This is a preprint of: “An inverse approach to the center problem”, Jaume Llibre, Rafael Orlando Ramı́rez, Valentı́n Ramı́rez, Rend. Circ. Mat. Palermo (2), vol. 68, 29–64, 2019. DOI: [10.1007/s12215-018-0342-1] AN INVERSE APPROACH TO THE CENTER PROBLEM JAUME LLIBRE1 , RAFAEL RAMÍREZ2 AND VALENTÍN RAMÍREZ3 Abstract. We consider analytic or polynomial vector fields of the form ∂ ∂ X = (−y + X) + (x + Y ) , ∂x ∂y where X = X(x, y)) and Y = Y (x, y)) start at least with terms of second order. It is well–known that X has a center at the origin if and only if X has a 1 Liapunov-Poincaré local analytic first integral of the form H = (x2 + y 2 ) + 2 ∑∞ Hj , where Hj = Hj (x, y) is a homogenous polynomial of degree j. j=3 The classical center-focus problem already studied by Poincaré consists in distinguishing when the origin of X is either a center or a focus. In this paper we study the inverse center problem, i.e. for a given analytic function H of the previous form defined in a neighborhood of the origin, we determine the analytic or polynomial vector field X for which H is a first integral. Moreover, ∑∞ given an analytic function V = 1 + Vj in a neighborhood of the origin, j=1 where Vj is a homogenous polynomial of degree j, we determine the analytic or polynomial vector field X for which V is a Reeb inverse integrating factor. We study the particular case of centers  which have a local analytic first ∑∞ 1 2 integral of the form H = (x + y 2 ) 1 + Υj  , in a neighborhood of the 2 j=1 origin, where Υj is a homogenous polynomial of degree j for j ≥ 1. These centers are called weak centers, they contain the uniform isochronous centers and the isochronous holomorphic centers, but they do not coincide with the class of isochronous centers. We extended to analytic or polynomial differential systems the weak con- ditions of a center given by Alwash and Lloyd for linear centers with homo- geneous polynomial nonlinearities. Furthermore the centers satisfying these weak conditions are weak centers. Finally as an application we obtain the necessary and sufficient conditions for the existence of a weak center in a class of polynomial differential systems of degree four. 1. Introduction Let ∂ ∂ (1) X =P +Q , ∂x ∂y 2010 Mathematics Subject Classification. 34C05, 34C07. Key words and phrases. center-focus problem, analytic planar differential system, Liapunov’s constants, isochronous center, Darboux’s first integral, weak condition for a center, weak center. 1 2 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ be the real planar polynomial vector field associated to the real planar polynomial differential system (2) ẋ = P (x, y), ẏ = Q(x, y), where the dot denotes derivative with respect to an independent variable here called the time t, and P and Q are real coprime polynomials in R[x, y]. We say that the polynomial differential system (2) has degree m = max {degP, degQ}. In what follows we assume that origin O := (0, 0) is a singular point, i.e. P (0, 0) = Q(0, 0) = 0. The singular point O is a center if there exists an open neighborhood U of O where all the orbits contained in U \ {O} are periodic. The study of the centers of analytical or polynomial differential systems (2) has a long history. The first works are due to Poincaré [23] and Dulac [11] . Later on were developed by Bendixson [4], Frommer [12], Liapunov [20] and many others. Assume that the origin of the analytic or polynomial differential system (2) is a center. It is well–known that, after a linear change of variables and a constant scaling of the time variable (if necessary), system (2) can be written in one of the next three forms: ẋ = −y + X(x, y), ẏ = x + Y (x, y), (3) ẋ = y + X(x, y), ẏ = Y (x, y), ẋ = X(x, y), ẏ = Y (x, y), where X(x, y) and Y (x, y) are analytic or polynomials without constant and linear terms defined in a neighborhood of the origin. Then the origin O of the analytical or polynomial differential system (2) is called linear type, nilpotent or degenerate if after a linear change of variables and a scaling of the time it can be written as the first, second and third system of (3), respectively. In this paper we shall study the differential system of the linear type (4) ẋ = −y + X, ẏ = x + Y, where X = X(x, y) and Y = Y (x, y) are real analytic functions in an open neigh- borhood of O whose Taylor expansions at O do not contain constant and linear terms. For X, Y polynomials of a given degree, the Poincaré center-focus problem asks about conditions on the coefficients of X and Y under which O is a center. In the study of the center-focus problem the following theorems play a very important role (see for instance [20, 23, 25]) Theorem 1. For the analytic differential system (4) there exists a formal power series ∞ ∑ ∑∞ 1 2 W = Wn := (x + y 2 ) + Wn (x, y), n=2 2 n=3 AN INVERSE APPROACH TO THE CENTER PROBLEM 3 where Wj = Wj (x, y) is a homogenous polynomial of degree j such that ( ) dW ∂W3 ∂W4 = x+ + + . . . (−y + X(x, y)) dt ∂x ∂x ( ) ∂W3 ∂W4 + y+ + + . . . (x + Y (x, y)) ∂y ∂y ∞ ∑ = vj (x2 + y 2 )j+1 , j=1 where vj are the Poincaré-Liapunov constants. Assume that the formal power series W converges. If the constants vj = 0 for ∑∞ 1 j ∈ N then there exists a first integral H := (x2 + y 2 ) + Wj , and consequently 2 j=3 the origin is a center. If there exists a first non–zero Liapunov constant vj , then the origin is a stable focus if vj < 0 and unstable if vj > 0. Poincaré and Liapunov proved the next two results, see for instance [23, 20, 13, 26]. Theorem 2. A planar polynomial differential system m ∑ ∑m (5) ẋ = −y + Xj (x, y), ẏ = x + Yj (x, y), j=2 j=2 of degree m has a center at the origin if and only if it has a first integral of the form ∑ ∞ ∑∞ 1 (6) H= Hj (x, y) = (x2 + y 2 ) + Hj (x, y), j=2 2 j=3 where Xj , Yj and Hj are homogenous polynomials of degree j. The analytic function (6) is called the Poincaré-Liapunov local first integral. Theorem 3. An analytic planar differential system ∑∞ ∑∞ (7) ẋ = −y + Xj (x, y), ẏ = x + Yj (x, y), j=2 j=2 has a center at the origin if and only if it has a first integral of the form (6). Theorem 2 is due to Poincaré, and Theorem 3 is due to Liapunov. From Theorems 1, 2 and 3 it is clear that an analytic or polynomial differential system (4) has a center at the origin if and only if the Poincaré-Liapunov constants vk = 0 for k ≥ 1 (Poincaré’s criterion). Moreover, the vk ’s are polynomials over Q in the coefficients of the polynomial differential system. A necessary and sufficient condition to have a center is then the annihilation of all these constants. In view of the Hilbert’s basis theorem this occurs if and only if for a finite number of k, k < j and j sufficiently large, vk = 0. Unfortunately, trying to solve the center problem computing the Poincaré-Liapunov constants is in general not possible due to the huge computations. 4 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ Although we have an algorithm for computing the Poincaré-Liapunov constants for linear type center, we have no algorithm to determine how many of them need to be zero to imply that all of them are zero for cubic or higher degree polynomial differential systems. Bautin [2] showed in 1939 that for a quadratic polynomial differential system, to annihilate all vk ’s it suffices to have vk = 0 for i = 1, 2, 3. So the problem of the center is solved for quadratic systems. This problem was solved for the cubic differential systems with homogenous nonlinearities (see for instance [25, 31, 32]). We recall the following definition. Let U be an open and dense set in R2 . We say that a non-constant C r with r ≥ 1 function F : U → R is a first integral of the analytic or polynomial vector field X on U , if F (x(t), y(t)) is constant for all values of t for which the solution (x(t), y(t)) of X is defined on U . Clearly F is a first integral of X on U if and only if X F = 0 on U . Now we shall introduce another criterion for solving the center problem due to Reeb. We need the following definitions and notions. A function V = V (x, y) is an inverse inverse integrating factor of system (2) in an open subset U ⊂ R2 if V ∈ C 1 (U ), V ̸≡ 0 in U and ( ) ( ) P Q ∂ ∂ ( ) V V ∂V ∂V ∂P ∂Q + = 0 ⇐⇒ P +Q =V + . ∂x ∂y ∂x ∂y ∂x ∂y The first integral F associated to the inverse inverse integrating factor V is given by the line integral or path integral ∫ ( ) P Q F (x, y) = − dy + dx , γ V V We note that {V = 0} is formed by orbits of system (2). The function 1/V defines an inverse integrating factor in U \{V = 0} of system (2) which allows to compute a first integral for (2) in U \{V = 0}. We consider now the relation between the existence of a center and that of an inverse integrating factor for analytic or polynomial vector fields. The main result is given by the following theorem which is analogous to Theorems 2 and 3. Theorem 4. [Reeb ’s criterion] (see for instance [29]). The analytic differential system (7) has a center at the origin if and only if there is a local nonzero analytic inverse integrating factor of the form V = 1 + h.o.t. in a neighborhood of the origin. An analytic inverse integrating factor having the Taylor expansion at the origin V = 1 + h.o.t. is called a Reeb inverse integrating factor. Darboux gave his geometric method of integration in his seminal work [8] of 1878. The geometric method of Darboux uses algebraic invariant curves of a poly- nomial differential system for computing a first integral of the system. There were numerous publications on the problem of the center using the Darboux method during the last part of the 20th century and the beginning of the 21st century (see for instance [6, 16, 34]). In fact there is the following conjecture due to Zoladek [34], see also [6]. See these papers for more details on this conjecture. AN INVERSE APPROACH TO THE CENTER PROBLEM 5 Conjecture 5. Suppose that the polynomial differential system (4) has a center at the origin. Then this system has a Darboux first integral or an algebraic symmetry. To show that a singular point is a center for system (4) we have two basic mechanisms: we either apply Poincaré–Liapunov Theorem and we show that we have a local analytic first integral, or we apply the Reeb inverse