Nilpotent centres via inverse integrating factors

In this paper, we are interested in the nilpotent centre problem of planar analytic monodromic vector fields. It is known that the formal integrability is not enough to characterize such centres. More general objects are considered as the formal inverse integrating factors. However, the existence of a formal inverse integrating factor is not sufficient to describe all the nilpotent centres. For the family studied in this paper, it is enough.


Introduction
We consider planar differential systems that can be written aṡ x = P (x, y),ẏ = Q(x, y), (1.1) where P and Q are analytic functions that vanish at the origin. It is known that system (1.1) has a centre at a singular point only if it is monodromic and it has either linear part of centre type, i.e. with imaginary eigenvalues (non-degenerate point), or nilpotent linear part (nilpotent point) or null linear part (degenerate point). A non-degenerate singular point is a centre if and only if it has a formal (actually analytic) first integral around the singular point. In contrast, the formal integrability does not characterize the nilpotent or degenerate centres, see for instance [16,18] although some nilpotent or degenerate centres have analytic first integral, see for instance [6]. On the other hand, a non-degenerate singular point is a centre if, and only if, there is a non-zero analytic inverse integrating factor around the singular point, see [24] and [17].
In this work, we focus on the nilpotent centre problem which consists in characterizing when a monodromic nilpotent singular point is a centre. In fact this problem is solved and several methods can be used. One is blowing-up the singularity by means of generalized polar coordinates, see for instance [9]. Another one is applying normal form theory up to some order to transform the system into a suitable Liénard normal form, see [10]. Other method to find the centres of a family is to study the orbitally reversibilitity due to the result that any nilpotent system has a centre at the origin if and only if it is orbitally reversible, see [12]. This method has been used in [1] to study certain families of nilpotent systems. Finally, the last method is embedding the nilpotent singularity as the limit of a family of non-degenerate centres, see [16,18] and the final corrected version in [14]. Moreover, in [4], it is proved that in order to determine certain nilpotent centres we can use the existence of a C ∞ Lyapunov function. However, no version is based in the construction of a formal power series which gives the obstructions to the existence of a nilpotent centre. This is because neither the first integral nor the inverse integrating factors are, in general, formal power series for nilpotent centres, see [2,19]. In [19], it is proved that any (possibly degenerate) centre of an analytic planar system admits a C ∞ inverse integrating factor and a C ∞ Lie symmetry in a neighbourhood of the singularity.
The monodromy problem consists in characterizing when a singular point is either a focus or a centre. Andreev in [11] solved this problem for nilpotent singular points. Theorem 1.1 (Andreev ) Consider the analytic systeṁ where P and Q are analytic functions around the origin without constant and linear terms and with an isolated singularity at the origin. Let y = φ(x) be the solution of y +P (x, y) = 0 passing through the origin. Consider the functions with a 0, α > 2 and b 0,β > 1 or Δ(x) ≡ 0. Then, the origin of (1.2) is monodromic if and only if a < 0, α = 2ñ − 1 is an odd number and one of the following conditions holds: The Andreev number n of a monodromic singular point at the origin of system (1.2) is the integerñ > 2 given in Theorem 1.1. Recently in [13], using the Andreev number n is proved that if n is even and there exists an inverse integrating factor then the system (1.1) has a centre at the singular point. If n = 2, then the existence of an inverse integrating factor characterizes all the centres. Finally, if there is an inverse integrating factor with minimum "vanishing multiplicity" at the singularity then, generically, the system (1.1) has a centre at the singular point.
In this work, we are interested in deepening knowledge of the methods based on formal power series for the nilpotent centre problem. We prove that for certain nilpotent systems the existence of a centre is equivalent to the existence of a formal inverse integrating factor generalizing the results obtained in [20,21]. For more general families, we do not know if there exists a formal power series to characterize if a nilpotent singular point is a centre or not.

