The main objective of this paper is to investigate topological properties from the view point of compact warped product submanifolds of a space form with the vanishing constant sectional curvature. That is, we prove the non-existence of stable integral p-currents in a compact oriented warped product pointwise semi-slant submanifold Mn in the Euclidean space Rp+2q which satisfies an operative condition involving the Laplacian of a warped function and a pointwise slant function, and show that their homology groups are zero on this operative condition. Moreover, under the assumption of extrinsic conditions, we derive new topological sphere theorems on a warped product submanifold Mn, and prove that Mn is homeomorphic to Sn if n=4, and Mn is homotopic to Sn if n=3. Furthermore, the same results are generalized for CR-warped products and our results recovered [
Citation: Ali H. Alkhaldi, Akram Ali, Jae Won Lee. The Lawson-Simons' theorem on warped product submanifolds with geometric information[J]. AIMS Mathematics, 2021, 6(6): 5886-5895. doi: 10.3934/math.2021348
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The main objective of this paper is to investigate topological properties from the view point of compact warped product submanifolds of a space form with the vanishing constant sectional curvature. That is, we prove the non-existence of stable integral p-currents in a compact oriented warped product pointwise semi-slant submanifold Mn in the Euclidean space Rp+2q which satisfies an operative condition involving the Laplacian of a warped function and a pointwise slant function, and show that their homology groups are zero on this operative condition. Moreover, under the assumption of extrinsic conditions, we derive new topological sphere theorems on a warped product submanifold Mn, and prove that Mn is homeomorphic to Sn if n=4, and Mn is homotopic to Sn if n=3. Furthermore, the same results are generalized for CR-warped products and our results recovered [
We focus in this work on families of planar real analytic vector fields X=P(x,y;λ)∂x+Q(x,y;λ)∂y defined in a neighborhood of a monodromic singularity that can be placed at the origin without loss of generality. Here λ∈Rp denotes the finite number of parameters in the family. We recall that a monodromic singular point of X is a singularity of X such that the associated flow rotates around it, and therefore a Poincaré map Π is well defined in a sufficiently small transversal section with an end point at the singularity. Centers and foci are examples of monodromic singularities. We will restrict the family to the monodromic parameter space Λ⊂Rp defined as the parameter subset for which the origin is a monodromic singularity of X. The common characterization of Λ is via the blow-up procedure developed by Dumortier in [1]; see also Arnold [2]. It is worth emphasizing that Algaba and co-authors present in [3,4] an algorithmic procedure to determine the parameter restrictions that defines Λ in terms of the Newton diagram N(X) of the vector field X; see also [5].
In [6], an explicit analytic first-order ordinary differential equation was obtained for the Poincaré map Π associated with a monodromic singularity (without local zero angular speed curves) of X under the assumption of the existence of a Laurent inverse integrating factor V of X. Later on, in [7], an example of focus without Puiseux (a generalization of Laurent) inverse integrating factor is presente. This example shows that the differential equation obtained in [6] for Π is not universal.
Initially, our first main objective in this work was to find an explicit analytic ordinary differential equation f(ρ,Π,Π′,…,Πk)=0 for the Poincaré map Π(ρ) associated to a monodromic singularity of X where f only depends on objects that do not include the flow of X. If this objective were achieved, an alternative proof of the fact that Π admits an asymptotic Dulac series can be constructed via Bruno's theory [8,9] of asymptotic solutions of analytic differential equations at singularities. In this work we take the former universal object to be the analytic invariant curve F(x,y)=0 through the monodromic singularity that there always exists, as it was proved in [10]. Using this idea, we find a first order ordinary differential equation for Π only depending on F; see Theorem 1.1 in Section 1.1, although we have been able to prove its analyticity only under some assumptions; see, for example, Theorem 1.6. This equation generalizes the one obtained in [6] in the sense that both coincides in the particular case that F becomes an inverse integrating factor of X, as we show in Section 2.