integrating factor. Another mechanism for detecting centers has been given by Mikonenko see [28]. The main objective of the present paper is to analyze the center problem from the inverse point of view (see for instance [17, 30]). Indeed, either given an analytic function H of the form (6) we shall determine the analytic functions X and Y in (4) in such a way that the function H is a first integral of the differential system ∑∞ (4), or given an analytic function 1+ Vj in a neighborhood of the origin we shall j=1 determine the analytic functions X and Y in (4) in such a way that the analytic differential system (4) has the function V as a Reeb inverse integrating factor. We say that a center at the origin of an analytic differential system is a weak center if in a neighborhood  of  the origin it has an analytic first integral of the form ∑∞ 1 H = (x2 + y 2 ) 1 + Υj  , where Υj is a homogenous polynomial of degree 2 j=1 j. We have characterized the expression of an analytic or polynomial differential system having a weak center at the origin, see Theorem 15. Moreover we prove that the uniform isochronous centers and isochronous holomorphic centers are weak centers. We have extended the weak conditions of a center given by Alwash and Lloyd in [1] for linear center with homogenous polynomial nonlinearities (see Proposition 10), to a general analytic and polynomial differential system see Theorem 25. Fur- thermore the centers satisfying the generalized weak conditions of a centers are weak centers. Finally as an application we obtain the necessary and sufficient conditions for the existence of a weak center in a class of polynomial differential systems of degree four. 2. Preliminary concepts and results In the proofs of the results that we provide in this paper it plays an important role the following results. As usual the Poisson bracket of the functions f (x, y) and g(x, y) is defined as ∂f ∂g ∂f ∂g {f, g} := − . ∂x ∂y ∂y ∂x We will need the following result. Proposition 6. The next relation holds ∫ 2π {H2 , Ψ}|x=cos t, y=sin t dt = 0 0 for arbitrary C 1 function Ψ = Ψ(x, y) defined in the interval [0, 2π]. 6 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ Proof. Indeed, if we change x = cos t, y = sin t then it is easy to show that ∂Ψ ∂Ψ dΨ(cos t, sin t) {H2 , Ψ}|x=cos t, y=sin t = x −y = . ∂y ∂ x x=cos t, y=sin t dt Hence, ∫ 2π t=2π {H2 , Ψ}|x=cos t, y=sin t dt = Ψ(cos t, sin t)|t=0 = 0. 0  The following result is due to Liapunov (see Theorem 1, page 276 of [20]). Theorem 7. If all the roots λ1 , . . . , λn of the equation p11 − λ p21 ... pn1 p12 p22 − λ ... pn2 ... ... ... ... p1n p2n ... pnn − λ are such that the relation λ = m1 λ1 + . . . + mn λn , is not vanishing for an arbitrary non-negative integers m1 , . . . , mn linked by the expression m = m1 + . . . + mn ̸= 0. Then for an arbitrary given homogenous polynomial U = U (x1 , . . . , xn ) of degree m there exists a unique homogenous polynomial V = V (x1 , . . . , xn ) of degree m which is a solution of the equation ∑ n ∂V (pj1 x1 + . . . + pjn xn ) = U. j=1 ∂x j In particular, for n = 2 the partial differential equation ∂V ∂V (8) x −y := {H2 , V } = U, ∂y ∂x has a unique solution V if and only if λ1 m1 + λ2 m2 = i(m1 − m2 ) ̸= 0 with m = m1 + m2 ̸= 0. As a simple consequence of Theorem 7 we have the next result. Corollary 8. Let U = U (x, y) be a homogenous polynomial of degree m. The linear partial differential equation (8) 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 homogenous polynomial solution is of the form V + c(x2 + y 2 )m/2 with c ∈ R. Moreover, for m even these solutions exist if and only if ∫ 2π U (x, y)|x=cos t, y=sin t dt = 0. 0 In what follows some examples of planar vector fields having a center are studied. 2.1. Hamiltonian system. When system (4) is Hamiltonian, i.e. there exists a function F = F (x, y) such that ∂F (x, y) ∂F (x, y) −y + X(x, y) = − , x + Y (x, y) = . ∂y ∂x 1 2 Hence F = (x + y 2 ) + h.o.t. is a first integral. 2 AN INVERSE APPROACH TO THE CENTER PROBLEM 7 2.2. Reversible system. Besides Hamiltonian systems there is another class of systems (4) for which the origin is a center, namely the reversible systems satisfying the following definition. We say that system (4) is reversible with respect to the straight line l through the origin if it is invariant with respect to reversion about l and a reversion of time t (see for instance [7]). The following criterion goes back to Poincaré see for instance [24], p.122. Theorem 9. The origin of system (4) is a center if the system is reversible. In particular this theorem is applied for the case when (4) is invariant under the transformations (x, y, t) −→ (−x, y, −t) or (x, y, t) −→ (x, −y, −t). 2.3. Weak condition for a center. The following condition weak condition for a center was due to Alwash and Lloyd [1, 18], see also [18]. Proposition 10. The origin is a center of a polynomial differential system of the form (9) ẋ = −y + Xm , ẏ = x + Ym , where Xm and Ym are homogenous polynomial of degree m, if there exists µ ∈ R such that ( ) ∂Xm ∂Ym (x2 + y 2 ) + = µ (xXm + yYm ) , ∂x ∂y and either m = 2k is even; or m = 2k − 1 is odd and µ ̸= 2k; or m = 2k − 1 is odd, µ = 2k and ∫ 2π ( ) ∂Xm ∂Ym + |x=cos t, y=sin t dt = 0. 0 ∂x ∂y In [9] the author proved that if µ = 2m then system (9) has the rational first integral x2 + y 2 − 2 (xYm − yXm ) . (x2 + y 2 )m 2.4. Cauchy-Riemann condition for a center. Another particular case of dif- ferential systems with a center are the systems satisfing the Cauchy–Riemann con- ditions (see for instance [7]). Proposition 11. (Cauchy-Riemann condition for a center) Let O be a center of (2). Then O is isochronous center if P and Q satisfy the Cauchy-Riemann equations ∂P ∂Q ∂P ∂Q (10) = , =− ∂x ∂y ∂y ∂x A center of system (4) for which (10) holds is called a holomorphic center, which is also an isochronous center, see for more details [21] and [22]. We recall that a center of system (4) located at the origin is an isochronous center if all the periodic solutions in a neighborhood of the origin have the same period. 8 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ 3. Statement of the main results The main results are stated in the following four subsections. 3.1. Analytic and polynomial vector fields with a linear type center. We state and solve the following inverse problems for the centers of analytic and poly- nomial vector fields. Inverse Poincaré-Liapunov’s Problem Determine the analytic (polynomial) planar vector fields k ∑ ∑k ∂ ∂ (11) X = (−y + Xj ) + (x + Yj ) , for k ≤ ∞, j=2 ∂x j=2 ∂y for which the given function (6) is a local analytic first integral where Xj = Xj (x, y), Yj = Yj (x, y) for j ≥ 2 are homogenous polynomials of degree j. Inverse Reeb Problem Determine the analytic (polynomial) planar vector fields ∞ ∑ (11) for which the V = 1 + Vj is the Reeb inverse integrating factor, i.e. j=1 ( ) ∂V ∂V ∂ X (x) ∂ X (y) (12) X (x) + X (y) =V + ∂x ∂y ∂x ∂y The inverse Poincaré-Liapunov’s problem and inverse Reeb problem for the an- alytic (k = ∞) planar vector fields has been solved in the following theorem which provides the expressions of the analytic differential systems (7) in function of its first integral (6) or in function of its Reeb inverse integrating factor. Theorem 12. Consider the analytic vector field X . Then this vector field has a Poincaré-Liapunov local first integral if and only if it has a Reeb inverse integrating factor. Moreover, (i) the analytic differential system associated to the vector field X∞ for which H = (x2 + y 2 )/2 + h.o.t. is a local first integral can be written as (13) ∞ ∑ ∞ ∑ ẋ = −y + Xj = −y + ({Hj+1 , x} + g1 {Hj , x} + . . . + gj−1 {H2 , x}) , j=2 j=1   ∞ ∑ = 1 + gj  {H, x} j=1 ∞ ∑ ∞ ∑ ẏ = x + Yj = x + ({Hj+1 , y} + g1 {Hj , y} + . . . + gj−1 {H2 , y}) , j=2 j=1   ∞ ∑ = 1 + gj  {H, y} j=1 AN INVERSE APPROACH TO THE CENTER PROBLEM 9 where gj = gj (x, y) is an arbitrary homogenous polynomial of degree j which ∑∞ we choose in such a way that the series gj converge in the neighborhood j=1 of the origin. (ii) The differential system associated to the vector field X∞ for which V = ∑∞ 1+ Vj is a Reeb integrating factor can be written as j=1     ∞ ∑ ∞ ∑ ẋ = 1 + Vj  {F, x}, ẏ = 1 + Vj  {F, y}, j=1 j=1 ∞ ∑ where F = Fj and F2 = (x2 + y 2 )/2, Fj = Fj (x, y) for j > 2 is an j=2 arbitrary homogenous polynomial of degree j which we choose in such a ∞ ∑ way that Fj converges, i.e. F is an arbitrary Poincaré-Liapunov local j=2 first integral. In fact, in the proof of Theorem 12 we provide the expression of the vector fields having a given Poincaré-Liapunov local first integral and the expression of the vector fields having a given Reeb inverse integrating factor. The inverse Poincaré-Liapunov’s problem and inverse Reeb problem for the poly- nomial planar vector fields (k = m < ∞) has been solved in the following theorem which provides the expressions of the analytic differential systems (7) in function of its first integral (6) or in function of its Reeb integrating factor. Theorem 13. Consider the polynomial vector field Xm . Then this polynomial vec- tor field has a Poincaré-Liapunov local first integral if and only if it has a Reeb inverse integrating factor. Moreover, the differential system associated to the vec- tor field Xm for which H = (x2 + y 2 )/2 + h.o.t. is a local first integral can be written as   ∞ ∑ ẋ = 1 + gj  {H, x} j=1 = {Hm+1 , x} + (1 + g1 ){Hm , x} + . . . + (1 + g1 + . . . + gm−1 ){H2 , x}, (14)   ∞ ∑ ẏ = 1 + gj  {H, y} j=1 = {Hm+1 , y} + (1 + g1 ){Hm , y} + . . . + (1 + g1 + . . . + gm−1 ){H2 , y}. 