The Poincaré map of degenerate monodromic singularities with Puiseux inverse integrating factor

1 Introduction

We consider families of real analytic planar differential systems

depending analytically on the parameters λ ∈ R p together with its associate family of vector fields X λ = P ( x , y ; λ ) ∂ x + Q ( x , y ; λ ) ∂ y . Throughout the work, we restrict the family to the parameter space Λ ⊂ R p so that the origin ( x , y ) = ( 0 , 0 ) is made sure to be a monodromic singularity of the whole family (1). That means that P ( 0 , 0 ; λ ) = Q ( 0 , 0 ; λ ) = 0 and the local orbits of X λ turn around the origin for any λ ∈ Λ . The characterization of Λ is usually done by means of the blow-up procedure described by Dumortier in [10], see also Arnold [7] and Medvedeva [30], for example. In [4,5], Algaba et al. give a new algorithmic criterium to determine the parameter constrains defining the set Λ , which is based on the analysis of the Newton diagram of the vector field X λ . This alternative algorithm can be easily checked and improves the previous schemes in some technical aspects.

In this monodromic scenario, Il’yashenko [26] and Écalle [11] show independently that the origin only can be either a center or a focus since X λ is analytic. We recall that a center possesses a punctured neighborhood (period annulus) foliated by periodic orbits of X λ , while in a neighborhood of the focus, the orbits spiral around it. The characterization of the subsets of Λ for which one of these two possible behaviors occurs is called the center-focus problem associated to the origin of X λ .

When the differential matrix D X λ ( 0 , 0 ) of X λ at the origin has nonzero pure imaginary eigenvalues, the center-focus problem was already solved in the Poincaré and Lyapunov works, and the reader may consult the modern text books [12,32]. Later, Moussu [31] solved the center-focus problem in the nilpotent case that consists in the situation in which D X λ ( 0 , 0 ) is not identically zero, but its two eigenvalues are zero.

To give algorithms to know the stability of the monodromic singular points in the degenerate case corresponding to have D X ( 0 , 0 ) identically zero is a challenging problem [21,23,24], and this article tries to advance in this direction under additionally restrictions specified below. So we deal with the very difficult degenerate center-focus problem even though our approach also works in the other less degenerate situations. The desingularization blow-up procedure transforms the origin into a monodromic polycycle Γ with associated Poincaré return map Π on the one side. After Il’yashenko’s work [26] we know that, in general, the Poincaré map Π : Σ ⊂ ( R + , 0 ) → ( R + , 0 ) is no longer differentiable at the origin but it is a semiregular map. In consequence Π can be expressed as a Dulac asymptotic expansion with linear leading term. In this setting the center case is characterized by Π equal to the identity map.

Several degrees of difficulty arise depending on the nature of Γ . The easiest case corresponds to a Γ without singularities because there are no characteristic directions after the first (weighted) polar blow-up, so that Γ is a periodic orbit. We denote by Mo ( p , q ) this monodromic class, see Definition 12 to be more precise. In the class Mo ( p , q ) , the Poincaré map Π is analytic, and therefore, the classical scheme proposed by Poincaré and Lyapunov to solve the center problem works with minor modifications, and it is known as Bautin method. The reader may consult, for instance, see [22] and the references therein.

Due to the analyticity of X λ in λ , an important step was taken by Medvedeva [30] who prove that the Poincaré map Π has a Dulac asymptotic expansion of the form

where η 1 > 0 , the exponents ν j > 1 are independent of λ and grow to infinity, and the coefficients of the P j are polynomials whose coefficients depend analytically on the coefficients of X λ . In particular, Π ( x ) = η 1 x + o ( x ) and Medvedeva in [28] shows how to compute the linear coefficient η 1 after the desingularization of Γ . Under some technical conditions, in [29], Medvedeva and Batcheva shows that when η 1 = 1 , then Π ( x ) = x ( 1 + η 2 x 1 / n + o ( x 1 / n ) ) for some positive integer n . As far as we know, no other results about the structure of Π with specific restrictions on X λ are given in the literature, and in this work, we explore this idea.

