Nonlinear

We generalize the method of construction of an integrating factor for Abel differential equations, developed in Briskin et al. (1998), for any generic monodromic singularity. Here generic means that the vector field has not characteristic directions in the quasi-homogeneous leading term in certain coordinates. We apply this method to some degenerate differential systems. © 2021TheAuthor(s).PublishedbyElsevierLtd.Thisisanopenaccessarticleunder theCCBYlicense(http://creativecommons.org/licenses/by/4.0/).


Introduction
We consider an autonomous system of the form, where F is an analytic planar vector field defined in a neighborhood of the origin U ⊂ R 2 having an equilibrium point at the origin, i.e., F(0) = 0 and where P and Q are analytic in U.
The so-called monodromy problem consists in characterize when a vector field has a well-defined return map in a neighborhood of an isolated singularity, that is, if the vector field has or not characteristic orbits passing through this isolated singularity, see [1][2][3][4][5]. Ilyashencko [6] and Ecalle [7] proved simultaneously that a singular point of an analytic differential system cannot be an accumulation point of limit cycles. Consequently any monodromic singular point of an analytic system is a focus or a center.
Once we know that the singular point is monodromic appears another classic problem called the center problem or stability problem which consists in distinguish if this singularity is a focus or a center, see [8][9][10]. If the linear part of the vector field at the origin is nondegenerate, the Poincaré-Lyapunov method solves the center problem, see the seminal works [11][12][13][14]. In this method are introduced the focal values also called Poincaré-Lyapunov constants. These constants are polynomials in the parameters of the system with rational coefficients. The vanishing of all these quantities is a necessary and sufficient condition to have a nondegenerate center. Some particular cases are studied for instance, in [15], a necessary and sufficient condition is given for perturbations of quasi-homogeneous polynomial Hamiltonian systems having a center. The nilpotent case is similar to the nondegenerate one (see [8,9,[16][17][18][19][20]). The degenerate case is more involved (see [10,[21][22][23][24][25]). For instance, Ilyashencko [26] proved that the degenerate center problem for polynomial vector field is not algebraically solvable, that is, the center conditions are not, in general, polynomials in the parameters of system (1.1). In short the degenerate center problem for polynomial vector fields is a well posed problem but, in general, algebraically unsolvable. However it is proved that it is analytic solvable, see [27].
We denote by P n and Q n the leading terms of the same order n of P and Q respectively in the homogeneous order, eventually one of these two polynomials can be zero. We say that θ 0 is a characteristic direction in the homogeneous order of the singular point, in this case located at the origin of system (1.1), if it is verified cos θ 0 Q n (cos θ 0 , sin θ 0 ) − sin θ 0 P n (cos θ 0 , sin θ 0 ) = 0 A characteristic orbit of system (1.1) is a trajectory that enters or leaves the singular point located at the origin tending to this point with a definite tangent and this tangent is given by a characteristic direction in the homogeneous order. Of course any monodromic singular point has not a characteristic orbit but it can have characteristic directions. In certain monodromic degenerate singular points a geometric method can be applied to determine the stability of singular points with characteristic directions in the homogeneous order, see [21]. Systems with characteristic directions in homogeneous order are also studied in [10,22] using the blow-up technique. Other methods are developed for some specific degenerate systems, see [18,[28][29][30][31][32][33]. The Bautin method [34], introduced to find the maximum number of limit cycles that bifurcate from the origin for quadratic systems with center-type linear part, can be also used to degenerate monodromic singular points without characteristic directions. The method consists in computing the derivatives of the Poincaré map using a recursive linear system of differential equations. In the present work using this method we have managed to find the first terms of the Poincaré map for monodromic singular points without characteristic directions in certain coordinates (using a quasi-homogeneous order which generalized the homogeneous order). Hence the first step is to find the coordinates where the system does not have characteristic directions. We denote such vector fields as generic degenerate vector fields and these vector fields have a monodromic quasi-homogeneous first component.
For the computation of the generalized focal values we generalize the method of Briskin, Françoise and Yomdin [35], and we compute the integrating factor of the associated Abel equation which gives a linear recursive system of differential equations in order to obtain the focal values.