10 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ where ∑∞ 1 2 2 H= (x + y ) + Hj = τ1 Hm+1 + τ2 Hm + . . . + τm H2 2 j=2 (15) ∫ ( ) = Ω dHm+1 + (1 + g1 )dHm + . . . + (1 + g1 + . . . + gm−1 )dH2 γ  −1 ∞ ∑ where Ω := 1 + gj  , γ is an oriented curve (see for instance [33]), τj = j=1 τj (x, y) is a convenient analytic function in the neighborhood of the origin such that τj (0, 0) = 1, and gj = gj (x, y) is an arbitrary homogenous polynomial of degree j which we choose in such a way that Ω is the inverse Reeb inverse integrating factor which satisfies the first order partial differential equation (16) {Hm+1 , Ω} + {Hm , (1 + g1 )Ω} + . . . + {H2 , (1 + g1 + . . . + gm−1 )Ω} = 0. Remark 14. From the proof of Theorem 13 it follows that (16) is equivalent to the infinite number of first order partial differential equations 0 = {Hm+1 , g1 } + {Hm , g2 } + . . . + {H3 , gm−1 } + {H2 , gm }, 0 = {Hm+1 , g12 − g2 } + {Hm , g1 g2 − g3 } + . . . + {H3 , g1 gm−1 − gm } (17) +{H2 , g1 gm − gm+1 }, .. .. .. . . . with unknowns the homogenous polynomials gj of degree j ≥ m. Hence by Corollary 6 we obtain the conditions ∫ 2π ({Hm+1 , g1 } + {Hm , g2 } + . . . + {H3 , gm−1 })|x=cos t,y=sin t dt = 0, 0 ∫ 2π ( ) {Hm+1 , g12 − g2 } + {Hm , g1 g2 − g3 } + . . . + {H3 , g1 gm−1 − gm } x=cos t,y=sin t dt = 0, 0 .. .. .. . . . The first condition, by Corollary 8 guarantees the existence of the solution gm of first equation of (17), the second condition, again by Corollary 8, guarantees the existence of the solution gm+1 of the second equation of (17), and so on. 3.2. Analytic and polynomial vector fields with local analytic first inte- 1 gral of the form H = (x2 + y 2 ) (1 + h.o.t.). We say that a differential system 2 (4) has a weak center at the origin if it has a local analytic first integral of the form   ∑ ∞ 1 2 H = (x + y 2 ) 1 + Υj (x, y) := H2 Φ(x, y), 2 j=1 where Υj is a convenient homogenous polynomial of degree j. The aim of this section is to study the weak centers for analytic and polynomial differential systems. AN INVERSE APPROACH TO THE CENTER PROBLEM 11 In the study of the weak centers plays a fundamental role the differential systems of the form (18) ẋ = −y(1 + Λ) + xφ, ẏ = x(1 + Λ) + yφ, where Λ = Λ(x, y) and φ = φ(x, y) are convenient analytic functions, as we can show from the following theorem. Theorem 15. An analytic differential system (7) has a weak center at the origin if and only if this system can be written as   ∑∞ ( ) j+1 j 3 ẋ = −y 1 + Υj−1 + g1 Υj−2 + . . . + gj−2 Υ1 + gj−1  j=2 2 2 2 x ∑( ) ∞ + {Υj−1 , H2 } + g1 {Υj−2 , H2 } + . . . + gj−2 {Υ1 , H2 } 2 j=2 := −y(1 + Λ) + xφ (19)   ∑∞ ( ) j+1 j 3 ẏ = x 1 + Υj−1 + g1 Υj−2 + . . . + gj−2 Υ1 + gj−1  j=2 2 2 2 y ∑( ) ∞ + {Υj−1 , H2 } + g1 {Υj−2 , H2 } + . . . + gj−2 {Υ1 , H2 } 2 j=2 := x(1 + Λ) + yφ, where Υ0 = 1, g0 = 1, gj and Υj   polynomial of degree j for j ≥ 1 are homogenous ∞ ∑ and has the first integral H = H2 1 + Υj  . Moreover assuming that j=2 j+1 j 3 Υj−1 + g1 Υj−2 + . . . + gj−2 Υ1 + gj−1 = 0, 2 2 2 {Υj−1 , H2 } + g1 {Υj−2 , H2 } + . . . + gj−2 {Υ1 , H2 } = 0, for j ≥ m + 1, we obtain necessary and sufficient conditions under which the poly- nomial differential system (19) of degree m and has the first integral (20) H = H2 Φ = H2 (1 + µ1 Υ1 + . . . + µm−1 Υm−1 ), where µj = µj (x, y) is a convenient analytic function in the neighborhood of the origin for j = 1, . . . , m − 1. The singular point of system (4) located at the origin is an isochronous center if all the periodic solutions in a neighborhood of it has the same period. Corollary 16. The weak center of a polynomial differential system (18) is an isochronous center if and only if ∫ 2π dθ (21) = 2π, 0 1 + Λ(r cos θ, r sin θ) 12 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ where (r, θ) are the polar coordinates, and r satisfies that   ∞ ∑ H(r cos θ, r sin θ) = r2 /2 1 + rj Υj (cos θ, sin θ) j=1 is a constant on any periodic solution surrounding the isochronous center. A center O of system (2) 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. Corollary 17. The weak center of an analytic differential system (18) is a uniform isochronous center if and only if 1 ( ) ∑∞ ẋ = −y + x {Hj , g1 } + . . . + {H2 , gj−1 } , j=2 j+1 (22) 1 ( ) ∑∞ ẏ = x + y {Hj , g1 } + . . . + {H2 , gj−1 } . j=2 j+1 Moreover the weak center of polynomial differential system of degree m (18) is a uniform isochronous center if and only if (22) holds and {Hj , g1 } + . . . + {H2 , gj−1 } = 0, holds for j ≥ m + 1. In particular for quasi-homogenous differential system (9) we have that (22) becomes x ẋ = −y + {H2 , gm−1 }, m+1 y ẏ = x + {H2 , gm−1 }, m+1 ( )2/(1−m) m−1 and has the Poincaré-Liapunov first integral F = H2 1 + gm−1 . m+1 The inverse approach to study the uniform isochronous center was given in [19]. Theorem 15 has the following additional corollary. Corollary 18. Assume that the planar differential system (5) has a center at the origin. Then this center is a holomorphic isochronous center if and only if system (5) can be written as (18), i.e. is a weak center, with the function Λ and φ satisfying the Cauchy–Riemann conditions ∂φ ∂Λ ∂φ ∂Λ − = 0, + = 0. ∂x ∂y ∂y ∂x Hence φ+i(1+Λ) = f (z) where z = x+iy, and f = f (z) is a holomorphic function on C. Moreover, a polynomial differential system (18) with a holomorphic center at the origin is Darboux integrable. Remark 19. From Corollaries 18 and 17 it follows that all the uniform isochronous centers and all the holomorphic isochronous centers for polynomial differential sys- tems are always weak centers. AN INVERSE APPROACH TO THE CENTER PROBLEM 13 It is important to observe that there is not a relation between isochronous centers and weak centers, i.e. there exist isochronous centers which are not weak centers and weak centers which are not isochronous centers. Then for instance the quadratic isochronous center 4x2 16y ẋ = −y − , ẏ = x(1 − ), 3 3 is not a weak center because it has the first integral H = (9 − 24y + 32x2 )2 /(3 − 16y) for more details see [5]. On the other hand the quadratic system ẋ = −y − x2 − 3y 2 , ẏ = x + 2xy, has a weak center at the origin because it has the first integral H = (1 + 2y)(x2 + y 2 ) but it is not isochronous see [5]. In fact in [18] we provide all the quadratic system with weak centers. We observe that any linear type center after an analytic change of variables is locally a weak center. This follows from the following theorem which goes back to Poincaré and Liapunov, see [23, 20, 27]. Theorem 20 (Poincaré normal form of a nondegenerate center). For a polynomial differential system (5) with a center at the origin, there exists a local analytic change of coordinates (23) u = x + h.o.t., v = y + h.o.t., 2 2 and an analytic function Ψ = Ψ(u + v ) such that the coordinate change (23) transforms system (5) into the form ∂H ∂H u̇ = − , v̇ = , ∂v ∂u ∫ 1 ( ) where H = 1 + Ψ(u2 + v 2 ) d(u2 + v 2 ). Without loss of generality we can 2 assume that Ψ(0, 0) = 0. Now we introduce the following definitions and notations. Let R[x, y] be the ring of all real polynomials in the variables x and y, and let X be the polynomial vector field (2) of degree m. Let g = g(x, y) ∈ R[x, y]\R. Then g = 0 is an invariant algebraic curve of X if ∂g ∂g Xg = P +Q = Kg, ∂x ∂y where K = K(x, y) is a polynomial of degree at most m − 1, which is called the cofactor of g = 0. A function g = g(x, y) satisfying that g = 0 is an invariant curve (i.e. formed by orbits of the vector field X ) is called partial integral. If g ∈ R[x, y]\R then g is called a polynomial partial integral or a Darboux polynomial. If the polynomial g is irreducible in R[x, y], then we say that the invariant algebraic curve g = 0 is irreducible, and that its degree is the degree of the polynomial g. A first integral F of the polynomial vector field (1) is called Darboux if F = ek(x,y)/h(x,y) g1λ1 (x, y) . . . grλr (x, y), where k, h, g1 , ..., gr are polynomials and λ1 , . . . , λr are complex constants. For more details on the so–called Darboux theory of integrability see for instance Chap- ter 8 of [10]. 14 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ We introduce the following definition. We say that a polynomial vector field X of degree m is quasi–Darboux integrable if there exist r polynomial partial integrals g1 , . . . , gr and s non-polynomial partial integrals f1 , . . . , fs analytic in D ⊆ R2 satisfying ∂fj ∂fj X (fj ) = P +Q = Kj fj , ∂x ∂y where Kj = Kj (x, y) is a convenient polynomials of degree m − 1, 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), is a first integral, where k = k(x, y), h = h(x, y) are polynomials, and λ1 , . . . , λr , κ1 , . . . , κs , are complex constants. We observe that a generalization of the Darboux theory was developed in the paper [14], which evidently contains the above defini- tion with another name, but for our aim we shall use the name of quasi–Darboux integrable. We have the following conjecture. Conjecture 21. A polynomial differential system (18) having a weak center at the origin is quasi-Darboux integrable. This conjecture is supported by several facts which we give below. Proposition 22. A polynomial differential system (18) with a weak center at the origin is quasi–Darboux integrable in a neighborhood of the origin with the first integral   m+1 ∑ 1 2 H = (x + y 2 ) 1 + τj (x, y)Υj (x, y) := H2 f (x, y), 2 j=1 where H2 = 0 is an invariant algebraic curve and f = 0 is an analytic (non polynomial) invariant curve with cofactor 2φ and −2φ respectively. 3.3. Center problem for analytic or polynomial vector fields with a gen- eralized weak condition of a center. First we prove the following two proposi- tions. Proposition 23. Assume that a differential system (4) satisfies the relation ( ) ∂(−y + X) ∂(x + Y ) (24) (x2 + y 2 ) + = µ (x(−y + X) + y(x + Y )) , ∂x ∂y with µ ∈ R\{0}. Then the system can be written as in (18) with ( ) 2 ∂(−y + X) ∂(x + Y ) φ(x, y) = + , µ ∂x ∂y and Λ = Λ(x, y) an arbitrary analytic function in a neighborhood of the origin. Moreover system (4) has the inverse integrating factor (x2 + y 2 )µ/2 , and it can be written as (25) ẋ = (x2 + y 2 )µ/2 {F, x} ẏ = (x2 + y 2 )µ/2 {F, y}, AN INVERSE APPROACH TO THE CENTER PROBLEM 15 with ∫ ( ) −Xdy + Y dx d(x2 + y 2 ) F = + γ (x2 + y 2 )µ/2 2(x2 + y 2 )µ/2  ∫ (26)  1 −Xdy + Y dx   (x2 + y 2 )(µ−2)/2 + if µ ̸= 2, 2−µ 2 2 µ/2 ∫ γ (x + y ) = √ −Xdy + Y dx   log x2 + y 2 + , if µ = 2.  γ (x2 + y 2 ) Note that if in (24) we have that µ = 0, then system (4) is a Hamiltonian system. Proposition 24. Consider the polynomial differential system (2) of degree m which satisfy the relations ∫ 2π ( ) ∂P ∂Q (27) + |x=cos t, y=sin t dt = 0. 