This is a preprint of: “The center problem for the class of Λ–Ω differential systems”, Jaume Llibre, Rafael Orlando Ramı́rez, Valentı́n Ramı́rez, Rend. Circ. Mat. Palermo (2), 2020. DOI: [10.1007/s12215-020-00568-5] The center problem for the class of Λ − Ω differential systems Jaume Llibre · Rafael Ramı́rez · Valentı́n Ramı́rez Abstract The center problem, i.e. distinguish between a focus and a center is a classical problem in the qualitative theory of planar differential equations which go back to Darboux, Poincaré and Liapunov. Here we solve the center problem for the class of planar analytic or polynomial differential systems k X k X ẋ = −y + X = −y + Xj , ẏ = x + Y = x + Yj , k ≤ ∞, j=2 j=2 where Xj = Xj (x, y) and Yj = Yj (x, y) are homogenous polynomials of degree j > 1, under the condition   2 2 ∂X ∂Y (x + y ) + = µ (xX + yY ) with µ ∈ R \ {0}. ∂x ∂y This work is supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER) and 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. J. Llibre is the corresponding author J. Llibre Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain. Tel.: +34 5811303 Fax: +34 5812790 E-mail: jllibre@mat.uab.cat R. Ramı́rez Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Avin- guda dels Paı̈sos Catalans 26, 43007 Tarragona, Catalonia, Spain. E-mail: rafaelorlando.ramirez@urv.cat V. Ramı́rez Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain. E-mail: valentin.ramirez@e-campus.uab.cat 2 J. Llibre, R. Ramı́rez, V. Ramı́rez Moreover we prove that these centers are weak centers, additionally, we provide their first integrals. Keywords First integral · Poincaré-Liapunov first integral · analytic planar differential system · polynomial differential system · weak center Mathematics Subject Classification (2010) 34C05 · 34C25 1 Introduction and main results ∂ ∂ Let X = (−y + X) + (x + Y ) be the real planar analytic or polynomial ∂x ∂y vector field associated to the real planar differential system k X k X ẋ = −y + X = −y + Xj , ẏ = x + Y = x + Yj , for k≤∞ (1) j=2 j=2 where Xj = Xj (x, y) and Yj = Yj (x, y) are homogenous polynomials of degree j. The Poincaré center-focus problem asks about conditions on the coefficients of X and Y under which all trajectories of system (1) contained in a small open neighborhood of the origin are closed, of course with exception of the origin. One of the mechanism to solve the center-focus problem is the following result due to Poincaré and Liapunov. Theorem 1 A planar analytic or polynomial differential system (1) has a center at the origin if and only if it has a first integral of the form ∞ X X∞ 1 2 H= Hj (x, y) = (x + y 2 ) + Hj (x, y), (2) j=2 2 j=3 where Hj are homogenous polynomials of degree j. The first integral H is called the Poincaré-Liapunov first integral. A center is called a weak center if the Poincaré-Liapunov first integral satisfies x2 + y 2 H= (1 + h.o.t.) := H2 Φ. 2 The next theorem is proved in [4]. Theorem 2 A center of an analytic (polynomial) (2) differential system is