T. Date, Classification and analysis of two-dimensional real homogeneous quadratic differential equation systems, J. Differential equations 32 (1979), 311–334. [5] F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differ- ential Systems, Springer-Verlag, Berlin, 2006. [6] H. Poincaré, Mémoire sur les courbes définies par les équations différentielles, Oeu- vreus de Henri Poincaré, vol. 1, Gauthier–Villars, Paris, 1951, pp. 95–114. [7] D. Schlomiuk, Algebraic particular integrals, integrability and the problem of the center, Trans. Amer. Math. Soc. 338 (1993), 799–841. [8] N.I. Vulpe, Affine invariant conditions for the topological distinction of quadratic systems with a center, (Russian) Differentsialńye Uravneniya 19 (1983), 371–379; translation in Differential equations 19 (1983), 273–280. 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, 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.ist.utl.pt