0 ∂x ∂y m+1 ∑ m−1 ∑ Then there exist polynomials H̃ = Hj and G = Gj of degree m + 1 and j=3 j=1 m − 1 respectively such that system (2) can be written as ẋ = P = {H̃, x} + (1 + G){H2 , x}, (28) ẏ = Q = {H̃, y} + (1 + G){H2 , y}. Note that we have extended the definition of “weak condition for a center” given in subsection 2.5 for a quasi-homogenous polynomial differential system to a general analytic differential system. Proposition 10 can be generalized as follows. Theorem 25. [Generalized weak condition of a center of an analytic (polynomial) differential systems] We consider an analytic (polynomial) differential system (7). Then the origin is a weak center if there exists µ ∈ R\{0} such that (61) hold. Moreover this differential system can be written as (25) with the first integral: ∫ ( ) (1 + (1 − 1/λ)Υ + q(H2 ))dH2 + H2 dΥ F = 1/λ , γ H2 with λ = 2/µ and Υ = Υ(x, y) and q = q(H2 ) are a convenient analytic functions, or which is equivalent ( ( )) 1 ∂Υ ∂Υ x ẋ = −y 1 + q(H2 ) + (1 − 1/λ)Υ + x +y + {Υ, H2 }, 2 ∂x ∂y 2 (29) ( ( )) 1 ∂Υ ∂Υ y ẏ = x 1 + q(H2 ) + (1 − 1/λ)Υ + x +y + {Υ, H2 }. 2 ∂x ∂y 2 Moreover, if (29) is a polynomial differential system of degree m, i.e. Υ = Υ(x, y) [(m−1)/2] ∑ is a polynomials of degree m − 1 and q(H2 )) = αj H2j−1 , here [(m − 1)/2] is j=1 the integer part of (m − 1)/2, αj is a constant for j = 1, . . . , [(m − 1)/2] such that λ−1 1 + α1 + Υ(0, 0) ̸= 0. λ then the system (29) is quasi Darboux-integrable with the first integral F which is given in what follows 16 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ [m/2] ∏ (i) If λ ̸= 1 and (n − 1/λ) ̸= 0, then n=2 H2 (30) F =( )λ/(λ−1) . [(m−1)/2]−1 1 + α1 α2 H2 αm H2 Υ+ + + ... + 1 − 1/λ 2 − 1/λ [(m − 1)/2] − 1/λ The algebraic curves H2 = 0 and [(m−1)/2]−1 1 + α1 α2 H2 αm H2 g =Υ+ + + ... + =0 1 − 1/λ 2 − 1/λ [(m − 1)/2] − 1/λ are invariant curves with cofactors {g, H2 } and (1 − 1/λ){g, H2 }, respec- tively. (ii) If λ = 1 and 1 + α1 ̸= 0, then ( ) [(m−1)/2]−1 − Υ+α2 H2 +α3 H22 /2+...+αm H2 /([(m−1)/2]−1) /(1+α1 ) (31) F = H2 e . The algebraic curves H2 = 0 is invariant with cofactor {H2 , Υ}.   [(m−1)/2] ∏ (iii) If 1/[m/2] ≤ λ = 1/k < 1 and αk (n − k) ̸= 0, then n=2,n̸=k H2 (32) F = 1/(k−1) . [(m−1)/2] ∑ Υ + 1 + α1 + αj H j−1 + αk H2k−1 log H2  1−k j−k 2 j=2, j̸=k The algebraic curve H2 = 0 and non-polynomial curve [(m−1)/2] 1 + α1 ∑ αj f =Υ+ + H j−1 + αk H2k−1 log H2 = 0, 1−k j−k 2 j=2, j̸=k are invariant curves with cofactors {Υ, H2 } and (1−k){Υ, H2 }, respectively. We observe that lim f (x, y) = 0. (x,y)→(0,0) [(m−1)/2] ∏ (iv) If 1/[(m − 1)/2] ≤ λ = 1/k < 1, αk = 0 and (n − k) ̸= 0, then n=2,n̸=k H2 (33) F = 1/(k−1) . [(m−1)/2] ∑ Υ + 1 + α1 + αj H j−1  1−k j−k 2 j=2, j̸=k The algebraic curves H2 = 0 and [(m−1)/2] 1 + α1 ∑ αj g =Υ+ + H j−1 = 0 1−k j−k 2 j=2, j̸=k are invariant algebraic curves with cofactors {g, H2 } and (1 − k){g, H2 } respectively. AN INVERSE APPROACH TO THE CENTER PROBLEM 17 The given first integrals has the following Taylor extension at the origin F = H2 (1+ h.o.t.) Consequently the origin is a weak center. In an analogous way we can study the analytic case. 3.4. Linear centers with degenerate infinity. We shall study the following class of differential systems m−1 ∑ m−1 ∑ (34) ẋ = −y + Xj + xRm−1 , ẏ = x + Yj + yRm−1 , j=2 j=2 where Rm−1 = Rm−1 (x, y) is a convenient nonzero homogenous polynomial of degree m − 1. Such system are polynomial differential systems with a degenerate infinity. This name is due to the fact that in the Poincaré compactification of (34) the line at infinity is filled with singular points. Proposition 26. Assume that a polynomial differential system (5) has a center at the origin with a first integral H given in (6). Then this system has a degenerate infinity if it can be written as m ∑ m−1 x ∑ ẋ = gm+1−j {Ψ̄j , x} + {Hm+1−j , gj }, j=2 m + 1 j=1 m ∑ m−1 ∑ y ẏ = gm+1−j {Ψ̄j , y} + {Hm+1−j , gj }, j=2 m + 1 j=1 j−1 ∑ where Ψ̄j = Hk . k=2 Proposition 26 characterizes the polynomial differential systems having a degen- erate infinity and a linear type center at the origin. Proposition 27. Polynomial differential system (18) has a degenerate infinity if it can be written as m−2 ∑ m−2 ∑ ẋ = −y(1 + Λj ) + x φj + xφm−1 , j=2 j=2 (35) m−2 ∑ m−2 ∑ ẏ = x(1 + Λj ) + y φj + yφm−1 , j=2 j=2 i.e. Λm−1 = 0. All the results of this subsection are proved in section 7. 4. The Proofs of Subsection 3.1 Proof of Theorem 12 . First we prove the “only if part”.Assume that the analytic differential system (7) has a Poincaré-Liapunov local first integral. Then we shall see that it can be written as (13). 18 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ Consider a general analytic vector field with a singular point at the origin. Then it can be written as that we write as     ∑∞ ∑∞ ∂ ∂ X = Xj (x, y) + Yj (x, y) , j=1 ∂x j=1 ∂y where Xj and Yj for j = 0, 1, . . . are homogenous polynomials of degree j. Since the analytic first integral H starts with H2 = (x2 +y 2 )/2, without loss of generality this implies that X1 (x, y) = −y and Y1 (x, y) = x. Hence the following infinite number of equations must be satisfied ( ) dH ∂ H3 0= = x+ + . . . (−y + X2 + X3 + . . .) dt ∂x ( ) ∂ H3 + y+ + . . . (x + Y2 + Y3 + . . .) ∂y = xX2 + yY2 + {H2 , H3 } ∂ H3 ∂ H3 +xX3 + yY3 + X2 + Y2 + {H2 , H4 } ∂x ∂y ∂ H3 ∂ H3 ∂ H4 ∂ H4 +xX4 + yY4 + X3 + Y3 + X2 + Y2 + {H2 , H5 } + . . . ∂x ∂y ∂x ∂y .. .. .. .. . . . . ∂ H3 ∂ H3 ∂ Hn ∂ Hn +xXn + yYn + Xn−1 + Yn−1 + . . . + X2 + Y2 + {H2 , Hn+1 } ∂x ∂y ∂x ∂y .. .. .. .. . . . . Consequently (36) xX2 + yY2 + {H2 , H3 } = 0, ∂ H3 ∂ H3 xX3 + yY3 + X2 + Y2 + {H2 , H4 } = 0, ∂x ∂y ∂ H3 ∂ H3 ∂ H4 ∂ H4 xX4 + yY4 + X3 + Y3 + X2 + Y2 + {H2 , H5 } = 0, ∂x ∂y ∂x ∂y .. .. .. .. .. .. . . . . . . ∂ H3 ∂ Hn ∂ Hn xXn + yYn +Xn−1 + . . . + X2 + Y2 + {H2 , Hn+1 } = 0, ∂x ∂x ∂y ∂ H3 ∂ Hn+1 ∂ Hn+1 xXn+1 + yYn+1 + Xn + . . . + X2 + Y2 + {H2 , Hn+2 } = 0, ∂x ∂x ∂y .. .. .. .. .. .. . . . . . .. First, we introduce the notations k ∑ k ∑ X0 = Xj = ({Hj+1 , } + g1 {Hj , } + . . . + gj−1 {H2 , }) , j=2 j=2 AN INVERSE APPROACH TO THE CENTER PROBLEM 19 where k ≤ ∞ and gj = gj (x, y) is a homogenous polynomial in the variables x and y of degree j, for j = 1, 2, . . . , k. The first equation of (36) can be rewritten as follows ( ) ( ) ∂ H3 ∂ H3 x X2 + + y Y2 − = 0. ∂y ∂x Solving it with respect to X2 and Y2 we obtain ∂ H3 X2 = − − yg1 = {H3 , x} + g1 {H2 , x} := X2 (x), ∂y ∂ H3 Y2 = + xg1 = {H3 , y} + g1 {H2 , y} := X2 (y), ∂x where g1 = g1 (x, y) is an arbitrary homogenous polynomial of degree one. By substituting these polynomials into the second equation of (36) we get ( ) ( ) ∂ H4 ∂ H3 ∂ H4 ∂ H3 x X3 + + g1 + y Y3 − − g1 = 0. ∂y ∂y ∂x ∂x By solving this equation with respect to X3 and Y3 we have ∂ H4 ∂ H3 X3 = − − g1 − yg2 = {H4 , x} + g1 {H3 , x} + g2 {H2 , x} := X3 (x), ∂y ∂y ∂ H4 ∂ H3 Y3 = + g1 + xg2 = {H4 , y} + g1 {H3 , y} + g2 {H2 , y} := X3 (y), ∂x ∂x where g2 = g2 (x, y) is an arbitrary homogenous polynomial of degree two. By continuing this process we obtain X4 , Y4 , . . . , Xn , Yn , i.e. Xn = {Hn+1 , x} + g1 {Hn , x} + . . . + gn−1 {H2 , x} := Xn (x), (37) Yn = {Hn+1 , y} + g1 {Hn , y} + . . . + gn − 1{H2 , y} := Xn (y), where gn = gn (x, y) is an arbitrary homogenous polynomial of degree n. Hence, ∞ ∑ since gj converges in a neighborhood of the origin, we get that j=1 ∞ ∑ ẋ = −y + X2 + X3 + . . . + Xj + . . . = −y + X (x) = −y + Xj (x) j=2   ∞ ∑ = 1 + gj  {H, x}, j=1 ∞ ∑ ẏ = x + Y2 + Y3 + . . . + Yj + . . . = x + X (y) = x + Xj (y) j=2   ∞ ∑ = 1 + gj  {H, y}. j=1 ∞ ∑ Note that the function 1+ gj is an analytic integrating factor of the differential j=1 system (13) i.e. it is a Reeb inverse integrating factor. 20 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ Now we prove the “ if ” part. We assume that system (5) has a Reeb inverse integrating factor. From the equation (12), i.e. ( ) ∂ V1 ∂ V2 ∂ V3 (X1 + X2 + X3 + . . .) + + + ... ∂x ∂x ∂x ( ) ∂ V1 ∂ V2 ∂ V3 + (Y1 + Y2 + Y3 + . . .) + + + ... ∂y ∂y ∂y ( ) ∂ X1 ∂ Y1 ∂ X2 ∂ Y2 ∂ X3 ∂ Y3 = (1 + V1 + V2 + . . .) + + + + + + ... ∂x ∂y ∂x ∂y ∂x ∂y if follows that (38) ∂ X1 ∂ Y1 0= + , ∂x ∂y ∂ V1 ∂ V1 ∂ X2 ∂ Y2 Y1 + X1 = + , ∂y ∂x ∂x ∂y ( ) ∂ V2 ∂ V2 ∂ V1 ∂ V1 ∂ X3 ∂ Y3 ∂ X2 ∂ Y2 Y1 + X1 + X2 + Y2 = + + V1 + , ∂y ∂x ∂x ∂y ∂x ∂y ∂x ∂y .. .. .. .. .. .. . . . . . . From the first equation of (38) we get that ∂ F2 ∂ F2 X1 = − , Y1 = , ∂y ∂x where F2 = F2 (x, y) is an arbitrary homogenous polynomial of degree 2. From the second equation of (38) we obtain ( ) ( ) ∂ ∂ F2 ∂ ∂ F2 X2 + V1 + Y2 − V1 = 0, ∂x ∂y ∂y ∂x hence ∂ F3 ∂ F2 ∂ F3 ∂ F2 (39) X2 = − − V1 , Y2 = + V1 , ∂y ∂y ∂x ∂x where F3 = F3 (x, y) is an arbitrary homogenous polynomial of degree 3. From the third equation of (38) we obtain ( ) ( ) ∂ ∂ F3 ∂ F2 ∂ ∂ F3 ∂ F2 X3 + V 1 + V2 + Y3 − V1 − V2 = 0, ∂x ∂x ∂y ∂y ∂y ∂x thus ∂ F4 ∂ F3 ∂ F2 ∂ F4 ∂ F3 ∂ F2 X3 = − − V1 − V2 , Y3 = + V1 + V2 , ∂y ∂y ∂y ∂x ∂x ∂x where F4 = F4 x, y) is an arbitrary homogenous polynomial of degree 4. By contin- uing this process we get ∂ Xj ∂ Yj {Fj , V1 } + . . . + {F2 , Vj−1 } = + , ∂x ∂y AN INVERSE APPROACH TO THE CENTER PROBLEM 21 ( ) ( ) ∂ ∂Fj ∂ ∂Fj by considering that {Fk , Vn } = Vn − Vn we deduce the rela- ∂y ∂x ∂x ∂y tion ( ) ( ) ∂ ∂ Fj ∂ F2 ∂ ∂ Fj ∂ F2 Xj + V1 + . . . + . . . + Vj−1 + Xj − V1 − − . . . − Vj−1 = 0. ∂x ∂y ∂y ∂y ∂x ∂x By using the notation k ∑ k ∑ X̃0 = X̃j = ({Fj+1 , } + V1 {Fj , } + . . . + Vj−1 {F2 , }) , j=12 j=12 where k ≤ ∞, Vj = Vj (x, y) and Fj = Fj (x, y) are homogenous polynomial in the variables x and y of degree j, for j = 1, 2, . . . , k, we get that X̃1 (x) = {F2 , x} = X1 , X̃1 (y) = {F2 , y} = Y1 , X̃2 (x) = {F3 , x} + V1 {F2 , x} = X2 , X̃2 (y) = {F3 , y} + V1 {F2 , y} = Y2 , (40) .. .. .. .. .. .. . . . . . . X̃m (x) = {Fm+1 , x} + V1 {Fm , x} + . . . + Vm−1 {F2 , x} = Xm , X̃m (y) = {Fm+1 , y} + V1 {Fm , y} + . . . + Vm−1 {F2 , y} = Ym , (41) X̃m+k−1 (x) = {Fm+k , x} + V1 {Fm−1+k , x} + . . . + Vm+k−2 {F2 , x} = Xm+k−1 , X̃m+k−1 (y) = {Fm+k , y} + V1 {Fm−1+k , y} + . . . + Vm+k−2 {F2 , y} = Ym+k−1 , where k > 1 and Fk = Fk (x, y) is an arbitrary homogenous polynomial of degree ∑∞ j, for j ≥ 3, such that the series F = Fj , converges at neighborhood of the j=2 origin. By considering that we are interesting in studying the linear type center then X1 = −y and X1 = x then we have that F2 = (x2 + y 2 )/2. Therefore F = (x2 + y 2 )/2 + F3 + F4 + . . . , is a Poincaré-Liapunov local first integral this prove the “ if ” part of the theorem. 