Only few works (some of them are [19,20]) give explicit restrictions on X λ to compute η 1 when X λ ∉ Mo ( 1 , 1 ) , see also [3] and the references therein. It is also worth to mention that in [20], where Gasull et al. define a kind of monodromic singularities (called the class S k ω in [20]) such that each singular point on the first polar blow-up polycycle Γ desingularizes into two well-posed hyperbolic saddles, and they prove in that case that η 1 can be written as follows:

where R k and F k are homogeneous polynomials of degree k + 1 defined as follows:

with X k = P k ( x , y ; λ ) ∂ x + Q k ( x , y ; λ ) ∂ y being the homogeneous vector field of degree k defining the leading part of X λ , i.e., X λ = X k + ⋯ , where the dots denote a sum of homogeneous vector fields of degree greater than k . In formula (3), the symbol PV means Cauchy principal value and is defined as follows. Given a continuous function f defined in I ⊂ [ 0 , 2 π ] \ Ω with Ω = { θ 1 ∗ , … , θ ℓ ∗ } , the Cauchy principal value of ∫ I f ( θ ) d θ is defined as the following limit (if it exists):

where I ε = I \ J ε with J ε = ∪ i = 1 ℓ ( θ i ∗ − ε , θ i ∗ + ε ) . Improper integrals with more than one singularity are convergent if, splitting the interval of integration, each of the split integrals is convergent. On the other hand, the Cauchy principal value arises when the singularities are handled in a balanced way. Clearly, if an improper integral is convergent, it converges to its principal value, but the latter may exist though the corresponding improper integral does not.

The same expression (3) for η 1 was obtained earlier in [19] for another class of monodromic singularities called the class G in that work. Also Medvedeva in [30] found (3) under different restrictions on X λ .

In this work, we reobtain the same expression (3) under our specific restrictions on X λ . These restrictions are related with the existence of an inverse integrating factor v ( x , y ; λ ) in a neighborhood of the origin for X λ , which can be written in Puiseux series using weighted polar coordinates, see the expression (12). Recall that in this situation the differential 1-form ω = P ( x , y ; λ ) d y − Q ( x , y ; λ ) d x satisfies that ω / v is closed (the exterior differential d ( ω / v ) = 0 ) off the set v − 1 ( 0 ) . The inverse integrating factor with Laurent expansion was used in [18] to obtain the coefficients of the Taylor expansion of Π in the simplest situation when Γ is a periodic orbit, and therefore, X λ ∈ Mo ( p , q ) . In the recent work [16], the center conditions of the degenerate center-focus problem for X λ in Λ was considered using inverse integrating factors whose expression in polar coordinates has the exceptional divisor of the first polar blow-up as isolated zero-set. The present work is a wide generalization of several results included in [18] and [16].

The work is structured as follows. The basic background and the main results are given in Section 2. Section 3 is dedicated to summarize well-known results about the characterization of Λ , while in Section 4, we develop the concepts associated to the weighted polar blow-up. In the next section, we make a brief break to explain what is the algebraic origin of Puiseux inverse integrating factors. In Section 6, we give the proofs of all the results. Section 7 is dedicated to present some nontrivial examples illustrating how the theory works. We also write an Appendix where some technical details that are implicit throughout this work are given.

2 Main results

Hereinafter, we omit the dependency on the parameters λ in all the formulas except when it is relevant.

Developing in Taylor series at ( 0 , 0 ) both components of the analytic system (1), we obtain

where P i and Q i are homogeneous polynomials of degree i and P k 2 + Q k 2 ≢ 0 .

Taking polar coordinates ( x , y ) ↦ ( θ , r ) with x = r cos θ , y = r sin θ , and rescaling the time (dividing the vector field by r k − 1 ) system (4) becomes the polar system

with

We say that θ = θ ∗ is a characteristic direction for the origin of system (4) if F k ( θ ∗ ) = 0 , and we define Ω as the set of all the characteristic directions:

We recall that we can move a characteristic direction just by taking a linear change of coordinates in system (1).