Nilpotent centres and inverse integrating factors
A scalar function f is quasi-homogeneous of type t = (t 1 , t 2 ) ∈ N 2 and degree k if The vector space of quasi-homogeneous scalar function of type t and degree k is denoted by P t k . A vector field F = (P , Q) T is quasi-homogeneous of type t and degree k if P ∈ P t k+t 1 and Q ∈ P t k+t 2 . The vector space of quasi-homogeneous vector field of type t and degree k is denoted by Q t k . Given an analytic vector field F, we can write it as a quasi-homogeneous expansion corresponding to a fixed type t: where x ∈ R 2 , r ∈ Z and F k ∈ Q t k i.e. each term F k is a quasi-homogeneous vector field of type t and degree k. Any F k ∈ Q t k can be uniquely written as In this work, we consider analytic nilpotent differential system of the form: with d ∈ R, F j ∈ Q t j , t = (1, 2), and where the conservative-dissipative decomposition of the first quasi-homogeneous component of (2.4) is given by A non-null C 1 class function V is an inverse integrating factor of systemẋ = F(x) on U if satisfies the linear partial differential equation ∇V · F = div (F) V , being div (F) := ∂P /∂x + ∂Q/∂y the divergence of F. We will say that V is a formal inverse integrating y]] is the algebra of the power series in x and y with coefficients in C, convergent or not.
Proof The proof of statement (a) is given in [7,Theorem 16]. Following the same ideas it is possible to prove (b).
Remark In Proposition 2.5, it is not necessary neither the convergence of the transformation nor the convergence of the transformed vector field. In fact, we only need the formal part of the transformed vector field since when system (2.4) is monodromic the Poincaré map is analytic, see [23]. More specifically, in this case, we consider a vector field F ∈ C ω with a monodromic nilpotent singular point at the origin. The transformed system is given byF + τ(x, y), wherẽ F is a formal vector field and where τ is a flat vector field at the origin. By Moussu, see [23], we know that the Poincaré map for F is given by Π(x) = x + n>2 a n x n ∈ C ω and the Poincaré map ofF is the formal mapΠ(x) = x + n>2ã n x n .
Moreover, if a 2 = . . . = a n−1 = 0 and a n 0, that is F has a focus at the origin this implies thatã 2 = . . . =ã n−1 = 0 andã n 0. Otherwise ifã n = 0 for n > 2, thenΠ(x) = x that implies a n = 0 for n > 2 and Π(x) = x, that is, the origin is a centre for F and hence we do not need the convergence of the transformation.
The following proposition is proved in [7, Theorem 6].

Proposition 2.6 System (2.4) with d = 0 has a formal inverse integrating factor if and only
The next proposition characterizes when system (2.4) has a centre at the origin.

Proposition 2.7 System (2.4) has a centre at the origin if and only if F is orbitally equivalent to
Proof We give the proof for each statement.
Case d = 0 Applying Proposition 2.5 statement (a), the origin of F is a centre if and only if the origin of (ẋ,ẏ) = G : The sufficient condition is trivial because, if f 0 ≡ f 2 ≡ 0, then G is monodromic and is invariant by the symmetry (x, y, t) → (−x, y, −t) and therefore has a centre at the origin.
To prove the necessity of the condition, we assume that f 2 0 + f 2 2 ) 0 and we definê G = (y, −x 3 ) T + xhf 1 (h)D 0 whose origin is a centre. The wedge product G ∧Ĝ measures the direction of the orbits ofẋ = G(x) crossing the ovals of the centreẋ =Ĝ(x).
We claim that this wedge product is semi-definite. To prove our claim, we consider k = min{i ∈ N ∪ {0} : a (2) i 0 or a (0) i 0}. If the minimum is a (0) k , then G ∧Ĝ = 4h k+2 (a (0) k + · · · ) which is definite. If the minimum is a (2) k , then G ∧Ĝ = 4x 2 h k+1 (a (2) k + · · · ) which is semi-definite. In both cases, the orbits ofẋ = G(x) cross the ovals of the centre ofẋ =Ĝ(x) in the same direction which implies that the origin of system (2.4) cannot be a centre.
Case d 0 Applying Proposition 2.5 statement (b), the origin of F is a centre if and only if the origin of (ẋ,ẏ) = G := (y + dx 2 The sufficient condition, as before, is trivial because if f 0 ≡ f 2 ≡ 0 then G is monodromic and is invariant by the symmetry (x, y, t) → (−x, y, −t) and therefore has a centre at the origin.
To prove the necessity of the condition, we assume that f 2 0 + f 2 2 ) 0 and we definê G = (y + dx 2 , −x 3 + 2dxy) T whose origin is a centre. We compute the wedge product G ∧Ĝ that is given by We are going to see that this expression is also semi-definite. We define, as before, In both cases, the orbits of systemẋ = G(x) cross the ovals of the centre ofẋ =Ĝ(x) in the same direction which implies that the origin of system (2.4) cannot be a centre.
The next theorems are the main results of this work and they give a new characterization of the centres of the nilpotent differential system (2.4). This characterization provides a scalar procedure to determine the centres of system (2.4). Proof Applying Proposition 2.7 statement (a), we have that the origin of system (2.4) is a centre if and only if F is orbitally equivalent to G := (y, −x 3 ) + xf 1 (h)D 0 and by Proposition 2.6, this condition is equivalent to have a formal inverse integrating factor. Theorem 2.8 is not satisfied for others nilpotent vector fields different from system (2.4). For instance, we provide an example where the necessary condition of Theorem 2.8 is not satisfied, that is, we give a vector field which has a centre at the origin but it does not have a formal inverse integrating factor. Proposition 2.9 The origin of system (ẋ,ẏ) T = F := (y, −x 5 ) + d x 3 (x, 3y) T is a centre but it does not have any formal inverse integrating factor.
Proof The origin of the system is a centre because the system is monodromic and is invariant by the symmetry (x, y, t) → (−x, y, −t). However, the system has not a formal inverse integrating factor. If we define h = − 1 6 3y 2 + x 6 and D 0 = (x, 3y) T , then F = X h + dx 3 D 0 and V = h 7/6 is a non-formal inverse integrating factor. In [7] is proved that this inverse integrating factor is unique up to a constant multiplicative because any invariant curve has the form h α u, where u ∈ R[[x, y]] is a unity element. Moreover, in order to be h α u an inverse integrating factor it is enough to take α = 7/6 and u = 1. Now we give an example where the sufficient condition of Theorem 2.8 is not satisfied for nilpotent systems different from system (2.4), that is, we find a vector field that has a formal inverse integrating factor. Nevertheless, this vector field has not a centre at the origin.
Proof The system has a formal inverse integrating factor V = h 2 . However, the origin of the system is not a centre because the wedge product is definite and therefore the origin is a focus.
For these last systems, we do not have a characterization of the centres using an scalar algorithm and these systems require further studies.