Perturbations of quasi-homogeneous systems
We now introduce some notation in order to present the main results. A scalar polynomial f is quasihomogeneous of type t = (t 1 , t 2 ) ∈ N 2 and degree k if f (ε t 1 x, ε t 2 y) = ε k f (x, y). The vector space of quasi-homogeneous scalar polynomials of type t and degree k is denoted by P t k . A polynomial 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 polynomial quasi-homogeneous vector fields of type t and degree k is denoted by Q t k . Given a vector field F = (P, Q) T , we define the divergence of F as div(F) = ∂P ∂x + ∂Q ∂y . We denote X h = (− ∂h ∂y , ∂h ∂x ) T the Hamiltonian vector field with Hamilton function h. We define the wedge product of two vector fields as F ∧ G := PQ −P Q, where F = (P, Q) T and G = (P ,Q) T .
The vector field (1.1) can be written as the sum of quasi-homogeneous terms of type t: where F k ∈ Q t k for all k, and r ∈ Z. If we select the type t = (1, 1), we are using in fact the Taylor expansion, but in general, each term in the above expansion involves monomials with different degrees. The main tool we use is a type of decomposition for quasi-homogeneous vector fields. This decomposition will provide notable simplifications in the computation of the normal form. The following proposition provides the decomposition of any quasi-homogeneous vector field, see for more details [36].
r , then there exist unique polynomials µ r ∈ P t r (F r ) dissipative part of and h r+|t| ∈ P t r+|t| (F r ) conservative part of such that: This decomposition generalizes those given, for the homogeneous case, by Baider and Sanders [37] and Collins [38]. Our goal is to characterize when a singular point of system (2.2) is a center. To do this, we must first know if the singular point is monodromic.
The next result characterizes when the origin of a quasi-homogeneous system is monodromic (see [39, Corollary 1 and Theorem 2]).
Next we present a sufficient condition in order that the origin of (2.2) be monodromic (see [39,Theorem 2]). Theorem 2.3. If the origin of systemẋ = F r (x) with F r ∈ Q t r is monodromic then the origin of system (2.2) is also monodromic.
From now on we consider vector fieldṡ where the origin of F r is monodromic. The following result gives a consequence respect to the systeṁ x = F r (x) if the origin of system (2.4) is a center.
Theorem 2.4. If the origin of (2.4), with F r monodromic, is a center, then the origin of systemẋ = F r (x) is also a center.
Proof . Assume that the origin of (2.4) is a center. If the origin ofẋ = F r (x) is a focus then by using by [39,Theorem 5] the origin of (2.4) is also a focus and this gives a contradiction. ■ In [40,Theorem 3.3] the necessary and sufficient conditions so that a quasi-homogeneous system has a center at the origin are determined.
From now on, applying Theorem 2.4 we can assume that the system (2.4) iṡ whereẋ = F r (x) has a center at the origin.
There are degenerate centers whose first quasi-homogeneous component F r is not monodromic, that is, the systemẋ = F(x) = F r (x) + F r+1 (x) + · · · has a center at the origin and F r = X h r+|t| + µ r D 0 is not monodromic. In this case the Hamiltonian function h r+|t| associated to the first quasi-homogeneous component F r is not generic and have real multiple roots in its decomposition in C[x, y]. For instance, consider the following vector fieldẋ The vector field (2.6) can be decomposed as sum of two quasi-homogeneous vector fields of degree 2 and 4 respect to the type t = (1, 1), where The origin of (2.6) is monodromic because F = X H , where H(x, y) = y 4 + x 2 y 2 + x 6 > 0 for all (x, y) ∈ U \ {(0, 0)}). Moreover the origin of (2.6) is a center since F is a Hamiltonian vector field. On the other hand F 2 (x) = X h with h(x, y) = y 2 (y 2 + x 2 ) is not monodromic because does not satifies the condition h(x, y) ̸ = 0 for all (x, y) ∈ U \ {(0, 0)} and h has real multiple roots. These cases, where the first quasi-homogeneous component is non-monodromic, are non-generic and are not subject of our study in this work.
In the following Proposition we apply to system (2.5) the change of coordinates x = ρ t 1 Cs(θ), y = ρ t 2 Sn(θ) in order to obtain the Poincaré map for such generic differential systems. In what follows and for simplicity, we denote by f (θ) := f (Cs(θ), Sn(θ)) where f is a scalar function of f : R 2 → R.

8)
where h r+j+|t| and µ r+j are the conservative-dissipative decomposition Proof . Applying the proposed change of coordinates we have which is the same as the vectorial equatioṅ Therefore we have where T is the minimal period of Cs(θ) and Sn(θ), becauseẋ = F r has a center at the origin.
Moreover, using the Euler formula ∇µ k D 0 = kµ k we get On the other hand for each j ≥ 0 we have From here we deduce that the transformed system iṡ for ρ > 0. Applying now the scaling of time dt = (r+|t|)h r+|t| (θ) ρ r dτ , we get the result. The condition that h r+|t| (θ) ̸ = 0 ∀θ ∈ [0, T ] is fundamental in our study. This hypothesis makes possible the existence of a generalized Abel differential equation with coefficients g i (θ) polynomial in Cs(θ), Sn(θ). ■

The associated generalized Abel equation
System (2.8) can be written as Developing in power series of ρ, the equation of the orbits of this system in generalized polar coordinates is given by the generalized Abel equation 10) The following Lemma determines the functions g i (θ) of the generalized Abel equation.
Lemma 2.6. Consider the generalized Abel equation (2.10), then we have that g i (θ) for all i ≥ 2 are defined by So we get: Applying a classical result of power series (see [41]) we have an expression of g i (θ) in terms of a determinant given by From here we deduce the result. ■ We know that the power series (2.10) converges for ρ sufficiently small, i.e., |ρ| ≪ 1. We denote by ρ(θ, ρ 0 ) = ∑ n≥1 a n (θ)ρ n 0 , the solution of Eq. (2.10) satisfying ρ(0, ρ 0 ) = ρ 0 . The Poincaré map is given by defined for ρ 0 > 0 sufficiently small. The coefficient a n (T ) are called the generalized Poincaré-Lyapunov constants. The following result gives the stability of the singular point.