Considering system (5), and assuming F k ( θ ) ≥ 0 , we have Θ ( θ , r ) ≥ 0 for 0 < r ≪ 1 sufficiently small. The point ( θ , r ) = ( θ ∗ , 0 ) with θ ∗ ∈ Ω is a singularity of the map Θ because, though Θ ( θ ∗ , 0 ) = 0 , the hypothesis of the implicit function theorem fail at the singularity since ∂ r Θ ( θ ∗ , 0 ) = 0 (and also ∂ θ Θ ( θ ∗ , 0 ) = 0 ) by the first monodromic conditions in Remark 10. We say that a branch of solutions of the equation Θ ( θ , r ) = 0 bifurcates from the characteristic direction θ ∗ of (4) if there is a continuous function r ∗ ( θ ) ≥ 0 , defined for all θ in a half-neighborhood of θ ∗ , such that r ∗ ( θ ∗ ) = 0 and Θ ( θ , r ∗ ( θ ) ) ≡ 0 . In short, to each characteristic direction θ i ∗ ∈ Ω , with i = 1 , … , ℓ , of (4), there can be associated several (or none) branches.

We define the sufficiently small cylinder

and the critical set

Notice that S ≠ ∅ because Ω ≠ ∅ although it may happen that

just when no branches appear. In [19], it is proved that the latter condition (8) always occurs when system (4) is polynomial and defined by the sum of two homogeneous vector fields. Of course, S depends on λ .

We define the critical parameters as those belonging to the subset Λ ∗ ⊂ Λ of the parameter space of X λ where (8) does not hold, that is,

We consider the ordinary differential equation of the orbits of (5):

where ℱ ( θ , r ) = ℛ ( θ , r ) / Θ ( θ , r ) is a function well defined in C \ S .

We say that a not locally null real C 1 ( C \ S ) function V ( θ , r ) is an inverse integrating factor of (9) if it is a solution of the linear partial differential equation:

From now on, we will fix our attention on a special class of functions V . We say that an inverse integrating factor V ( r , θ ) of (9) in C \ { S ∪ { r = 0 } } is a Puiseux inverse integrating factor if it can be expanded in convergent Puiseux series about r = 0 of the form

whose coefficients, for a fixed λ , are C 1 functions v i : S 1 \ Ω → R , the leading coefficient v m ( θ ; λ ) ≢ 0 , and ( m , n ) ∈ Z × N ∗ are fixed numbers called multiplicity and index. Here, we have used the notation N ∗ = N \ { 0 } . The particular case n = 1 leads to a Laurent inverse integrating factor, while n = 1 and m ≥ 0 produces an analytic inverse integrating factor in case that the v i are all analytic. It may happen that ( m , n ) are dependent of λ , and in this case, our results are no longer valid for all the family X λ with λ ∈ Λ rather they are still valid for specific vector fields X λ ˆ with λ ˆ ∈ Λ ˆ being Λ ˆ = { λ ∈ Λ : ( m ( λ ) , n ( λ ) ) ∈ Z × N ∗ } . In the particular case that ( m , n ) is independent of λ , then Λ ˆ = Λ .

The following first result is valid in all the monodromic parameter space Λ and shows a wide class of systems (1) having a center at the origin when a Puiseux inverse integrating factor with nonpositive multiplicity exists.

Our second main result is only valid in the restricted parameter space Λ ˆ \ Λ ∗ . It characterizes the structure of the asymptotic Dulac expansion of Π in Λ ˆ \ Λ ∗ under the assumption that family (1) possesses a Puiseux inverse integrating factor in terms of its multiplicity m and index n . In particular, its statement (iii) extends (in Λ ˆ \ Λ ∗ ) the scope of Theorem 4.

From Theorem 5(ii) and expression (2), it follows the next important corollary.