22 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ Moreover, by summing we get ∞ ∑ ∞ ∑ ẋ = −y + Xj = ({Fj+1 , x} + V1 {Fj , x} + . . . + Vj−1 {F2 , x}) j=2 j=2   ∞ ∑ = 1 + Vj  {F, x} j=2 ∞ ∑ ∞ ∑ ẏ = x + Xj = ({Fj+1 , y} + V1 {Fj , y} + . . . + Vj−1 {F2 , y}) j=2 j=2   ∞ ∑ = 1 + Vj  {F, y}, j=2 Thus the proof of the theorem follows.  Remark 28. From the “only if ” part follows that the arbitrariness which we deter- mine the vector fields with the given Poincaré-Liapunov local first integral is related ∑∞ with the Reebs inverse integrating factor V = 1 + gj and from the ” if” part fol- j=2 lows that the arbitrariness which we determine the vector fields with the given Reebs inverse integrating factor is related with the Poincaré-Liapunov local first integral F = (x2 + y 2 )/2 + F3 + F4 + . . .. Proof of Theorem 13. Now we assume that the vector field X is polynomial of de- gree m. First we prove the ”only if“ part. From (37) it follows that if Xn = Yn = 0 for n > m + 1, then (42) ∞ ∑ ∑m ẋ = −y + Xj (x) = −y + Xj (x) j=2 j=2 = −y + {Hm+1 , x} + (1 + g1 ){Hm , x} + . . . + (1 + g1 + . . . + gm−1 ){H2 , x}, ∞ ∑ m ∑ ẏ = x + Xj (y) = x + Xj (y) j=2 j=2 = x + {Hm+1 , y} + (1 + g1 ){Hm , y} + . . . + (1 + g1 + . . . + gm−1 ){H2 , y}. Clearly, if Xn = Yn = 0 for n ≥ m + 1, then (43) Xm+1 (x) = {Hm+2 , x} + g1 {Hm+1 , x} + . . . + +gm−1 {H3 , x} + gm {H2 , x} = 0, Xm+1 (y) = {Hm+2 , y} + g1 {Hm+1 , y} + . . . + +gm−1 {H3 , y} + gm {H2 , y} = 0, Xm+k (x) = {Hm+k+1 , x} + g1 {Hm+k , x} + . . . + gm+k−1 {H2 , x} = 0, Xm+k (y) = {Hm+k+1 , y} + g1 {Hm+k , y} + . . . + gm+k−1 {H2 , y} = 0, AN INVERSE APPROACH TO THE CENTER PROBLEM 23 for k > 2. This system of partial differential equations of first order is compatible if and only if the following relations hold {Hm+1 , g1 } + {Hm , g2 } + . . . + {H3 , gm−1 } + {H2 , gm } = 0, (44) {Hm+k , g1 } + {Hm+k−1 , g2 } + . . . + {H2 , gm+k−1 } = 0, for k ≥ 2. Hence, in view of Proposition 8 we get that ∫ 2π ( ) {Hm+1 , g1 } + {Hm , g2 } + . . . + {H3 , gm−1 } dt = 0. 0 x=cos t, y=sin t We shall study partial differential equations (43) under the conditions (44). For k = 2 from (43) we get (45) dHm+2 = −g1 dHm+1 − g2 dHm + . . . − gm−1 dH3 − gm dH2 , where gm = gm (x, y) is an arbitrary homogenous polynomial of degree m which satisfies the first order partial differential equations (see (44) for n = m + 1.) Hence the two first partial differential system (43) is compatible, consequently integrating the 1-form (45) we obtain ∫ Hm+2 = − (g1 dHm+1 + g2 dHm + . . . + gm−1 dH3 + gm dH2 ) . γ On the other hand from (43) and using that Hj are homogenous polynomial of degree j, we get that 1 Hm+2 = − ((m + 1)g1 Hm+1 + . . . + 3gm−1 H3 + 2gm H2 ) . m+2 For k = 3 system (43) becomes {Hm+3 , x} + g1 {Hm+2 , x} + . . . + gm {H3 , x} + gm+1 {H2 , x} = 0, (46) {Hm+3 , y} + g1 {Hm+2 , y} + . . . + gm {H3 , y} + gm+1 {H2 , y} = 0, which in view of (45) system (46) can be written as ( ) {Hm+3 , x} + g12 − g2 {Hm+1 , x} + (g1 g2 − g3 ) {Hm , x} + . . . + (g1 gm−1 − gm ){H3 , x} + (g1 gm + gm+1 ) {H2 , x} = 0, (47) ( ) {Hm+3 , y} + g12 − g2 {Hm+1 , y} + (g1 g2 − g3 ) {Hm , y} + . . . + (g1 gm−1 − gm ){H3 , y} + (g1 gm + gm+1 ) {H2 , y} = 0, where gm+1 = gm+1 (x, y) is an arbitrary homogenous polynomial of degree m + 1 which satisfies the first order partial differential equation ( ) { g12 − g2 , Hm+1 } + {(g1 g2 − g3 ) , Hm } (48) + . . . + {(g1 gm−1 − gm ), H3 } + {(g1 gm − gm+1 ) , H2 } = 0. Hence, in view of Proposition 8 we get that ∫ 2π ( ( ) − { g12 − g2 , Hm+1 } + {(g1 g2 − g3 ) , Hm } 0 ) + . . . + {(g1 gm−1 − gm ), H3 } dt = 0. x=cos t, y=sin t 24 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ On the other hand, from (47) and in view of the fact that Hj are homogenous polynomial of degree j we get that 1 ( ( ) Hm+3 = − (m + 1) g12 − g2 Hm+1 − m (g1 g2 − g3 ) Hm m+3 ) − . . . + 3(g1 gm−1 − gm )H3 − 2 (g1 gm − +gm+1 ) H2 . Under the condition (48) system (47) is compatible, consequently after the integra- tion de 1-form ( ) dHm+3 = − g12 − g2 dHm+1 − (g1 g2 − g3 ) dHm − . . . + (g1 gm−1 − gm )dH3 − (g1 gm − gm+1 ) dH2 , we get that ∫ ( 2 ) Hm+3 = − g1 − g2 dHm+1 − (g1 g2 − g3 ) dHm γ − . . . + (g1 gm−1 − gm )dH3 − (g1 gm − gm+1 ) dH2 1 ( ( ) = − (m + 1) g12 − g2 Hm+1 − m (g1 g2 − g3 ) Hm m+3 ) − . . . + 3(g1 gm−1 − gm )H3 − 2 (g1 gm − gm+1 ) H2 For k = 4 system (43) becomes {Hm+4 , x} + g1 {Hm+3 , x} + . . . + gm {H4 , x} + gm+1 {H3 , x} + gm+2 {H2 , x} = 0, {Hm+4 , x} + g1 {Hm+3 , y} + . . . + gm {H4 , y} + gm+1 {H3 , y} + gm+2 {H2 , y} = 0, which in view of (45) system (46) can be written as (49) ( ) {Hm+4 , x} + −g13 + 2g1 g2 − g3 {Hm+1 , x} ( ) ( ) + −g12 g2 + g3 g1 + g22 − g4 {Hm , x} + . . . + g1 gm+1 − g12 gm−1 − gm+2 {H2 , x} = 0, ( ) {Hm+4 , y} + −g13 + 2g1 g2 − g3 {Hm+1 , y} ( ) ( ) + −g12 g2 + g3 g1 + g22 − g4 {Hm , y} + . . . + g1 gm+1 − g12 gm−1 − gm+2 {H2 , y} = 0, where gm+2 = gm+2 (x, y) is an arbitrary homogenous polynomial of degree m + 2 which satisfies the first order partial differential equation 0= {−g13 + 2g1 g2 − g3 , Hm+1 } + {−g12 g2 + g3 g1 + g22 − g4 , Hm } (50) + . . . + {(g2 − g12 )gm + g1 gm+1 − gm+2 , H2 }. ∫ 2π ( ( ) { −g13 + 2g1 g2 − g3 , Hm+1 } 0 ( ) ) + . . . + { (g2 − g12 )gm−1 + g1 gm − gm+1 , H3 } dt = 0. x=cos t, y=sin t Under this condition system (49) is compatible, consequently after the integration of the 1-form ( ) ( ) dHm+4 = − −g13 + 2g1 g2 − g3 dHm+1 −. . .− (g2 − g12 )gm + g1 gm+1 − gm+2 dH2 AN INVERSE APPROACH TO THE CENTER PROBLEM 25 and using the property of homogenous polynomial we get that ∫ ∑ m ( ) Hm+4 = gj (g2 − g12 ) + g1 gj+1 − gj+2 dHm+2−j . γ j=1 m 1 ∑ ( ) = (m + 2 − j)( gj (g2 − g12 ) + g1 gj+1 − gj+2 Hm+2−j . m + 4 j=1 By continuing this process we deduce that dHm+k = −βk−1 dHm+1 − βk dHm − . . . − (βm+k−2 + gm+k−2 )dH2 for k ≥ 5, where βj = βj (x, y) are homogenous polynomial of degree j, and gm+k−2 is an arbitrary homogenous polynomial of degree m + k − 2 which we choose as a solution of the first order partial differential equation (51) {βk−1 , Hm+1 } + . . . + {βm+k−2 + gm+k−2 , H2 } = 0 for k ≥ 5, Thus ∫ Hm+k = (−βk−1 dHm+1 − βk dHm − . . . − (βm+k−2 + gm+k−2 )dH2 ) γ 1 = ((m + 1)αk Hm+1 + mαk+1 Hm + . . . + 2αm+k−2 H2 ) m+k for k ≥ 5, where αk is a convenient homogenous polynomial of degree k. From these results it follows that the homogenous polynomials Hj+1 and gj−1 for j > m we determine by the line integral and as a solution of the linear partial differential equation respectively. By summing we finally obtain (52) ∞ ∑ ∞ ∑ H= Hj = (x2 + y 2 )/2 + Hj j=2 j=3 ∫ ( ) = 1 − g1 − g2 − . . . − gm−1 − gm . . . + g12 + . . . + gm−1 2 2 + gm + . . . dHm+1 γ ∫ ( ) + 1 − g2 − g3 − . . . − gm−1 − gm . . . − g1 g2 − g1 g3 + . . . dHm γ .. .. .. .. . . . . ∫ ( ) + 1 − gm − gm+1 − . . . − 2g1 g2 − 2g1 g3 − . . . − 2g1 gm−1 − g12 − g22 . . . dH2 γ = τm+1 Hm+1 + τm Hm + . . . + τ2 H2 , where τj = τj (x, y) is a convenient analytical function, for j = 2, . . . , m + 1. 26 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ ∞ ∑ Hence, if gj converges in a neighborhood of the origin, then in view of the j=3 Taylor expansion  2 ∞ ∑ ∞ ∑ 1 Ω := ∞ =1− gj +  gj  + . . . , ∑ j=3 j=1 1+ gj j=1 we get that Ω= 1 − g1 − g2 − . . . − gm−1 − gm . . . + g12 + . . . + gm−1 2 2 + gm + ..., (1 + g1 )Ω = 1 − g2 − g3 − . . . − gm−1 − gm . . . − g1 g2 − g1 g3 + . . . , .. .. .. .. . . . . (1 + g1 + . . . + gm−1 )Ω = 1 − gm − gm+1 − . . . − 2g1 g2 − 2g1 g3 − . . . − 2g1 gm−1 − g12 − g22 . . . . Therefore the function (52) can be written as follows ∞ ∑ 2 2 H = (x + y )/2 + Hj = τm+1 Hm+1 + τm Hm + . . . + τ2 H2 j=3 ∫ ( ) = Ω dHm+1 + (1 + g1 )dHm + . . . (1 + g1 + . . . gm−1 )dH2 , γ On the other hand, by summing (44), (48), (50), (51) and etc. we get { } 2 −g1 − g2 − . . . − gm−1 − gm . . . + g12 + . . . + gm−1 2 + gm + . . . , Hm+1 +{1 − g2 − g3 − . . . − gm−1 − gm . . . − g1 g2 − g1 g3 + . . . , Hm } .. .. .. .. . . . . +{1 − gm − gm+1 − . . . − 2g1 g2 − 2g1 g3 − . . . − 2g1 gm−1 − g12 − g22 . . . , H2 } = {Ω, Hm+1 } + {Ω(1 + g1 ), Hm } + . . . + {Ω(1 + g1 + . . . + gm−1 ), H2 } = 0. Hence we obtain that the polynomial differential system (42) of degree m can be ∞ ∑ written as (14) where 1 + gj is the Reeb inverse integrating factor. In short the j=2 proof of the “only if part” and the statement (i) follows. This proves the “only if part” of the theorem. ∞ ∑ Now we prove the ” if” part. We assume that V = 1 + Vj is the Reeb inverse j=2 integrating factor. From (40) and (41) it follows that If Xj = Yj = 0 for j > m + 1, then X̃k (x) = {Fk+1 , x} + V1 {Fk , x} + . . . + Vk−1 {F2 , x} = 0, (53) X̃k (y) = {Fk+1 , y} + V1 {Fk , y} + . . . + Vk−1 {F2 , y} = 0, for k ≥ m + 1. AN INVERSE APPROACH TO THE CENTER PROBLEM 27 System of partial differential equations of first order (53) is compatible if and only if (54) {V1 , Fk } + {V2 , Fk−1 } + . . . + {Vk−1 , F2 } = 0, where k ≥ m + 1. The proof of statement (ii) can be obtained analogously to the proof of statement (i), if we take gj = Vj and Hj+1 = Fj+1 for j = 1, . . . , m. Finally we observe that from (15) it follows that ∂F = Ω ({Hm+1 , x} + (1 + g1 ){Hm , x} + . . . + (1 + g1 + . . . + gm−1 ){H2 , x}) , ∂y ∂F = Ω ({Hm+1 , y} + (1 + g1 ){Hm , y} + . . . + (1 + g1 + . . . + gm−1 ){H2 , y}) , ∂x ∂2 H ∂2 H From the condition = , we get the condition (16). In short the theorem ∂ x∂ y ∂ y∂ x is proved.  Example 29. We shall determine the quadratic system having the Reeb integrating factor V = (1a + Ay)2b/A−1 = 1 + (2b − A)y + (A − 2b)(A − b)y 2 1 + (A − 2b)(A − b)(3A − 2b)y 3 + . . . 