Theorem 2.7. If the origin of system (2.5) is a focus then there exists a Poincaré-Lyapunov constant for
n > 1 different from zero. In fact if a 2 (T ) = · · · = a n−1 (T ) = 0, and a n (T ) ̸ = 0, the focus is stable for sig(h r+|t| (θ))a n (T ) < 0 and unstable for sig(h r+|t| (θ))a n (T ) > 0. If a n (T ) = 0 for all n ≥ 2 then the origin is a center.

Computation of the generalized Poincaré-Lyapunov constants
To obtain the generalized Poincaré-Lyapunov constants just replace θ = T in the functions a i (θ) which in turn are obtained by imposing that the function ρ(θ, ρ 0 ) = ∑ n≥1 a n (θ)ρ n 0 be a solution of Eq. (2.10). This is known as Bautin method [34]. More specifically we have Developing the right hand side of (2.12) we can get the expression of the Poincaré-Lyapunov constants in a recursive form. More concretely we have For n = 1 results that and for n > 1 the differential equation (2.13) results (2.14) 3. The integrating factor for Eq. (2.10) In [35] Briskin, Françoise and Yomdin showed that there exists an integrating factor for the Abel equation associated to a polynomial perturbation of any lineal center i.e. for vector fields of the form (−y + · · · , x + · · · ) T , see also [42]. In this paper we generalize this result finding an integrating factor for vector fields of the form (2.5).

Proposition 3.8. There exists an unique formal power series
is an integrating factor of (2.10) with Φ(ρ, 0) ≡ 0, where the coefficients Φ i (θ), i ≥ 1, verify the following initial value problem defined in a recurrence form: Proof . If R is an integrating factor of (2.10) and we denote by which completes the proof. ■ The next result provides the expression of the inverse of the Poincaré map.
is the integrating factor of (2.10) then there exists a Hamiltonian function as we would to proof. ■ The following result relates the Poincaré-Lyapunov constants with the integrating factor constants. Proof . Applying Proposition 3.9 we get and being a 1 (θ) ≡ 1, for each k ≥ 2 we have 0 = a k (T ) The next result is a consequence of Proposition 3.9 and provides another method to compute the Poincaré-Lyapunov constants and consequently another resolution of the center-focus problem.
Proof . If the origin of system (2.5) is a focus then there exists k ∈ N such that a 2 (T ) = · · · = a k−1 (T ) = 0 and a k (T ) ̸ = 0, using Proposition 3.10 we obtain that Φ k−1 (T ) = −ka k (T ) ̸ = 0. If the origin of system (2.5) is a center then a k (T ) = 0 ∀k ∈ N and using Proposition 3.10 we get Φ k (T ) = 0, for all k ≥ 1.

Sufficient conditions for a center
It is well-known that if a system is formally integrable or orbitally reversible and the origin is monodromic then the origin is a center, see for instance [43][44][45] and references therein.
We can simplify the calculation of the focal values Φ k (T ) if we know that a part of the system to study is a center as the following result shows. Proposition 4.12. Assume that the system (2.5) can be written into the formẋ =X(x) + X(x), wherē X = F r + · · · , X = ∑ j>r G j , G j ∈ Q t j and the origin of systemẋ =X(x) is a center. Letḡ j (θ) be the coefficients of the generalized Abel equation , for i ≥ 0, then the following statements hold: Proof . The proof of statement (a) is trivial because as the systemẋ =X(x) has a center at the origin then Φ n (T ) = 0, for n ≥ 1. Now we prove statement (b). Taking into account that (1−Φ(θ, ρ)) −1 , are integrating factors of (2.10) and (4.16), respectively, the following recurrence relations are satisfied therefore we have that ) which completes the proof. ■ The function Φ 1 (θ) is given by an integral, the function Φ 2 (θ) is obtained through an integral whose integrant has also an integral. In general in the computation of Φ n (θ) we need integrate n times. For this reason it is convenient to reduce the terms in Φ n (θ) in which appear high subindexes of Φ i (θ) with i < n. This can be done applying the following result obtained using integration by parts. This property is used in the applications. Lemma 4.13. Consider Φ n ,Φ n ,Φ n ,ḡ n andĝ n of Proposition 4.12 then the following statements hold.
Proof . The proof follows using the derivative rule, the definition of integrating factor of the generalized Abel equation that appears in Proposition 3.8 and using the definition of Φ n ,Φ n ,Φ n ,ḡ n andĝ n of Proposition 4.12. ■
We apply to systems of the form (5.17) the change of generalized polar coordinates where (Cs(θ), Sn(θ)) are the solutions of the initial value problem

Degenerate system of type t = (2, 3)
In this subsection we study the center problem for the monodromic system which corresponds to (5.17) with t = (2, 3).