a weak center if and only if (2) can be written as ẋ = −y(1 + Λ(x, y)) + xΩ(x, y), (3) ẏ = x(1 + Λ(x, y)) + yΩ(x, y). Differential system (3) is called Λ − Ω differential equation. Another well- know mechanism to solve the center-focus problem is the Reeb criterium. Center problem for the Λ − Ω differential systems 3 Theorem 3 [Reeb’s criterion] (see for instance [8]). The analytic differential system (1) has a center at the origin if and only if there is a local nonzero X∞ analytic inverse integrating factor of the form V = 1 + gj (x, y), called j=1 in what follows the Reeb inverse integrating factor, in a neighborhood of the origin, where gj = gj (x, y) is the homogenous polynomial of degree j > 0. The following result is proved in [7]. Corollary 1 Assuming that differential system (1) has a center at the origin. Then the Poincaré-Liapunov first integral H can be written as ∞ ∞ ! X 1 X Tk H= (xYn − yXn ) − , (4) n=1 n+1 k+n+1 k=1 where Tk is a homogenous polynomial for k ≥ 1 such that ∞ X k−1 X 1 1 = ∞ =1− Tj , Tk = gk − gj Tk−j , V X j=1 j=1 1+ gj j=2 for k ≥ 1 where T1 = g1 , and V is the Reeb inverse integrating factor. In particular if the vector field X is polynomial of degree m then (4) becomes m ∞ ! X 1 X Tk H= (xYn − yXn ) − . n=1 n+1 k+n+1 k=1 Moreover if the center is a weak center then ∞ ∞ ! X 1 X Tk H = H2 2Λn−1 (x, y) − , n=1 n+1 k+n+1 k=1 where Λ0 = 1. A partial integral of the vector field X is a differentiable function G : D −→ R where D is an open subset of R2 satisfying ∂G ∂G X (G) = P +Q = KG, ∂x ∂y with K a function of the same class than G. We say that an analytic (polynomial) vector field X is quasi–Darboux in- tegrable if there exist r polynomial partial integrals g1 , . . . , gr ∈ R[x, y] and s non-polynomial C r with r > 0 partial integrals where Kj = Kj (x, y) is a convenient analytic (polynomial) for j = 1, . . . , s such that the function F = ek(x,y)/h(x,y) g1λ1 (x, y) . . . grλr (x, y)f1κ1 (x, y) . . . fsκs (x, y), 4 J. Llibre, R. Ramı́rez, V. Ramı́rez is a first integral, where k = k(x, y), h = h(x, y) are analytic (polynomial) functions, and λ1 , . . . , λr , κ1 , . . . , κs , are complex constants. In [4] the following conjecture was stated. Conjecture 1. Any analytic (polynomial) vector field with a weak center at the origin is quasi–Darboux integrable. This conjecture is supported in particular by the results given in the proof of the next Theorem 5 and the following two results. Note that the first result is a local one. Theorem 4 Any analytic (polynomial) vector field with a weak center at the origin is locally quasi–Darboux integrable. Theorem 4 is proved in section 2 The weak conditions provided by Alwash and Lloyd [1] give sufficient con- ditions for the existence a center. More precisely, they proved: Proposition 1 The origin of coordinates of the polynomial differential system ẋ = −y + Xm , ẏ = x + Ym , (5) where Xm and Ym are homogenous polynomial of degree m is a center if there exists µ ∈ R \ {0} such that   2 2 ∂(−y + Xm ) ∂(x + Ym ) (x + y ) + = µ (x(−y + Xm ) + y(x + Ym )) , (6) ∂x ∂y and Z 2π either  m = 2k; or m= 2k − 1 and µ 6= 2k; or m = 2k − 1, µ = 2k and ∂X2k−1 ∂Y2k−1 + dt = 0. 