We end the main results of this work focusing in families of vector fields in the monodromic class Mo ( 1 , 1 ) . For such families, we define the quantity ξ as follows:

and it will play a fundamental role in the results presented below.

From the Puiseux series ℱ ( r , θ ) / V ( r , θ ) = r ( n − m ) / n ∑ i ≥ 0 g i ( θ ) r i / n , we also define the following relevant quantities

The explicit expressions of the ξ i in terms of the coefficients of system (5) and the Puiseux inverse integrating factor (12) are rather complicated. As a sample, in the particular case n = 1 , after some extensive but routine calculations, we obtain

while

when n = m = 2 .

The next main result generalizes the following well-known fact about centers in monodromic singularities of homogeneous vector fields (indeed, it can be stated in a more general form for quasi-homogeneous vector fields). The homogeneous vector field X k = P k ( x , y ) ∂ x + Q k ( x , y ) ∂ y with a monodromic singularity at the origin ( F k ( θ ) ≠ 0 ) has the inverse integrating factor x Q k − y P k , which is transformed into the Puiseux inverse integrating factor r 2 of the corresponding differential equation (9) with m = 2 and n = 1 . Moreover, ξ 0 = ∫ 0 2 π R k ( θ ) / F k ( θ ) d θ and ξ 1 = 0 , and the origin is a center of X k if and only if ξ 0 = 0 .

3 Monodromic conditions

In this section, we summarize some well-known results about necessary and some sufficient conditions for the origin of (1) be monodromic. In other words, we will present some restriction on the parameters of family (1) in order that they lie in Λ .

The characteristic direction for the origin of system (4) appears from the linear factors in R [ x , y ] of the homogeneous polynomial x Q k ( x , y ) − y P k ( x , y ) . Therefore, unless that polynomial be identically zero, the number of characteristic directions is less than or equal to k + 1 . It is well known that necessary monodromy conditions for the origin of system (4) are that k be odd, F k ≢ 0 , and F k ( θ ) does not change sign, hence F k ( θ ) ≥ 0 reversing the time if necessary. Therefore, without loss of generality, we focus in the case F k ( θ ) ≥ 0 , so that the flow of (4) rotates counterclockwise around the origin, and when we use the term monodromic, it is implicit the above direction of rotation.

We have that the invariant circle Γ = { r = 0 } is a polycycle of system (5) and that ( r , θ ) = ( 0 , θ ∗ ) is a critical point of (5) for any θ ∗ ∈ Ω whose linear part is identically zero, and hence, { r = 0 } is a nonelementary polycycle.

Moreover, if θ ∗ is a characteristic direction, then θ ∗ + π is too because ( ℛ ( θ + π , r ) , Θ ( θ + π , r ) ) = ( ℛ ( θ , r ) , Θ ( θ , r ) ) . Moreover, since ( ℛ ( θ + π , − r ) , Θ ( θ + π , − r ) ) = ( − 1 ) k ( − ℛ ( θ , r ) , Θ ( θ , r ) ) , it suffices to consider only the characteristic direction in [ 0 , π ) to know the topological behavior of the orbits of (5) around the rest of characteristic direction.

We will assume that x Q k ( x , y ) − y P k ( x , y ) ≢ 0 so that the characteristic directions are isolated, that is, the set Ω ¯ ⊂ [ 0 , π ) of half characteristic directions has cardinality # Ω ¯ ≤ ( k + 1 ) / 2 .

The main tool in the analysis of the monodromy of a degenerate singularity is the blow-up technique, see, for example, [7]. Dumortier proved in 1977 that after a finite number of blow-ups, any planar real analytic vector with real isolated singularity can be transformed into an analytic field of directions on a smooth manifold where the singular point becomes an elementary polycycle, that is, a polycycle with a finite number of elementary (hyperbolic or degenerate elementary) singularities different from a focus or a center.