Characterization of the centres of system (2.4)
The following results give the characterization of the nilpotent centres for system (2.4).

Lemma 3.11
Let F be the associated vector field to system (2.4). There exist a formal function V = h + · · · and some numbers α ∈ R, α 0 m ∈ N and A ∈ N ∪ {0} ∪ {+∞} such that one of the following conditions is satisfied: Proof We divide the proof in two cases.

(i) If d 0, by Proposition 2.5 statement (b), F is orbitally equivalent to
(i.1) If g 2 (h) ≡ g 4 (h) ≡ 0, then we are in the same situation (a) for A = 0.
(i.2) If g 2 (h) 2 + g 4 (h) 2 ) 0, we consider then h is a first integral of G and F is also integrable with a first integral of the form V = h + · · · and the statement (a) with A = +∞ is satisfied.
Assume now that f 2 2 (h) + f 2 4 (h) + f 2 5 (h) ) 0 in this case, we consider • If i 0 = 5, we will see that it is possible to choose a value of A and a function b We must eliminate the odd terms in h.
The odd terms in h of the right-hand side are Taking into account that (5) m 0 h m 0 +··· = m 0 +· · · := m 0 +f 5 (h) withf 5 (0) = 0 and imposing 1 + A − 2 − m 0 = 0, that is, A = m 0 + 1, the odd terms in h are In order to vanish these terms, we just take b(h) = exp (m 0 +2)h . Now we come back to the functions g 2 (h) and g 4 (h). If g 2 (h) ≡ g 4 (h) ≡ 0, we have that statement (a) is satisfied. If (g 2 (h)) 2 + (g 4 (h)) 2 ) 0 considering the nonzero terms of lower degree, we have that statement (b) or (c) would be satisfied. Proof Applying Theorem 2.8, the origin of system (2.4) is a centre if and only if its vector field associated F, has a formal inverse integrating factor. The factors of the inverse integrating factor are invariant curves. Furthermore, the first quasi-homogeneous component of the formal invariant curve must be invariant curve of the first quasihomogeneous component of F, that is, of F 1 . Notice that the unique irreducible invariant curve of F 1 are the factors of h. Therefore, the inverse integrating factor is of the form W = h 1+A + · · · := V 1+A with A ∈ N ∪ {0} and V = h + · · · . Consequently, and from here the statement of the theorem follows. We can include the case A = +∞ and in this case ∇V · F = 0, that is, F would be integrable. Applying Lemma 3.11, we have the condition A = 0 if and only if d 0.
We note that Lemma 3.11 and Theorem 3.12 give us a method to compute the necessary conditions to have a nilpotent centre for system (2.4).  Proof To see that statement (a) is equivalent to statement (b), it is sufficient to apply Proposition 2.7 statement (b). The equivalence between statement (a) and (c) follows from Theorem 3.12. Finally, the equivalence between statements (a) and (d) follows from Proposition 2.7 statement (b) and Theorem 1.3 of [5].

Applications
In this section, we apply the scalar algorithm obtained in order to find nilpotent centres.