0 ∂x ∂y x=cos t, y=sin t The main objective of this paper is to extend and improve the result of Propo- sition 1 to analytic and polynomial differential systems. Thus our two main results are the following ones. Theorem 5 Differential system (1), satisfying the condition   2 2 ∂(−y + X) ∂(x + Y ) (x + y ) + = µ (x(−y + X) + y(x + Y )) , (7) ∂x ∂y where µ ∈ R \ {0}, has a weak center at the origin if and only if: For the analytic systems one of the following conditions holds (i) µ∈/ N \ {1}, (ii) µ = 2, (iii) µ = 2k + 1 ∈ N (iV) µ = 2k ∈ N\{2}, and Z 2π   ∂X2k−1 ∂Y2k−1 β2k−1 := + dt = 0. (8) 0 ∂x ∂y x=cos t, y=sin t Center problem for the Λ − Ω differential systems 5 For the polynomial systems of degree m one of the following conditions holds (i) µ∈/ {1, 3, . . . , m + 1} , (ii) µ = 2, (iii) µ = 2k + 1 ∈ {1, 2, . . . , m + 1}, (iv) µ = 2k ∈ {3, . . . , m + 1} and β2k−1 = 0. Moreover differential system (1) satisfying (7) and having a weak center at the origin is quasi-Darboux integrable. Corollary 2 Differential system (5) under condition (6) has a weak center at the origin if and only if (i) If µ∈/ {2, m + 1}, (ii) If µ = 2, (iii) If µ = 2k + 1 ∈ {1, m + 1}, (iv) If µ = 2k = m + 1 and β2k−1 = 0. Theorem 5 and Corollary 2 are proved in section 3. 2 Preliminary results In the proofs of our results it plays an important role the following propositions and corollaries. The Poisson bracket of two functions f and g is defined as ∂f ∂g ∂f ∂g {f, g} := − . ∂x ∂y ∂y ∂x Proposition 2 The next relation holds Z 2π {H2 , Ψ }|x=cos t, y=sin t dt = Ψ (cos 2π, sin 2π) − Ψ (cos 0, sin 0) 0 for arbitrary C 1 function Ψ = Ψ (x, y) defined in U ⊆ R2 . Proof Indeed, Z 2π Z 2π   ∂Ψ ∂Ψ {H2 , Ψ }|x=cos t, y=sin t dt = −yx dt 0 0 ∂y ∂ x x=cos t, y=sin t Z 2π   Z 2π ∂Ψ ∂Ψ d = ẋ + ẏ dt = (Ψ (cos t, sin t)) dt 0 ∂x ∂ y x=cos t, y=sin t 0 dt = Ψ (cos 2π, sin 2π) − Ψ (cos 0, sin 0). The following corollary is due to Liapunov (see Theorem 1, page 276 of [3]) 6 J. Llibre, R. Ramı́rez, V. Ramı́rez Corollary 3 Let U = U (x, y) be a homogenous polynomial of degree m. The linear partial differential equation ∂V ∂V x −y := {H2 , V } = U, ∂y ∂x has a unique homogenous polynomial solution V of degree m if m is odd; and if V is a homogenous polynomial solution when m is even then any other homoge- nous polynomial solution is of the form V +c(x2 +y 2 )m/2 with c ∈ R. Moreover, Z 2π for m even these solutions exist if and only if U (x, y)|x=cos t, y=sin t dt = 0. 