System (4) becomes (5) after performing the first polar blow-up. If # Ω ¯ = 0 , there appear no critical points on Γ and then the Poincaré-Lyapunov theory can be reproduced in a straightforward way, in particular, the Poincaré map Π is analytic. In this work, we assume that # Ω ¯ ≥ 1 so that Γ = { r = 0 } becomes a polycycle.

In this work, when we need to characterize Λ , that is, when we need necessary and sufficient monodromic conditions, we use Theorems 3 and 4 of [4] or Theorem 2 of [5].

4 The weighted polar blow-up

We take an analytic vector field X = P ( x , y ) ∂ x + Q ( x , y ) ∂ y with

and define the support of X by supp ( X ) = { ( i , j ) ∈ N 2 : ( a i j , b i j ) ≠ ( 0 , 0 ) } . Then the boundary of the convex hull of the set

where R + denote the set positive real numbers, is made up of two open rays and a polygon. Then the Newton diagram N ( X ) of the vector field X is composed by that polygon together with the rays that do not lie on a coordinate axis, if they exist. It is easy to see that selecting the weights ( p , q ) ∈ N 2 , with p and q coprimes, given by the tangent q / p of the angle between an edge of N ( X ) and the ordinate axis, one obtains an expansion

where r ≥ 1 and X j = P p + j ( x , y ) ∂ x + Q q + j ( x , y ) ∂ y are ( p , q ) -quasihomogeneous vector fields of degree j . We call X r the leading part of X , which clearly depends on the chosen weights ( p , q ) . We define the set W ( N ( X ) ) ⊂ N 2 whose elements are all the weights in N ( X ) .

The weighted polar blow-up ( x , y ) ↦ ( ρ , φ ) given by

wit Jacobian J ( φ , ρ ) = ρ p + q − 1 ( p cos 2 φ + q sin 2 φ ) , brings the ( p , q ) -quasihomogeneous vector field X j = P p + j ( x , y ) ∂ x + Q q + j ( x , y ) ∂ y of degree j into the system

where

Therefore, after removing the common factor ρ r / D ( φ ) > 0 , X is transformed into the following form

If X r has not a monodromic singularity at the origin, then G ˆ r ( φ ) has real roots in [ 0 , 2 π ) . Taking into account that ρ ˙ = O ( ρ ) , φ ˙ = G ˆ r ( φ ) + O ( ρ ) , we see that { ρ = 0 } is a polycycle.

We consider the ordinary differential equation of the orbits of (18), that is,

where ℱ ˆ 1 ( φ ) = F ˆ r ( φ ) / G ˆ r ( φ ) . Then ℱ ˆ is a function well defined in C \ S p q being the critical set

so that S p q ≠ ∅ if Ω p q = { φ ∗ ∈ S 1 : G ˆ r ( φ ∗ ) = 0 } ≠ ∅ . Analogously as in (8), it may happen that

With this notation, the set of characteristic directions Ω and the critical set S defined in (7) correspond to Ω 11 and S 11 , respectively.

For each weight ( p , q ) ∈ W ( N ( X ) ) , we define the subset Λ p q ⊂ Λ of the parameter space of X as follows:

and the critical parameters Λ ∗ ⊂ Λ as the intersection

We recall that X ∈ Mo ( p , q ) if and only if G ˆ r ( φ ) has no real roots in [ 0 , 2 π ) . Therefore, in particular, S p q = ∅ when X ∈ Mo ( p , q ) . Moreover, the origin is a center of X r if and only if, additionally, ∫ 0 2 π ℱ ˆ 1 ( φ ) d φ = 0 , see [21]. It can be checked (using the Bautin method or just by the reciprocal of Theorem 5 in [1]) that if the origin is a center of X ∈ Mo ( p , q ) , then it is also a center of X r . In other words, ∫ 0 2 π ℱ ˆ 1 ( φ ) d φ = 0 is a necessary center condition at the origin of X ∈ Mo ( p , q ) .

When X ∈ Mo \ Mo ( p , q ) we still may define, like it was done in (15) for the particular case ( p , q ) = ( 1 , 1 ) , the principal value

which may exist or not.