3 = 1 + V1 + V2 + V3 + . . . , where A and b are nonzero constants. The quadratic polynomial differential system (39) in this case becomes ∂F3 ∂F3 X2 = − − (2b − A)y 2 , Y2 = + (2b − A)xy, ∂y ∂x where F3 is a homogenous polynomial of degree 3, which satisfies the conditions (54) for k = 3. Hence we obtain that F3 = bx2 y + κ y 3 , where κ is a constant. Therefore X2 = −bx2 − (A − 2b + 3κ)y 2 , Y2 = 2bxy + (A − 2b)xy = Axy. 5. The Proofs of Subsection 3.2 Proof of Theorem 15. Necessity We suppose that system (4) has a weak center at the origin . Consequently there exists an analytic local first integral H = H2 (1 + ∑∞ Υj ) := H2 Φ. Then from Theorem 12 it follows the necessary and sufficient j=1 conditions on the existence of a linear type center for an analytic differential system 28 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ differential system. Thus (13) becomes (55) ( ) ( ) ∂Φ y ∂Φ x ∂Φ x ẋ = V {H, x} = −V yΦ + H2 = −V y Φ + + + V {Φ, H2 } ∂y 2 ∂y 2 ∂x 2    ∑∞ ∑∞ ∞ ( j+2 ) x∑j+2 = 1 + gj  −y 1 + Υj + {Υj , H2 } j=1 j=1 2 2 j=1 2 ( ∞ ( ∑ )) j+1 j 3 = −y 1 + Υj−1 + g1 Υj−2 + . . . + gj−2 Υ1 + gj−1 j=2 2 2 2 x ∑( ) ∞ + {Υj−1 , H2 } + g1 {Υj−2 , H2 } + . . . + gj−2 {Υ1 , H2 } 2 j=2 ∑∞ ∞ ( ) x∑ = −y 1 + Λj + Ωj j=2 2 j=2 ( ) ( ) ∂Φ y ∂Φ x ∂Φ y ẏ = V {H, y} = V xΦ + H2 =Vx Φ+ + + V {Φ, H2 } ∂x 2 ∂y 2 ∂x 2    ∑∞ ∑∞ ∞ ( j+2 ) y ∑j+2 = 1 + gj  x 1 + Υj + {Υj , H2 } j=1 j=1 2 2 j=1 2 ( ∑∞ ( )) j+1 j 3 = x 1+ Υj−1 + g1 Υj−2 + . . . + gj−2 Υ1 + gj−1 j=2 2 2 2 y ∑( ) ∞ + {Υj−1 , H2 } + g1 {Υj−2 , H2 } + . . . + gj−2 {Υ1 , H2 } 2 j=2 ∑∞ ∞ ( ) y∑ = x 1+ Λj + Ωj . j=2 2 j=2 Sufficiency Now we suppose that (19) holds and show that then the origin is a weak center. Indeed, from (55) we obtain that Ḣ2 = 2V H2 {Φ, H2 }, Φ̇ = dΦ Φ −2V Φ{Φ, H2 }, thus =− then H = H2 Φ is a first integral, consequently dH2 H2 the origin is a weak center. The second statement we prove as follows. Under the assumption ∑∞ ( ) j+1 j 3 −y Υj−1 + g1 Υj−2 + . . . + gj−2 Υ1 + gj−1 j=m 2 2 2 x ∑( ) ∞ + {Υj−1 , H2 } + g1 {Υj−2 , H2 } + . . . + gj−2 {Υ1 , H2 } = 0 2 j=m ∞ ( ∑ ) j+1 j 3 x Υj−1 + g1 Υj−2 + . . . + gj−2 Υ1 + gj−1 j=m 2 2 2 AN INVERSE APPROACH TO THE CENTER PROBLEM 29 y ∑( ) ∞ + {Υj−1 , H2 } + g1 {Υj−2 , H2 } + . . . + gj−2 {Υ1 , H2 } = 0 2 j=m which is equivalent to the equations j+1 j 3 Υj−1 + g1 Υj−2 + . . . + gj−2 Υ1 + gj−1 = 0 2 2 2 {Υj−1 , H2 } + g1 {Υj−2 , H2 } + . . . + gj−2 {Υ1 , H2 } = 0 for j > m + 1, from (55) we get the following polynomial differential equations of degree m.   m ( ∑ ) j+1 j 3 ẋ = −y 1 + Υj−1 + g1 Υj−2 + . . . + gj−2 Υ1 + gj−1  j=2 2 2 2 x ∑( ) m + {Υj−1 , H2 } + g1 {Υj−2 , H2 } + . . . + gj−2 {Υ1 , H2 } , 2 j=2   ∑m ( ) j + 1 j 3 ẏ = x 1 + Υj−1 + g1 Υj−2 + . . . + gj−2 Υ1 + gj−1  j=2 2 2 2 y ∑( ) m + {Υj−1 , H2 } + g1 {Υj−2 , H2 } + . . . + gj−2 {Υ1 , H2 } 2 j=2 Here Υj is a convenient homogenous polynomial of degree j, such that H2 = (x2 + y 2 )/2, Hj+2 = H2 Υj , for j = 1 . . . m + 1 and gj is an arbitrary homogenous polynomial of degree j satisfying (16). Consequently in view of (15) we obtain the local first integral (20). Thus the theorem is proved  Example 30. The following cubic polynomial differential system has a center at the origin (see [35]) 1( 2 ) y2 x ẋ = −y + x − xy − 2y 2 − xy 2 − y 3 = −y(1 + y + ) + (x − y − y 2 ) 2 2 2 1( ) y 2 y ẏ = x + 3xy − y 2 + xy 2 − y 3 = x(1 + y + ) + (x − y − y 2 ), 2 2 2 Consequently this system can be rewritten as (18) with the functions φ and Λ de- 1 termined as follows 1 + Λ = (1 + (y + 1)2 , φ = x − y − y 2 . and hence the center 2 is a weak center. In order to illustrate Theorem 13 we study the following quadratic systems. Example 31. For the quadratic differential system ẋ = −y − bx2 − dy 2 ẏ = x + Axy, d−b−A 3 the functions H3 and g1 are H3 = 2b y H2 + y and g1 = (A − 2b)y. It 3 is easy to show that the solution of equation (16) for m = 2, i.e. the equation 30 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ H3 , Ω} + {H2 , (1 + g1 )Ω} = 0, is Ω = (1 + Ay)2b/A−1 . Consequently from (15) the quadratic system has the following first integral ∫ h(x, y) = (1 + Ay)2b/A−1 (dH3 + (1 + g1 )dH2 ) ( ( )) = (1 + Ay)2b/A 2(b(A + b)(A + 2b)H2 + (d − A − b) 1 − 2by + by 2 (A + 2b) , A + b − d + h(x, y) and from it we obtain the Poincaré-Liapunov first integral H = = 2b(A + b)(A + 2b) H2 + h.o.t.. In particular if d − A − b = 0, then the quadratic polynomial differential system has a weak center at the origin with H3 = 2b y H2 , and Poincaré-Liapunov first integral H = (1 + Ay)2b/A−1 H2 . Proof of Corollary 17. From the equation Λ = 0 and in view of of (22) we get ∞ ( ∑ ) j+1 j 3 Υj−1 + g1 Υj−2 + . . . + gj−2 Υ1 + gj−1 = 0, j=2 2 2 2 hence by considering that Hj = H2 Υj−2 we obtain that 1 Hj+1 = − (j g1 Hj + . . . + 3 gj−2 H3 + 2gj−1 H2 ) , j+1 for j ≥ 2. On the other hand, in view of the previous relation it follows from (22) that ∞ ( ∑ ) 2φ = {Υj−1 , H2 } + g1 {Υj−2 , H2 } + . . . + gj−2 {Υ1 , H2 } j=2 1 ∑( ) ∞ = {Hj+1 , H2 } + g1 {Hj , H2 } + . . . + gj−2 {H3 , H2 } H2 j=2 1 ∑ ( −1 ) ∞ = {jg1 Hj + . . . + 2gj−1 H2 , H2 } + g1 {Hj , H2 } + . . . + gj−2 {H3 , H2 } H2 j=2 j + 1 1 ∑( 1 ) ∞ = − ({jg1 Hj + . . . + 2gj−1 H2 , H2 ) − g1 {Hj , H2 } − . . . − gj−2 {H3 , H2 } H2 j=2 j + 1 1 ∑( 1 ) ∞ = − ({jg1 Hj , H2 } − (j + 1)g1 {Hj , H2 } + . . .) H2 j=2 j + 1 1 ∑( 1 ) ∞ = − (jHj {g1 , H2 } − g1 {Hj , H2 } + . . .) H2 j=2 j + 1 ( )) 1 ∑( 1 ∞ ∂Hj ∂Hj ∂g1 ∂g1 = − (x +y ){g1 , H2 } − (x +y )({Hj , H2 } + . . .) H2 j=2 j + 1 ∂x ∂y ∂x ∂x AN INVERSE APPROACH TO THE CENTER PROBLEM 31 ∞ 1 ∑ 1 ( ) = (x2 + y 2 ) {Hj , g1 } + . . . + {H2 , gj−1 } H2 j=2 j + 1 1 ( ) ∞ ∑ = 2 {Hj , g1 } + . . . + {H2 , gj−1 } . j=2 j+1 Hence, if we assume that {Hj , g1 } + . . . + {H2 , gj−1 } = 0 for j > m + 1 then we obtain the conditions under which the polynomial differential system has a uniform isochronous center at the origin. Thus the proof of the corollary follows.  Proof of Corollary 16. First we observe that differential equations (18) in polar coordinates x = r cos θ, y = r sin θ becomes ṙ = rφ(r cos θ, r sin θ), θ̇ = 1 + Λ(r cos θ, r sin θ), hence in view of that the center is weak center, then the polar coordinates must be such that H(r cos θ, r sin θ) = C = constant. Hence we get that the weak center is an isochronous center if and only if (21) holds, thus the corollary is proved.  Proof of Proposition 22. Since at the origin of system (18) there is a weak center, we have an analytic first integral H = H2 f in a neighborhood of the origin. So clearly H2 = 0 and f = 0 are invariant curves of system (18). It is easy to check dH2 that = 2H2 φ. From the first integral H = H2 f we get that dt dH2 df df f + H2 = 2H2 φ f + H2 = 0. dt dt dt df Thus = −2φ f, and the proposition is proved.  dt Proof of Corollary 18. From the Cauchy–Riemann conditions it is easy to obtain condition ∂Λ ∂φ ∂Λ ∂φ ∂ (φ + iΛ) + = 0, − = 0 ⇐⇒ = 0, ∂x ∂y ∂y ∂x ∂ z̄ i.e. the functions Λ and φ are harmonic functions. Moreover differential system (18) in complex coordinates z = x + i y and z̄ = x − i y becomes (56) ż = iz + zΦ(z) := f (z), where f = f (z) is a holomorphic function, Φ(z) = φ + iΛ. Clearly if we have a differential system ż = zΨ(z) where Ψ = u(x, y) + iv(x, y) then this differential system can be rewritten as follows ẋ = −yv + xu ẏ = xv + yu, i.e. can be rewritten as (18) with 1 + Λ = v and φ = u. From equations (56) we get ( ) dz dz − = 0, f (z) f (z) hence after the integration the existence of the first integral ∫ ( ( )) (∫ ) dz dz dz (57) − = 2i Im = Const.. γ f (z) f (z) γ f (z) 32 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ follows. Now we prove that a holomorphic center for a polynomial differential ∫ system is dz Darboux integrable. Indeed from (57) it follows that F = 2 Im is a first f (z) integral. We shall study the case when m−1 ∏ ks f (z) = z(i+Φ(z)) = z (z − zs ) for ks ∈ N : k1 +k2 +. . .+km−1 = m−1, s=1 where zs are complex numbers for s = 1, . . . , m − 1, such that (58) Ref ′ (0) := 0, Imf ′ (0) = i. Under this condition the origin is a holomorphic isochronous center. We develop 1/f (z) and 1/f (z̄) as follows (1) m−1 ∑ (1) m−1 ∑ A(2) m−1 ∑ A(k s) 1 A0 As s s = + + + . . . + , f (z) z s=1 z − zs s=1 (z − zs )2 s=1 (z − zs )ks ( ) 1 1 = , f (z̄) f (z) (ks ) ks ! (k ) (k ) (1) (1) (k ) (k ) where As = = αs s +iβs s , α0 = 0, β0 = 1, with αs s , βs s ∈ f (ks ) (zs ) dn f (z) R and f (n) (z) = . We get after the integration dz n ∫ ( (1) m−1 ∑ A(1) m−1 ∑ A(2) m−1 ∑ A(k s) ) A0 s s s + + + ... + dz γ z s=1 z − zs s=1 (z − zs )2 s=1 (z − zs )ks ∫ ( (1) m−1 ∑ Ā(1) m−1 ∑ Ā(2) m−1 ∑ Ā(k s) ) Ā0 s s s − + + + ... + dz̄ γ z̄ s=1 z̄ − z̄s s=1 (z̄ − z̄s )2 s=1 (z̄ − z̄s )ks ∫ ( ( )) dz dz = − . γ f (z) f (z) After some computations we obtain the following expression for the first integral  m−1  ∏ A(1)  (z − zs ) s   s=0  F̃ = log   m−1   ∏ Ā(1)  (z̄ − z̄s ) s s=0 m−1 ∑ (2) m−1 ∑ (k ) As 1 As s − + ... + s=1 (z − zs ) s=1 (1 − ks ) (z − zs )−1+ks m−1 ∑ (2) m−1 ∑ (k ) Ās 1 Ās s − + ... + . s=1 (z̄ − z̄s ) s=1 (1 − ks ) (z̄ − z̄s )−1+ks AN INVERSE APPROACH TO THE CENTER PROBLEM 33 where z0 = 0. In view of the relation y − ys √ i arctan z − zs = (x − xs )2 + (y − ys )2 e x − xs . Hence  y − ys  m−1 ∏ i(A(1) (1) A(1) (1) s +Ās ) arctan s −Ās x − xs  + Ψ F̃ = log  (|z − zs |) e s=0 m−1 ∑ (2) m−1 ∑ (k ) As 1 As s Ψ := − + ... + s=1 (z − zs ) s=1 (1 − ks ) (z − zs )−1+ks m−1 ∑ (2) m−1 ∑ (k ) Ās 1 Ās s − − ... − . s=1 (z̄ − z̄s ) s=1 (1 − ks ) (z̄ −¯zs )−1+ks and by considering condition (58) after tedious computations we obtain that the first integral is a Darboux first integral: m−1 ∑ y − ys m−1 ∏ αs(1) arctan ( )β (1) x − xs F = e−iF̃ = e−iΨ (x − xs )2 + (y − ys )2 s e s=0 s=0  m−1  ∑ y − ys  αs(1) arctan x − xs  m−1 ∏(  )β (1)  = (x2 + y 2 ) e−iΨ (x − xs )2 + (y − ys )2 s e s=1 .  s=1  So the first integral F has a Taylor expansion in the neighborhood of the origin has the form F = (x2 + y 2 ) (1 + h.o.t.) . Thus the holomorphic isochronous center is a weak center. In short the proposition is proved.  We observe that the problem on the existence the first integral for the complex differential system was study in particular in [27] Proof of Proposition 23. From (24) it follows that x(−y + X − λ xφ) + y(x + Y − yφ) = 0, ∂(−y + X) ∂(x + Y ) where λ = 2/µ and φ = + . Thus ∂x ∂y −y + X = −ν y + xφ, x + Y = xν + yφ, where ν = ν(x, y) is an arbitrary function. Denoting ν = 1 + Λ we get that differential equations (4) coincide with (18). On the other hand in view of the relations ( ) ∂H2 ∂H2 ∂(−y + X) ∂(x + Y ) (−y + X) + (x + X) = λ H2 φ = λ H2 + , ∂x ∂y ∂x ∂y which is equivalent to ( ) ( ) ∂ −y + X ∂ x+Y (59) + = 0. ∂x (x2 + y 2 )µ/2 ∂y (x2 + y 2 )µ/2 34 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ i.e. H2λ is inverse integrating factor. Thus differential system (4) can be written as (25) with F given by the formula (26). In short corollary is proved.  