0 Proof (Proof of Theorem 4) Indeed, if the analytic vector field has a weak center at the origin then it admits a Poincaré-Liapunov local analytic first integral at the origin H = H2 (1 + h.o.t) = H2 Φ(x, y). From Theorem 2 we get that Ḣ2 = 2H2 Ω(x, y), i.e. H2 is a partial integral with analytic cofactor 2Ω(x, y). It is easy to show that the analytic function Φ(x, y) is an analytic partial integral with cofactor −2Ω(x, y). In short the theorem is proved. In the proof of Theorem 5 and Corollary 2 we need the following two propositions which also appeared in the Ph.D. [7], and for completeness we prove them here. Proposition 3 Let ẋ = P (x, y), ẏ = Q(x, y) (9) be an analytic (polynomial) differential system . Then this system can be writ- ten as ∂F ∂F ẋ = P = − − y G, ẏ = Q = + x G. (10) ∂y ∂x where F and G are convenient analytic (polynomial) functions, if and only if Z 2π   ∂P ∂Q + |x=cos t, y=sin t dt = 0. (11) 0 ∂x ∂y holds. Moreover if (11) holds then differential system (9) can be written as   ∂Φ Λ̃ x ∂P ∂Q ẋ = − −y + + , ∂y µ µ ∂x ∂y   (12) ∂Φ Λ̃ y ∂P ∂Q ẏ = +x + + , ∂x µ µ ∂x ∂y where Φ = µ F + 2H2 G, µ ∈ R\{0}, and Λ̃ is a convenient function. Center problem for the Λ − Ω differential systems 7 Proof Indeed, if (11) holds then there exist a function G such that ∂P ∂Q ∂G ∂G ∂P ∂Q {H2 , G} = + ⇐⇒ x −y = + ⇐⇒ ∂x ∂y ∂y ∂x ∂x ∂y ∂(Q − xG) ∂(P + yG) ∂F ∂F + = 0 ⇐⇒ P + yG = − , Q − xG = , ∂y ∂x ∂y ∂x consequently (10) holds. Assume that (10) holds then ∂P ∂Q ∂ 2 H̃ ∂ 2 H̃ + = {H2 , G} + − , ∂x ∂y ∂y∂x ∂x∂y ∂ 2 H̃ ∂ 2 H̃ and by considering that = we get that ∂y∂x ∂x∂y ∂P ∂Q {H2 , G} = + . ∂x ∂y Z 2π Hence in view of the relation {H2 , G} |x=cos t, y=sin t dt = 0 we obtain that 0 (11) holds. In short the proposition is proved. Now we prove formula (12) for differential system (9) under the condition (11). If (11) holds then from (10) we get that ∂P ∂Q {H2 , G} = + , {F, H2 } = xP + yQ. ∂x ∂y So   ∂P ∂Q {µ F + 2H2 G, H2 } = (x2 + y 2 ) + − µ (xP + yQ) . ∂x ∂y Thus         ∂P ∂Q ∂Φ ∂P ∂Q ∂Φ x µP − x + + + y µQ − y + − = 0. ∂x ∂y ∂y ∂x ∂y ∂x Consequently, by introducing the function Φ = µ F + 2H2 G, we get that   ∂P ∂Q ∂Φ µP − x + − = −Λ̃ y, ∂x ∂y ∂y   ∂P ∂Q ∂Φ µQ − y + − = Λ̃ x, ∂x ∂y ∂x Thus formula (12) is proved. In short the proposition is proved. 8 J. Llibre, R. Ramı́rez, V. Ramı́rez Proposition 4 The analytic differential system (2) satisfying condition (7) can be written as the Λ − Ω differential system   x ∂X ∂Y ẋ = −y(1 + Λ) + + = −y(1 + Λ) + xΩ(x, y), µ ∂x ∂y   (13) y ∂X ∂Y ẏ = x(1 + Λ) + + = x(1 + Λ) + yΩ(x, y). µ ∂x ∂y Where Λ = Λ(x, y) is a solution of the first order partial differential equation ∂Ω ∂Ω {H2 , Λ} = (µ − 2)Ω − x −y , (14) ∂x ∂y which in polar coordinates x = r cos θ, y = r sin θ becomes ! ∂   ∂ Ω̃ Λ̃ = ∂θ ∂r rµ−2 where Λ̃ = Λ(r cos θ, r sin θ) and Ω̃ = Ω(r cos θ, r sin θ). Proof Indeed, from (7) it follows that       ∂X ∂Y ∂X ∂Y x µX − x + + y µY − y + = 0. ∂x ∂y ∂x ∂y Hence     ∂X ∂Y ∂X ∂Y µX − x + = −µΛ, µY − y + = µ Λ, ∂x ∂y ∂x ∂y where Λ = Λ(x, y) is a convenient function. Thus we obtain that X = −yΛ + xΩ, Y = xΛ + yΩ, where   1 ∂X ∂Y Ω= + , (15) µ ∂x ∂y Thus we easily obtain (13). From these relations we get that ∂X ∂Λ ∂Ω = −y +Ω+x , ∂x ∂x ∂x ∂Y ∂Λ ∂Ω =x +Ω+y . ∂y ∂y ∂y Consequently the formula (14) it follows. Finally we observe that if   ∂Φ ∂Φ 1 ∂X ∂Y X=− −yΛ+xΩ, Y = +xΛ+yΩ, with Ω = + ∂y ∂x µ ∂x ∂y where Ω is