Example 1
We consider the planar differential system (4.6) The centre conditions of this family are studied, using a generalization of the Cherkas' method, in [15,Proposition 33]. We find the same centre conditions using the algorithm derived in the previous section. (ii) 3a 30 + b 21 = 2b 02 + a 11 = 0, a 30 0.
Proof First, we will see the necessity. We impose the condition ∇V · F − 1 1+A div (F) V = 0 of Theorem 3.12 degree by degree of quasi-homogeneity. We consider V = h + j>5 V j , where V j ∈ P (1,2) j and we chose V 5 in order to cancel the maximum number of terms and we obtain Taking into account that 4A − 1 0, the first condition is 3a 30 + b 21 = 0. Now we chose V 6 ∈ P (1,2) 6 . In this case, we can choose V 6 such that all the terms of degree 7 cancel in ∇V · F − 1 1+A div (F) V . Next, we consider V 7 ∈ P (1,2) 7 . After a good choice of V 7 , we have that Taking into account that 4A − 3 0, we have three possibilities either a 30 = 0, or 2b 02 + a 11 = 0, or b 02 − a 11 + b 40 = 0. The case a 30 = 0 corresponds to statement (i).
In the case (ii), that is, 3a 30 +b 21 = 2b 02 +a 11 = 0, we have that the divergence div (F) = 0 and consequently F is a Hamiltonian vector field and, as the origin is monodromic, the origin is a centre.
In the case (iv), system (4.6) takes the form Applying the change of variables u 40 2 u dt, and denoting by = d/dT , system (4.6) becomes which is monodromic and time-reversible because is invariant by the symmetry (u, v, t) → (−u, v, −t) and consequently the origin is a centre.

Proof
The Hamiltonian function of system (4.8) is h(x, y) = 1 4 If k 2 − 8 > 0, then h has two simple factor and consequently the origin of system (4.7) is not monodromic, see [8,Proposition 6].
If k 2 − 8 = 0, then h = − 1 2 v 2 has a double factor v which is not a factor of the divergence and applying [8,Proposition 6], we have that the origin of system (4.7) is not monodromic.
Finally, if k 2 − 8 < 0, then h is negative defined, does not vanish in a neighbourhood of the origin and hence, the origin of system (4.7) is monodromic, see [8,Proposition 5]. Proof Necessity: We recall that to have a monodromic singular point at the origin by Lemma 4.15, we have that k 2 < 8. From Theorem 3.12, we impose the condition ∇V · F − 1 1+A div (F) V = 0, where V = h + j>5 V j , V j ∈ P (1,2) j degree by degree of homogeneity in the equivalent system (4.8). We divide the study in the two case k 0 and k = 0.
In the case k 0, we have A = 0. We chose V 5 in order to cancel the maximum number of terms and we obtain ∇V · F − div (F) V = − 2ka 3k 2 −25 x 2 h + · · · .
Taking into account that k 0, we have that the first condition is a = 0. Now we chose V 6 ∈ P (1,2) 6 . In this case, we can choose V 6 in such way that all the terms of degree 7 in ∇V · F − div (F) V vanish. Now we consider V 7 ∈ P (1,2) 7 . After a good choice of V 7 , we have ∇V · F − div (F) V = (2k 2 −21)c 5k 2 −49 h 2 + · · · .
Taking into account that 2k 2 − 21 0, we get c = 0 which corresponds to the statement (i).
In the case k = 0, we have A > 0 and we can chose V 5 and V 6 in such way that all the terms of degree 6 and 7 vanish in ∇V · F − 1 1+A div (F) V . Now we chose V 7 in order to cancel the maximum number of terms and we obtain Taking into account that 4A − 3 0, we get that ab − 3c = 0. Next, we chose V 8 in order to cancel the maximum number of terms and we have ∇V · F − 1 1+A div (F) V = − (A−1)a 2 b 8(1+A) xh 2 + · · · .
In this case, we have three possibilities either a = 0 or b = 0 with a 0 or A = 1 with ab 0.
The case a = 0 corresponds to statement (ii). The case b = 0 with a 0 corresponds to statement (iii). In the case A = 1 with ab 0, we chose V 9 in order to cancel the maximum number of terms and we get Taking into account that a 3 b 0, we have that the origin cannot be a centre in this case.

Sufficiency:
In the case a = c = 0 with k 0, system (4.7) is time-reversible because it is invariant by the symmetry (x, y, t) → (−x, y, −t) and as the origin is monodromic then it is a centre. In the case a = c = k = 0, system (4.7) is also time-reversible because it has the same symmetry. In the case k = b = c = 0, system (4.7) is time-reversible because it is invariant by the symmetry (x, y, t) → (x, −y, −t) and as the origin is monodromic then it is a centre.