Proof of Proposition 24. Suppose that P and Q can be written as in (28) where H and G are polynomials, and we shall see that such polynomials exist when (27) holds. Then ∂ H̃ ∂ H̃ = −yG − P, = −xG + Q. ∂y ∂x ∂ 2 H̃ ∂ 2 H̃ Hence by considering that = , we get that ∂x∂y ∂y∂x ∂G ∂G ∂P ∂Q x −y = {H2 , G} = + . ∂y ∂x ∂x ∂y By considering that (27) holds, then in view of Corollary 8 we deduce that there m−1 ∑ exists a polynomial G = Gj such that j=1 ∂Gj ∂Gj ∂Pj ∂Qj (60) x −y = {H2 , Gj } = + . ∂y ∂x ∂x ∂y We can determine the function H as follows ∫ x m−1 ∑ ∫ y m−1 ∑ H̃ = (−x Gj + Q)dx − (y Gj + P ) dy, x0 j=1 y0 j=1 x=x0 where Gj = Gj (x, y) is the solution of equation (60). In short the proposition is proved.  We observe that from (28) it follows that ∂P ∂Q xP + yQ = {H̃, H2 }, + = {H2 , G}, ∂x ∂y thus in view of Proposition 6 we obtain that ∫ 2π (x P (x, y) + y Q(x, y)) |x=cos t, y=sin t dt = 0, 0 ∫ 2π ( ) ∂P ∂Q + |x=cos t, y=sin t dt = 0. 0 ∂x ∂y We consider an analytic differential system (7) under the assumptions ( ) 2 2 ∂X ∂Y (x + y ) + = µ (xX + yY ) , ∂x ∂y (61) ∫ 2π ( ) ∂X ∂Y + |x=cos t, y=sin t dt = 0, 0 ∂x ∂y where µ ∈ R\{0}. The previous result can be extended for the analytic vector field. Thus we have the following proposition. AN INVERSE APPROACH TO THE CENTER PROBLEM 35 Proposition 32. Let (2) be an analytic differential system which satisfies the re- lation ∫ 2π ( ) ∂P ∂Q + |x=cos t, y=sin t dt = 0. 0 ∂x ∂y ∞ ∑ ∞ ∑ Then there exist analytic functions H̃ = Hj and G = Gj such that j=2 j=1 ∂ H̃ ∂H ẋ = P = − −yG = {H̃, x}+G{H2 , x}, ẏ = Q = +xG = {H̃, y}+G{H2 , y}. ∂y ∂x Proof. It is analogous to the proof of Proposition 24.  Finally We have the following remarks related with differential system (18). Remark 33. (a) It is easy to observe that the singular points of differential system (18) lies on the intersection of the curves (x2 + y 2 ) φ(x, y) = 0 (x2 + y 2 ) (1 + Λ(x, y)) = 0. In particular if Λ = 0 then the only singular point is the origin. If the vector field is polynomial of degree m, then by Bezout Theorem the maximum number of singular points of system (18) is (m − 1)2 + 1. (b) If in (18) we assume that Λ = ω(x2 + y 2 ) − 1, and φ = λ(x2 + y 2 ), then system (18) becomes ẋ = −y ω(x2 + y 2 ) + x λ(x2 + y 2 ), ẏ = x ω(x2 + y 2 ) + y λ(x2 + y 2 ), which is called the lambda–omega system (see for instance [15]). (c) It is well known the following result. Let X be an analytic vector field associated to differential system (4). Then X has either the focus or a center at the origin, and under a formal change of coordinates differential system associated to X can be reduced to the Birkhoff normal form ẋ = −y(1 + S2 (x2 + y 2 )) + xS1 (x2 + y 2 ), ẏ = (1 + S2 (x2 + y 2 )) + yS1 (x2 + y 2 ), where Sj = Sj (x2 + y 2 ) for j = 1, 2 are formal series in the variable x2 + y 2 (see for instance [3]). Clearly these differential equations are particular case of (18). 6. The Proofs of Subsection 3.3 Proof of Theorem 25. We shall study only the case when the differential system is a polynomial differential systems of degree m. It is possible to show that condition (24) is equivalent to (59). Hence from the first of condition of (61) and in view of Proposition 23 we get that a polynomial differential system (4) can be written as (25) with F given in the formula (26). 36 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ On the other hand in view of Proposition 24 and the second of conditions (61) we get that there exist polynomials H̃ = H̃(x, y) and G = G(x, y) of degree m + 1 and m − 1 respectively, such that the following relations hold ẋ = −y + X = {H̃, x} + G{H2 , x}, (62) ẏ = x + Y = {H̃, y} + G{H2 , y}, Hence ∂X ∂Y + = {H2 , G}, xX + yY = {H̃, H2 }, ∂x ∂y consequently condition (24) becomes 2H2 {H2 , G} = µ{H̃, H2 }. Thus λH2 {H2 , G} = {H̃, H2 } ⇐⇒ {H2 , H̃ + λH2 G} = 0, where λ = 2/µ. Hence (63) H̃ = −λH2 G + λp(H2 ) := H2 Υ here p(H2 ) is a polynomial of degree [(m − 1)/2], where [(m − 1)/2] is the integer part of m − 1)/2 such that [m−1)/2] p(H2 ) = a1 H2 +a2 H22 +. . .+αm H2 := H2 q(H2 ) and G = −1/λ Υ+q(H2 ). Thus by putting (63) into differential system (62) we get ∂ H̃ ∂Υ ẋ = −y − − yG = −H2 − y(1 + Υ + G) ∂y ∂y ∂Υ λ−1 = −H2 − y(1 + Υ + q(H2 )), ∂y λ ( ( )) ∂Υ ∂Υ x = −y 1 + q(H2 ) + (1 − 1/λ)Υ + 1/2 x +y + {Υ, H2 }, ∂x ∂y 2 ∂ H̃ ∂Υ ẏ = x + + xG = H2 + x(1 + Υ + G) ∂x ∂x ∂Υ λ−1 = H2 + x(1 + Υ + +q(H2 )) ∂x λ ( ( )) ∂Υ ∂Υ y = x 1 + q(H2 ) + (1 − 1/λ)Υ + 1/2 x +y + {Υ, H2 }. ∂x ∂y 2 Consequently Ḣ2 = H2 {Υ, H2 }, ( ) λ−1 Υ̇ = − 1 + Υ + q(H2 ) {Υ, H2 }, λ hence dΥ 1−λ 1 + q(H2 ) = Υ− . dH2 λ H2 H2 After the integration this first order linear ordinary differential equations we have the following solution ( ∫ ) 1/λ−1 1 + q(H2 ) Υ = H2 C− 1/λ dH2 , H2 where C is an arbitrary constant. Consequently we have the following particular cases. AN INVERSE APPROACH TO THE CENTER PROBLEM 37 [(m−1)/2] ∏ (i) If λ ̸= 1 and (n − 1/λ) ̸= 0, then n=2 [(m−1)/2]−1 1/λ−1 1 + α1 α2 H2 αm H2 Υ = H2 C− − − ... − . 1 − 1/λ 2 − 1/λ [(m − 1)/2] − 1/λ (ii) If λ = 1, then [(m−1)/2]−1 α3 H22 αm H2 Υ = C − (1 + α1 ) log H2 − α2 H2 − − ... − . 2 [(m − 1)/2] − 1 [(m−1)/2] ∏ (iii) If 1 < λ = 1/k ≤ 1/[(m − 1)/2] and (n − k) ̸= 0, then n=2,n̸=k ( 1 + α1 1−k α2 Υ= −H2k−1 − C + H2 − H 2−k − . . . 1−k 2−k 2 αk+2 2 αm ) [(m−1)/2]−k −αk log H2 − αk+1 H2 − H2 − . . . − H2 . 2 [(m − 1)/2] − k [(m−1)/2] ∏ (iv) If 1/[(m − 1)/2] ≤ λ = 1/k < 1, αk = 0 and (n − k) ̸= 0, then n=2,n̸=k [(m−1)/2] 1 + α0 ∑ αj Υ= H2k−1 C − − H j−1 . 1−k j−k 2 j=2, j̸=k Excluding the constant C in the obtained solutions we deduce the first integrals F given in formula (30), (31), (32), (33).  λ−1 Remarks 34. (a) If in the equation (29) the relation 1+α1 + Υ(0, 0) = 0, λ holds then the origin is not linear type. Indeed, under this condition the given equation becomes   m−1 [(m−1)/2] ∑ j ∑ x ẋ = −y  ((1 − 1/λ) + )Υj + αj H2j−1  + {Υ, H2 }, j=1 2 j=2 2   m−1 [(m−1)/2] ∑ j ∑ y ẏ = x  ((1 − 1/λ) + )Υj + αj H2j−1  + {Υ, H2 }. j=1 2 j=2 2 m−1 ∑ where Υ = Υj , and Υj is a homogenous polynomial of degree j for j=1 j = 1, . . . , m − 1. (b) The first condition (61) is necessary as it follows from the next example. Consider the analytic differential system ( ) β H2 ∂G ( )β ẋ = −y 1 + G + + α x x2 + y 2 = −y + X, β+1 β + 1 ∂y (64) ( ) β H2 ∂G ( )β ẏ = x 1 + G − + α y x2 + y 2 = x + Y, β+1 β + 1 ∂x 38 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ where α (Liapunov constant) and β ≥ 0 are real numbers, G = G(x, y) is an analytic function in R2 . It is easy to prove that (64) satisfies the first of condition (61) for arbitrary α ̸= 0 with µ = 2(β + 1) and λ = 1/(β + 1), but the origin of this system is a focus . Moreover differential system (64) can be rewritten as ẋ = (x2 + y 2 )β+1 {F, x}, ẏ = (x2 + y 2 )β+1 {F, y}, where V = (x2 + y 2 )β+1 is an inverse integrating factor and a first integral is   β   1+ G  β+1 F = + α arctan(y/x) if β ̸= 0,  2β(x2 + y 2 )β    H2 e−2G−2α arctan (y/x) if β = 0, defined in R2 \ (0, 0). It is easy to prove that in this case ∂(−y + X) ∂(x + Y ) + = {H2 , G} + 2α(β + 1)(x2 + y 2 )β , ∂x ∂y consequently ∫ 2π ( ) ∂(−y + X) ∂(x + Y ) + |x=cos t, y=sin t dt = 2πα (β + 1) ̸= 0. 0 ∂x ∂y (c) The second of condition (61) for analytic differential system is necessary as it follows from the next example. If it holds then in view of Proposition 24 we get that analytic differential system (7) can be written as (28). In this case we have ∫ 2π ( ) ∂(−y + X) ∂(x + Y ) + |x=cos t, y=sin t dt 0 ∂x ∂y ∫ 2π = {H2 , G} |x=cos t, y=sin t dt = 0, 0 and that differential system (28) satisfies the second conditions of (61) for arbitrary G and H̃. Clearly there exist analytic function G and H̃ for which the origin is a focus and the first conditions of (61) does not hold. 7. The Proofs of Subsection 3.4 Proof of Proposition 26. From (14) and (34) it follows that ∂Hm+1 ∂ Hm + g1 + . . . + gm−1 y = −xRm−1 , ∂y ∂y (65) ∂Hm+1 ∂ Hm + g1 + . . . + gm−1 x = yRm−1 . ∂x ∂x Thus ( ) ( ) ∂Hm+1 ∂Hm+1 ∂Hm ∂Hm ∂H2 ∂H2 y +x + g1 y +x + . . . + gm−1 y +x = 0. ∂y ∂x ∂y ∂x ∂y ∂x Hence by considering that Hj is homogenous polynomial of degree j we get that 1 Hm+1 = − (mg1 Hm + (m − 1)g2 Hm−1 + . . . 2gm−1 H2 ) . m+1 AN INVERSE APPROACH TO THE CENTER PROBLEM 39 Substituting this polynomial into (65) we obtain 1 ( ∂Hm ∂g1 ∂Hm m g1 + mHm − (m + 1)g1 m+1 ∂y ∂y ∂y ∂H2 ∂gm−1 ∂H2 ) +... + 2 gm−1 + 2H2 − (m + 1)gm−1 ∂y ∂y ∂y (( ) ( ) ) 1 ∂Hm ∂Hm ∂g1 ∂ g1 ∂g1 ∂ Hm = x +y − x +y m+1 ∂x ∂y ∂y ∂x ∂y ∂y (( ) ( ) ) 1 ∂H2 ∂H2 ∂gm−1 ∂ gm−1 ∂gm−1 ∂ H2 +... + x +y − x +y m+1 ∂x ∂y ∂y ∂x ∂y ∂y m−1 x ∑ = {Hm+1−j , gj } = xRm−1 . m + 1 j=1 m−1 1 ∑ Thus Rm−1 = {Hm+1−j , gj }. So we proved the equation of ẋ in (35). m + 1 j=1 The proof for ẏ is similar.  Proof of Proposition 26. Follows from the equalities m−2 ∑ m−2 ∑ ẋ = −y(1 + Λ) + xφ = −y(1 + Λj ) + x Λj − yΛm−1 + xφm−1 j=1 j=1 m−2 ∑ m−2 ∑ ẏ = x(1 + Λ) + yφ = x(1 + Λj ) + y Λj + xΛm−1 + yφm−1 j=1 j=1 Thus the following relations must be hold −yΛm−1 + xφm−1 = xRm−1 , xΛm−1 + yφm−1 = yRm−1 , hence Λm−1 = 0 and φm−1 = Rm−1 . In short the proposition is proved.  8. Application of Theorem 12 and 13 In this section we study how to determine the Poincaré-Liapunov first integral and the Reeb inverse integrating factor for a given analytic or polynomial differential system. This problem is solved by applying Theorems 12 and 13. Given an analytic vector field X with a linear type center at the origin of coor- dinates, we shall use the expression of (13) to determine its first integral H and its Reeb inverse integrating factor. Thus, from (13) equating the terms of the same degree we get {Hj+1 , x} + g1 {Hj , x} + . . . + gj−1 {H2 , x} = Xj , {Hj+1 , y} + g1 {Hj , y} + . . . + gj−1 {H2 , y} = Yj , 40 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ for j ≥ 2. Hence ∂ H3 = −X2 − yg1 , ∂y ∂ H3 = Y2 − xg1 , ∂x ∂ H4 ∂ H3 = −X3 − g1 − yg2 , ∂y ∂y ∂ H4 ∂ H3 = Y3 − g1 − xg2 , ∂x ∂x (66) ∂ H5 ∂ H4 ∂ H3 = −X4 − g1 − g2 − yg3 , ∂y ∂y ∂y ∂ H5 ∂ H4 ∂ H3 = Y4 − g1 − g2 − xg3 , ∂x ∂x ∂x ∂ H6 ∂ H5 ∂ H4 ∂ H3 = −X5 − g1 − g2 − g3 − yg4 , ∂y ∂y ∂y ∂y ∂ H6 ∂ H5 ∂ H4 ∂ H3 = Y5 − g1 − g2 − g3 − xg4 , ∂x ∂x ∂x ∂x .. .. .. .. .. .. . . . . . . From the first two equation of (66) it follows that g1 must satisfy the first order partial differential equation ∂ X2 ∂ Y2 (67) {H2 , g1 } = + , ∂x ∂x which by Corollary 8 has a unique solution g1 . Substituting g1 into the first two equations of (66) and using the Euler’s Theorem for homogenous polynomial we 1 obtain H3 = (xY2 − yX2 − 2g1 H2 ) . 