given by formula (15), and Λ is a solution of equation (14). Thus the proposition is proved. Center problem for the Λ − Ω differential systems 9 A center O of system (1) is a uniform isochronous center if the equality xẏ − y ẋ = κ(x2 + y 2 ) holds for a nonzero constant κ; or equivalently in polar coordinates (r, θ) such that x = r cos θ, y = r sin θ, we have that θ̇ = κ. Clearly that from Theorem 2 it follows that uniform isochronous centers are weak center. Example 1 For differential system (5) under the condition (6) we get that differential system (13) and condition (14) becomes ẋ = −y(1 + Λm−1 ) + xΩm−1 , ẏ = x(1 + Λm−1 ) + yΩm−1 , (16) {H2 , Λm−1 } = (µ − m − 1)Ωm−1 , respectively. Consequently if µ − m − 1 6= 0, then system (16) can be written as {H2 , Λm−1 } {H2 , Λm−1 } ẋ = −y(1 + Λm−1 ) + x , ẏ = x(1 + Λm−1 ) + y , µ−m−1 µ−m−1 and if µ − m − 1 = 0, then ẋ = −y(1 + Λm−1 (H2 )) + xΩm−1 , ẏ = x(1 + Λm−1 (H2 )) + yΩm−1 . (17) Since the differential system (17) in polar coordinates writes ṙ = rΩm−1 (r cos θ, r sin θ), θ̇ = 1 + Λ(1/2), we get that the weak center in this case is a uniform center. 3 Proof of Theorem 5 and Corollary 2 Proof of Theorem 5. Multiplying (7) by (x2 + y 2 )(µ−2)/2 we get   ∂ ∂ ∂V ∂V V (−y + X) + (x + Y ) = (−y + X) + (x + Y ) , ∂y ∂y ∂x ∂y where V = (x2 + y 2 )µ/2 , or equivalently     ∂ −y + X ∂ x+Y + = 0. ∂x (x2 + y 2 )µ/2 ∂y (x2 + y 2 )µ/2 Hence there exists a function F̃ such that −y + X ∂ F̃ x+Y ∂ F̃ =− , = . (18) (x2 + y 2 )µ/2 ∂y (x2 + y 2 )µ/2 ∂x Thus ∂ F̃ ∂ F̃ −y + X = −(x2 + y 2 )µ/2 , x + Y = (x2 + y 2 )µ/2 . (19) ∂y ∂x 10 J. Llibre, R. Ramı́rez, V. Ramı́rez From this relation we easily obtain ∂X ∂Y + = µ(x2 + y 2 )(µ−2)/2 {F̃ , H2 }. ∂x ∂y Hence in view of Proposition 2 we get that Z 2π   ∂X ∂Y + dt = 0 ∂x ∂y x=cos t, y=sin t (20)   µ F̃ (cos(2π), sin(2π)) − F̃ (cos(0), sin(0)) . On the other hand, from (19) and in view of Proposition 4 it follows     x+Y y−X dF̃ = dx + dy (x2 + y 2 )µ/2 (x2 + y 2 )µ/2 (21) (x(1 + Λ(x, y)) + yΩ(x, y)) (y(1 + Λ(x, y)) − xΩ(x, y)) = dx + dy. (x2 + y 2 )µ/2 (x2 + y 2 )µ/2 ∂ 2 F̃ ∂ 2 F̃ From the condition = we get that the function Λ must be satisfies ∂y∂x ∂x∂y the equation (14). Clearly from (21) we obtain (1 + Λ(x, y))(xdx + ydy) Ω(x, y)(y dx − x dy) dF̃ = + . (22) (x2 + y 2 )µ/2 (x2 + y 2 )µ/2 ∞ X First we study the analytic case. Thus we assume that Λ(x, y) = Λj (x, y) j=1 ∞ X and Ω(x, y) = Ωj (x, y), where Λj (x, y) and Ωj (x, y) are homogeneous poly- j=1 nomials of degree j. Equation (22) in polar coordinates x = r cos θ, y = r sin θ becomes 1 + Λ(r cos θ, r sin θ) Ω(r cos θ, r sin θ ) df = dr − dθ, rµ−1 rµ−2 where f := f (r, θ) = F̃ (r cos θ, r sin θ) and Λj (r cos θ, r sin θ ) = rj αj (θ), Ωj (r cos θ, r sin θ ) = rj τj (θ), for j ∈ N, or equivalently     ∞ X ∞ X df = r1−µ + rj+1−µ αj (θ) dr −  rj+2−µ τj (θ) dθ. (23) j=1 j=1 Hence X∞ X∞ ∂f ∂f = r1−µ + rj+1−µ αj (θ), =− rj+2−µ τj (θ), ∂r j=1 ∂θ j=1 Center problem for the Λ − Ω differential systems 11 ∂2f ∂2f From the compatibility conditions = we get that ∂r∂θ ∂θ∂r ∂ (αj (θ)) = αj0 (θ) = −(j + 2 − µ)τj (θ), (24) ∂θ ∂  or equivalently rj αj (θ) = −(j + 2 − µ)rj τj (θ), which in cartesian coor- ∂θ dinates becomes {H2 , Λj } = −(j + 2 − µ)Ωj ∂Ωj ∂Ωj = (µ − 2)Ωj − x −y , ∂x ∂y / N \ {1} then in (24) j + 2 − µ 6= 0. Consequently from (26) Assuming that µ ∈ we obtain that     X∞ X∞ j+2−µ r df = r1−µ + rj+1−µ αj (θ) dr +  αj0 (θ) dθ j=1 j=1 j + 2 − µ   2−µ ∞ X j+2−µ r r = d + αj (θ) . 