3 We determine g2 as the solution of the first order partial differential equation ∂ X3 ∂ Y3 {H2 , g2 } = + − {H3 , g1 }, ∂x ∂x where g1 is the solution of (67). Then by Corollary 8 we get that under the condition ∫ 2π ( ) ∂ X3 ∂ Y3 + − {H3 , g1 } dt = 0, 0 ∂x ∂x x=cos t, y=sin t there exists g2 = g2 (x, y) of the form g̃2 (x, y)+cH2 where c is an arbitrary constant. Hence from the third and fourth equation of (66) we get 1 H4 = (xY3 − yX3 − 3g1 H3 − 2g2 H2 ) . 4 We determine g3 as the solution of the first order partial differential equation ∂ X4 ∂ Y4 {H2 , g3 } = + − {H4 , g1 } − {H3 , g2 }, ∂x ∂x where g1 , g2 , H3 and H4 are solutions of the previous differential equations. Then by Corollary 8 we get that there exists a unique solution g3 . Hence from the fith AN INVERSE APPROACH TO THE CENTER PROBLEM 41 and sixth equation of (66) we get 1 H5 = (xY4 − yX4 − 4g1 H4 − 3g2 H3 − 2g3 H2 ) . 5 By continuing this process we obtain the expression H and of the inverse integrating  −1 ∞ ∑ factor 1 + gj  . j=1 In order to illustrated this previous algorithm for computing the homogenous polynomials gj ’s and Hj ’s we have the following proposition. Proposition 35. The polynomial differential system of degree four ẋ = −y(1 + n1 x + n2 y) +x(a1 x + a2 y + a6 x3 + a7 y 3 + a8 x2 y + a9 xy 2 ), (68) ẏ = x(1 + n1 x + n2 y) +y(a1 x + a2 y + a6 x3 + a7 y 3 + a8 x2 y + a9 xy 2 ), has a weak center at the origin if and only if one of the two set of condition holds a21 + a22 ̸= 0, a 1 n1 + a 2 n2 = 0, (69) 2a22 a6 + a1 (a21 − 3a22 )a7 + a2 (a22 − a21 )a9 = 0, 2a22 a1 a8 + a2 (a22 − 3a21 )a9 + 3a1 (a21 − a22 )a7 = 0. or n21 + n22 ̸= 0, a1 n1 + a2 n2 = 0, (70) 2n31 n6 + a1 (3n21 − n22 )a7 + n1 (n21 − n22 )a9 = 0, 2n22 n1 a8 + 3n2 (n22 − n21 )a7 + n1 (3n22 − n21 )a9 = 0. Moreover system (68) under conditions (69) or (70) is invariant with respect to a straight line. Proof. Necessity. We suppose that the origin is a center of (68) and we shall prove that (69) or (70) holds. First we prove the necessity of the condition a1 n1 +a2 n2 = 0. Indeed, from Theorem 13 it follows that the differential system (68) can be written as {H5 , x} + (1 + g1 ){H4 , x} + (1 + g1 + g2 ){H3 , x} +(1 + g1 + g2 + g3 ){H2 , x} = −y(1 + n1 x + n2 y) + x(a1 x + a2 y + a6 x3 + a7 y 3 + a8 x2 y + a9 xy 2 ), (71) {H5 , y} + (1 + g1 ){H4 , y} + (1 + g1 + g2 ){H3 , y} +(1 + g1 + g2 + g3 ){H2 , y}, = x(1 + n1 x + n2 y) + y(a1 x + a2 y + a6 x3 + a7 y 3 + a8 x2 y + a9 xy 2 ) 42 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ We prove that this partial differential system has solution if and only if a1 n1 + a2 n2 = 0. Indeed, from (71) equating the terms of the same degree we obtain {H3 , x} + g1 {H2 , x} = −y(n1 x + n2 y) + x(a1 x + a2 y), {H3 , y} + g1 {H2 , y} = x(n1 x + n2 y) + y(a1 x + a2 y). By determining the homogenous polynomial g1 as the unique solution of the equa- tion {H2 , g1 } = (n2 +3a1 )x+(3a2 −n1 )y, we get that g1 = (n1 −3a2 )x+(n2 +3a1 )y. By the homogeneity we finally obtain that H3 = 2H2 (a2 x−a1 y). By inserting these polynomials into the system {H4 , x} + g1 {H3 , x} + g2 {H2 , x} = 0, (72) {H4 , y} + g1 {H3 , y} + g2 {H2 , y} = 0, Hence we get g2 from the equation {H2 , g2 } + {H3 , g1 } = 0. By Corollary 8 we deduce that this equation has solution if ∫ 2π (73) {H3 , g1 }|x=cos t,y=sin t dt = 4π(a1 n1 + a2 n2 ) = 0. 0 a 2 n2 We study the case when a2 a1 ̸= 0. Then we consider system (68) with n1 = − . a1 In view of (73) and by the homogeneity of H4 from (72) we get that 3a1 + n2 ( ) H4 = H2 y (a21 − a22 )y − 2a1 a2 x + c1 H22 , a1 3a1 + n2 g2 = (a1 y − a2 x)2 + c2 H2 , a1 where c1 and c2 are arbitrary constants. By inserting g1 , g2 , H3 and H4 into the equation {H5 , x} + g1 {H4 , x} + g2 {H3 , x} + g3 {H2 , x} = x(a6 x3 + a7 y 3 + a8 x2 y + a9 xy 2 ), (74) {H5 , y} + g1 {H4 , y} + g2 {H3 , y} + g3 {H2 , y} = y(a6 x3 + a7 y 3 + a8 x2 y + a9 xy 2 ) and by Corollary 8 we get that the equation {H4 , g1 } + {H3 , g2 } + {H2 , g3 } = 5(a6 x3 + a7 y 3 + a8 x2 y + a9 xy 2 ) has a unique solution g3 . From (74) we can obtain the homogenous polynomial H5 . Consequently we get that equations (71) have solutions if and only if te relation a1 n1 + a2 n2 = 0 holds. The case when a1 a2 = 0 it is easy to study. Thus the necessity of this condition is proved. Now we study the remain equations of (66), i.e. {Hj+1 , x} + g1 {Hj , x} + . . . + gj−1 {H2 , x} = 0 {Hj+1 , y} + g1 {Hj , y} + . . . + gj−1 {H2 , y} = 0, AN INVERSE APPROACH TO THE CENTER PROBLEM 43 for j ≥ 6. For j = 6 we get ∂ H6 ∂ H5 ∂ H4 ∂ H3 = −g1 − g2 − g3 − yg4 , ∂y ∂y ∂y ∂y (75) ∂ H6 ∂ H5 ∂ H4 ∂ H3 = −g1 − − g3 − xg4 , ∂x ∂x ∂x ∂x where H3 , H4 , H5 , g1 , g2 and g3 are homogenous polynomials obtained in the pre- vious equations. The homogenous polynomial H5 which satisfies (78) exists if and only if the homogenous polynomial g4 is a solution of the equation (76) {H5 , g1 } + {H4 , g2 } + {H3 , g3 } + {H2 , g4 } = 0, and by Corollary 8 we need that ∫ 2π 0= ({H3 , g3 } + {H4 , g2 } + {H5 , g1 })|x=cos t,y=sin t dt 0 (77) 3π (n1 + 2a2 ) = − (a1 λ2 − a2 λ1 ) , a2 where λ1 and λ2 are defined as 1 ( ) λ1 := 2 2a22 a6 + a1 (a21 − 3a22 )a7 + a2 (a22 − a21 )a9 , 2a2 1 ( 2 ) λ2 := 2 2a2 a1 a8 + a2 (a22 − 3a21 )a9 + 3a1 (a21 − a22 )a7 . 2a2 a1 Under the condition (78) differential equation (76) has a solution g4 which by Corollary 8 can be obtained with arbitrary term of the type c(x2 + y 2 )2 . By using the homogeneity of H6 we get 5 4 3 2 H6 = − g1 H5 − g2 H4 − g3 H3 − g4 H2 . 6 6 6 6 Since the integral of homogenous polynomial of degree 5 ∫ 2π ({H6 , g1 } + {H5 , g2 } + {H4 , g3 } + {H3 , g4 })|x=cos t,y=sin t dt ≡ 0, 0 then by Corollary 8 we obtain that there is a unique homogenous polynomial g5 satisfying {H6 , g1 } + {H5 , g2 } + {H4 , g3 } + {H3 , g4 } + {H2 , g5 } = 0. Partial differential equations of first order ∂ H7 ∂ H6 ∂ H5 ∂ H4 ∂ H3 = −g1 − g2 − g3 − g4 − yg5 , ∂y ∂y ∂y ∂y ∂y (78) ∂ H7 ∂ H6 ∂ H5 ∂ H4 ∂ H3 = −g1 − g2 − g3 − g4 − xg5 , ∂x ∂x ∂x ∂x ∂x have a solution if and only if ∫ 2π ({H7 , g1 } + {H6 , g2 } + {H4 , g3 } + {H3 , g4 })|x=cos t,y=sin t dt 0 π(3a2 − n1 ) = (µ1 λ1 + µ2 λ2 ) = 0 252a32 44 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ where Hj and gj−1 for j = 2, 3, 4, 5, 6 are homogenous polynomial of degree j, and µ1 and µ2 are 2781a21 a32 + 2673a21 a22 n1 + 690a21 a2 n21 + 819a52 + 927a42 n1 + 230a22 n21 − 216ca32 , 5301a31 a22 + 1305a31 a2 n1 − 414a31 n21 − 2457a1 a42 − 2781a1 a32 n1 − 690a1 a22 n21 + 648a1 a22 c, respectively, where c ia a constant. Under the previous condition the homogenous polynomial H7 can be calculated and we obtain 6 5 4 3 2 H7 = − g1 H5 − g2 H5 − g3 H4 − g4 H3 − g5 H2 . 7 7 7 7 7 By solving the system π(2a2 + n1 (a2 λ1 − 3a1 λ2 ) = 0, 2a2 π(3a2 − n1 ) (µ1 λ1 + µ2 λ2 ) = 0, 252a32 with respect to λ1 and λ2 and by considering that the determinant of the matrix of coefficients is (2a2 + n1 )(3a2 − n1 )(379a22 + 259a2 n1 + 46n21 ), and assuming that (2a2 + n1 )(3a2 − n1 ) ̸= 0 we deduce that λ1 = λ2 = 0. It is possible to study the case when (2a2 + n1 )(3a2 − n1 ) = 0 and the case when a1 a2 = 0. Thus we obtain the necessity of condition (69). In analogous way we can study the case n2 n1 ̸= 0. Sufficiency. We need the following results. Let (79) x = κ1 X + κ2 Y, y = −κ2 X + κ1 Y, be a non-degenerated linear transformation, i.e. κ21 + κ22 ̸= 0. Then differential system (18) becomes ( ) Ẋ = −Y 1 + Λ̃(X, Y ) + X Ω̃(X, Y ), (80) ( ) Ẏ = X 1 + Λ̃(X, Y ) + Y Ω̃(X, Y ), where Λ̃(X, Y ) = Λ(κ1 X +κ2 Y, −κ2 X +κ1 Y ) and φ̃(X, Y ) = φ(κ1 X +κ2 Y, −κ2 X + κ1 Y ). The proof of the next claim is easy. The differential system (80) is invariant under the transformation (X, Y, t) −→ (−X, Y, −t) if and only if it can be written as ( ) Ẋ = −Y 1 + Θ1 (X 2 , Y ) + X 2 Θ2 (X 2 , Y ), (81) ( ) Ẏ = X 1 + Θ1 (X 2 , Y ) + XY Θ2 (X 2 , Y ), and the differential system (80) is invariant under the transformation (X, Y, t) −→ (X, −Y, −t) if and only if it can be written as ( ) Ẋ = −Y 1 + Θ1 (X, Y 2 ) + XY Θ2 (X, Y 2 ), (82) ( ) Ẏ = X 1 + Θ1 (X, Y 2 ) + Y 2 Θ2 (X, Y 2 ), AN INVERSE APPROACH TO THE CENTER PROBLEM 45 Doing the change of variables (79) to system (68) we obtain system (80) for m = 4. From the claim system (68) written in the form (81) is invariant under the transformation (X, Y, t) −→ (−X, Y, −t) if and only if k1 n1 + k2 n2 = 0, k1 a2 − k2 a1 = 0, (83) 2κ22 a6 + κ1 (κ21 − 3κ22 )a7 + κ2 (κ22 − κ21 )a9 = 0, 2κ22 κ1 a8 + κ2 (κ22 − 3κ21 )a9 + 3κ1 (κ21 − κ22 )a7 = 0, and it is invariant under the transformation (X, Y, t) −→ (X, −Y, −t) if and only if k1 n2 − k2 n1 + k1 a1 + k2 a2 = 0, (84) 2κ31 a6 + κ1 (3κ21 − κ22 )a7 + κ1 (κ21 − κ22 )a9 = 0, 2κ22 κ1 a8 + 3κ2 (κ22 − κ21 )a7 + κ1 (3κ22 − κ21 )a9 = 0. We suppose that (69) or (70) hold and we claim that then the origin is a center of system (68). Now we prove this claim. First we study the case when a21 + a22 ̸= 0. Then after the change x = a1 X − a2 Y, y = a2 X + a1 Y, we get that this system coincides with system (81) for m = 4 and with κ1 = a1 and κ2 = a2 , and consequently system (68) is invariant under the change (X, Y, t) −→ (−X, Y, −t) i.e. it is reversible. 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Zoladek, The solution of the center-focus problem, Institute of Mathematics, University of Warsaw, Poland, (1992), preprint 80 pp. AN INVERSE APPROACH TO THE CENTER PROBLEM 47 [35] Z. Zhou, A new method for research on the center-focus problem of differential systems, Abstract and Applied Analysis 2014, (2014), 1–5. 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bel- laterra, Barcelona, Catalonia, Spain. E-mail address: jllibre@mat.uab.cat 2 Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Vir- gili, Avinguda dels Paı̈sos Catalans 26, 43007 Tarragona, Catalonia, Spain. E-mail address: rafaelorlando.ramirez@urv.cat 3 Universitat Central de Barcelona, Gran Vı́a de las Cortes Catalanas, 585 08007 Barcelona, Spain. E-mail address: vramirsa8@alumnes.ub.edu