2 − µ j=1 j + 2 − µ Therefore ∞ X rj+2−µ r2−µ f= + αj (θ), 2 − µ j=1 j + 2 − µ is a first integral, which in cartesian coordinates becomes ∞ X Λj (x, y) 1 + 2 − µ j=1 j + 2 − µ F̃ (x, y) = . (x2 + y 2 )(µ−2)/2 Therefore 1 2H2 H= = 2/(µ−2) , (25) F̃ (x, y)2/(µ−2) ∞ X  1 Λj (x, y)  + 2 − µ j=1 j + 2 − µ is an analytic first integral defined in a neighborhood of the origin, i.e. it is a Poincaré-Liapunov first integral. From the expression of this first integral 1 we obtain that the origin is a weak center. Clearly that H2 and + 2−µ X∞ Λj (x, y) are partial integrals with analytic cofactor 2Ω and (µ − 2)Ω j=1 j+2−µ respectively. 12 J. Llibre, R. Ramı́rez, V. Ramı́rez Now assume that µ ∈ N\{2}. Let j = µ−2, then αµ−2 (θ) = αµ−2 =constant. From (26) and in view of (24) we obtain that   X∞  1−µ αµ−2  df =  r + + rj+1−µ αj (θ)  dr r j=1  j6=µ−2  ∞ X   − τµ−2 (θ) + rj+2−µ τj (θ)  dθ (26) j=1 j6=µ−2   Z X∞ r2−µ rj+2−µ = d + τµ−2 (θ)dθ + αµ−2 log r + αj (θ) . 2−µ j=1 j + 2 − µ Thus Z X∞ r2−µ rj+2−µ f= + τµ−2 (θ)dθ + αµ−2 log r + αj (θ), 2−µ j=1 j+2−µ or equivalently the function f is   Z ∞ X 1  1 rj   µ−2 rµ−2 τµ−2 (θ)dθ + αj (θ) rµ−2  2 − µ + αµ−2 r log r + j+2−µ  j=1 j6=µ−2 Z 1  1 = + αµ−2 rµ−2 log r + Ωµ−1 (r cos θ, r sin θ)dθ rµ−2 2−µ Λj (r cos θ, r sin θ)  X∞ + . j=1 j+2−µ j6=µ−2 We have that Z y Ωµ−2 (r cos θ, r sin θ)dθ √ = βµ−1 arctan + ϕ, r= x2 +y 2 , θ=arctan(y/x) x where ϕ := ϕ(x, y) is a polynomial of degree µ − 1 and βµ−1 is a constant such that Z 2π Z 2π   1 ∂Xµ−1 ∂Yµ−1 βµ−1 = Ωµ−2 (cos t, sin t)dt = + |x=cos t, y=sin t dt 0 0 µ ∂x ∂y  0 Z if µ = 2k + 1, 2π   = 1 ∂X2k−1 ∂Y2k−1  + |x=cos t, y=sin t dt if µ = 2k, µ 0 ∂x ∂y Center problem for the Λ − Ω differential systems 13 This first integral in cartesian coordinates becomes 1  1 y F̃ (x, y) = + ϕ(x, y) + βµ−1 arctan (2H)(µ−2)/2 2 − µ x Λj (r cos θ, r sin θ)  X∞ Λµ−2 (2H2 ) + log H2 + , 2 j=1 j+2−µ j6=µ−2 Therefore from (20) we obtain that βµ−1 arctan(tan(2π)) F̃ (cos(2π), sin(2π)) − F̃ (cos(0), sin(0)) = . 2µ/2−1 Thus in order to obtain a center at the origin from (20) we need that βµ−1 = 0. From (24) we get that αµ−2 is a constant, i.e. X ak j cosk θ sinj θ = αµ−2 = constant, j+k=µ−2 µ−2 Hence Λµ−2 = r αµ−2 thus  (µ−2)/2 0 if µ 6= 2k, Λµ−2 = (2H) αµ−2 = (2H)k−1 α2k−2 if µ = 2k. Hence  2/(µ−2) 1 H̃ = F̃ (x, y) βµ−1 =0 2H2 =  2/(µ−2) . X∞  1 Λj Λµ−2 (H2 )   + + log (2H2 ) + ϕ(x, y) 2 − µ j+2−µ 2  j=1 j6=µ−2 Thus if µ 6= 2k H2 H= 2/(µ−2) , (27) X∞  1 Λj   + + +ϕ(x, y) 2 − µ j+2−µ  j=1 j6=µ−2 and if µ = 2k then H2 H̃ =  1/(k−1) . ∞ X  1 Λj (2H2 )k−1   + + log H2 + ϕ(x, y)  2 − 2k j + 2 − 2k 2  j=1 j6=2k−2 14 J. Llibre, R. Ramı́rez, V. Ramı́rez When µ = 2k > 2 the first integral H̃ is such that lim H̃ = 0 and (x,y)−→ (0,0) it is not analytic. So we have a center, by considering that is a center of a Λ − Ω equation we obtain that the origin is a weak center. Evidently that H2 X∞ 1 (2H2 )k−1 Λj and + log H2 + ϕ(x, y) + are partial integrals µ−2 2 j=1 j + 2−µ j6=2k−2 with analytic cofactor 2Ω and (µ − 2)Ω. If µ 6= 2k then H is a first integral an analytic at the origin. Thus the X∞ 1 Λj origin is a weak center. Clearly that H2 and + + ϕ(x, y) µ−2 j=1 j+2−µ j6=µ−2 are partial integrals with analytic cofactor 2Ω and (µ − 2)Ω. Finally we consider the case µ = 2. Then from (26) and (24) we have that     X∞ X∞ j r df (r, θ) = r−1 + rj−1 αj (θ) dr +  αj0 (θ) dθ j=1 j=1 j     X∞ j X∞ r Λj = d log r + αj  = d log r + . j=1 j j=1 j Therefore the first integral F̃ (x, y) becomes    X∞ X∞ 1 Λj 1 Λj  F̃ (x, y) = log 2H2 + = log 2H2 exp 2 . 2 j=1 j 2 j=1 j The Poincaré-Liapunov first integral in this case is   X∞ Λ j H = H2 exp 2 . j=1 j So if µ = 2 then the origin is a weak center, and Theorem 5 is proved for the analytic case. The condition in order that a polynomial differential system of degree m satisfying condition (7) has a center at the origin are obtained directly from conditions proved in the analytic case. Polynomial differential system of degree m satisfying condition (7) has a weak center at the origin if and only if (8) holds. We provide the expressions of the first integrals in the case of polynomial differential systems. If µ 6∈ {3, 4, . . . , m + 1} , then we get the analytic first integral in the neighborhood of the origin from (25) H2 H= 2/(µ−2) . m−1 X  1 Λj  + 2−µ j=1 j+2−µ Center problem for the Λ − Ω differential systems 15 If µ 6= 2k ∈ {3, 4, . . . , m + 1} , then from (27) we get that the Poincaré- Liapunov first integral is H2 H= 2/(µ−2) , m−1 X  1 Λj   + ϕ(x, y) + + 2 − µ j+2−µ  j=1 j6=µ−2 where ϕ(x, y) is a polynomial such that Z ! ϕ(x, y) = Ωµ−2 (r cos θ, r sin θ)dθ √ . r= x2 +y 2 , θ=arctan(y/x) βµ−1 =0 and if µ = 2k ∈ {3, 4, . . . , m + 1} , then H2 H̃ =  1/(k−1) . m−1 X  (2H2 )k−1  − 1 + ϕ(x, y) + Λj + log H2   2 − 2k j + 2 − 2k 2  j=1 j6=2k−2 The first integral H̃ is such that lim H̃ = 0 and is not analytic. So we (x,y)−→ (0,0) have a center, by considering that is a center of the Λ − Ω equation we obtain that the origin is a weak center. If µ = 2 the Poincaré-Liapunov first integral in this case is   m−1 X Λj H = H2 exp 2 , j=1 j then the origin is a weak center, and Theorem 5 is proved for polynomial case. Thus Theorem 5 is proved. Proof of Corollary 2. It is a simple consequence of Theorem 5 for the polyno- mial case the Poincaré-Liapunov first integral is 1 H2 H=  2/(m−1) =  2/(m−1) . 2 F̃ (x, y) 1 Λm−1 + m−1 m+1−µ 1 Λm−1 If µ ∈ / {2, m + 1}. Clearly that H2 and + are partial m−1 m+1−µ integrals with polynomial cofactor 2Ωm−1 and (m − 1)Ωm−1 respectively. If µ = m + 1 6= 2k then the Poincaré-Liapunov first integral is  2/(m−1) 1 2H2 H= = 1/(m−1) , F̃ (x, y) βm−1 =0 1 + ϕ(x, y) m−1 16 J. Llibre, R. Ramı́rez, V. Ramı́rez where ϕ(x, y) is a polynomial of degree m − 1 and such that Z ! ϕ(x, y) = Ωm−1 (r cos θ, r sin θ)dθ √ , r= x2 +y 2 , θ=arctan(y/x) βm−1 =0 Z 2π   ∂X2k−1 ∂Y2k−1 and βm−1 = + dt = 0. 0 ∂x ∂y x=cos t, y=sin t If µ = m + 1 = 2k then the first integral is H2 H̃ =  1/(k−1) . k−1 1 (2H2 ) − + log H2 + ϕ(x, y) 2 − 2k 2 The first integral H̃ is such that lim H̃ = 0 and it is not analytic. So (x,y)−→ (0,0) we have a center, by considering that is a center of the Λ − Ω equation we obtain that the origin is a weak center. If µ = 2 then we get the following Poincaré-Liapunov first integral   Λm−1 H = H2 exp 2 . m−1 In short the corollary is proved. 4 Example In [5, 6] we state the following conjecture Conjecture 2. The polynomial differential system of degree m ẋ = −y(1 + κ (a1 y − a2 x)) + x(a1 x + a2 y + Ωm−1 ) = −y + X, (28) ẏ = x(1 + κ (a1 y − a2 x)) + y(a1 x + a2 y + Ωm−1 ) = x + Y, where (κ + m − 2)(a21 + a22 ) 6= 0, and Ωm−1 = Ωm−1 (x, y) is a homogenous polynomial of degree m−1, has a weak center at the origin if and only if system (28) after a linear change of variables (x, y) −→ (X, Y ) is invariant under the transformations (X, Y, t) −→ (−X, Y, −t). Theorem 6 Conjecture 2 holds for m = 2, 3, 4, 5, 6. The center problem for the case when (κ + m − 2)(a21 + a22 ) = 0, we solve in the next proposition. Proposition 5 polynomial differential system of degree m (28) satisfies the condition (6) if (κ + m − 2)(a21 + a22 ) = 0. Center problem for the Λ − Ω differential systems 17 Proof After some computations we get that xX + yY = (x2 + y 2 )(a1 x + a2 y + Ωm−1 ), ∂X ∂Y ∂Ωm−1 ∂Ωm−1 + = (κ + 3)(a1 x + a2 y) + x +y + 2Ωm−1 ∂x ∂y ∂x ∂y κ=m−2 = (m + 1)(a1 x + a2 y + Ωm−1 ), hence we get that   2 2 ∂X ∂Y (x + y ) + = (m + 1) (xX + yY ) , ∂x ∂y which coincide with (6) with µ = m + 1. In this case we get that βm−1 = Z 2π Ωm−1 (cos t, sin t)dt. 0 To solution of the center problem for differential system (28) under the con- dition (κ + m − 2)(a21 + a22 ) = 0 is a simple consequence of Theorem 5 with µ = m + 1. The case when a1 = a2 = 0 was study in [2]. 5 Declarations 5.1 Funding This work is supported by the Ministerio de Ciencia, Innovación y Universi- dades, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER) and PID2019-104658GB-I00 (FEDER), the Agència de Gestió d’Ajuts Univer- sitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. 5.2 Conflicts of interest/Competing interests This paper has no conflicts of interest. 5.3 Availability of data and material Not applicable. 5.4 Code availability Not applicable. 18 J. Llibre, R. Ramı́rez, V. Ramı́rez References 1. M.A. Alwash and N.G. Lloyd, Non-autonomous equations related to polynomial two dimensional systems, Proc. Roy. Soc. Edinburgh 105 A (1987), 129–152. 2. R. Conti, Centers of planar polynomial systems. a review, Le Matematiche, Vol. LIII, Fasc. II, (1998), 207–240. 3. 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