We denote by N(X) the Newton diagram of X that is composed of edges joining both positive semi-axis; see [11] for a detailed account of this construction and also [3]. Each edge of N(X) has endpoints in N2 and slope −p/q with (p,q)∈N2 coprimes. From now on we denote by W(N(X))⊂N2 the set of all weights (p,q). We will use the weighted polar blow-up (x,y)↦(ρ,φ) given by
x=ρpcosφ, y=ρqsinφ, | (1.1) |
with Jacobian
J(φ,ρ)=ρp+q−1(pcos2φ+qsin2φ). |
In these coordinates, the system associated to X is written in the form ˙ρ=R(φ,ρ), ˙φ=Θ(φ,ρ) with R(φ,0)=0 and Θ(φ,ρ)=Gr(φ)+O(ρ). Therefore, the equations of the orbits of X near the origin are governed by the equation
dρdφ=F(φ,ρ):=R(φ,ρ)Θ(φ,ρ) | (1.2) |
well defined in C∖Θ−1(0) being the cylinder C={(θ,ρ)∈S1×R:0≤ρ≪1} with S1=R/(2πZ). the local set of zero angular speed is defined as Θ−1(0)={(φ,ρ)∈C:Θ(φ,ρ)=0,ρ≥0}, and the set of characteristic directions Ωpq={φ∗∈S1:Θ(φ∗,0)=Gr(φ∗)=0}. Notice that the set {ρ=0} is invariant for the flow of (1.2), and it becomes either a periodic orbit or a polycycle according to whether Ωpq=∅ or Ωpq≠∅, respectively.
We also define the (p,q)-critical parameters as the elements of the subset Λpq⊂Λ of the monodromic parameter space corresponding to vector fields with local curves of zero angular speed, that is,
Λpq={λ∈Λ:Θ−1(0)∖{ρ=0}≠∅}. |
We emphasize that Λpq=∅ when Ωpq=∅ but the converse is not true.
Given (p,q)∈W(N(X)), we take the (p,q)-quasihomogeneous expansion
X=∑j≥rXj, | (1.3) |
where r≥1 and Xj are (p,q)-quasihomogeneous vector fields of degree j, and Xr is called the leading vector field of X.
Let F(x,y)=0 be a real invariant analytic curve of X with F(0,0)=0, and denote by K(x,y) the cofactor of F, that is, X(F)=KF holds. We recall that F always exists, as it was proved in [10] using the separatrix theorem of Camacho-Sad. In weighted polar coordinates this equation is transformed into
ˆX(ˆF)=ˆKˆF, | (1.4) |
where ˆX=∂φ+F(φ,ρ)∂ρ, ˆF(φ,ρ)=F(ρpcosφ,ρqsinφ) and ˆK is the cofactor of the invariant curve ˆF=0 of ˆX whose expression is
ˆK(φ,ρ)=K(ρpcosφ,ρqsinφ)ρrΘ(φ,ρ). | (1.5) |
Let Φ(φ;ρ0) be the flow of ˆX with initial condition Φ(0;ρ0)=ρ0. In [10], it is proved that
Ipq(ρ0)=PV∫2π0ˆK(φ,Φ(φ;ρ0))dφ, | (1.6) |
exists for any initial condition ρ0>0 sufficiently small. Working in Λ∖Λpq we may also define the integral
ζpq(ρ0)=∫2π0∂F∂ρ(φ,Φ(φ;ρ0))dφ, | (1.7) |
and the difference
αpq(ρ0)=ζpq(ρ0)−Ipq(ρ0). |
A fundamental result in this work is the following one.
Theorem 1.1. The equation
ˆF(0,Π(ρ0))=ˆF(0,ρ0)exp(Ipq(ρ0)) | (1.8) |
holds. Moreover, in Λ∖Λpq, the relations
ζpq(ρ0)=log(Π′(ρ0)), | (1.9) |
and
ˆF(0,Π(ρ0))=ˆF(0,ρ0)exp(−αpq(ρ0))Π′(ρ0) | (1.10) |
are satisfied.
Remark 1.2. We believe that the fundamental equation (1.10) holds in the whole monodromic space Λ and not only in Λ∖Λpq, but we have no proof.
Remark 1.3. If X has ℓ>1 invariant analytic curves Fi=0 with cofactors Ki for i=1,…,ℓ, then it also has the ℓ-parameter invariant analytic curve F=∏iFmii=0 with arbitrary multiplicities mi∈N and cofactor K=∑imiKi. Sometimes we may find mi that simplifies the expression of K. As an example, the case when F becomes an inverse integrating factor of X, hence K=div(X), was studied in [6].
Il'Yashenko in [12] proves that Π has a Dulac asymptotic expansion possessing a linear leading term. More specifically, one has Π(ρ0)=η1ρ0+o(ρ0). The computation of the first (generalized) Poincaré-Lyapunov quantity η1 is quite involved and needs the use of cumbersome blow-ups computations; see for example [13]. Some sufficient conditions that guarantee the computation of η1 are stated below. We will user the following expansions. The functions Θ and R appearing in (1.2) have Taylor expansions at ρ=0 given by Θ(φ,ρ)=Gr(φ)+O(ρ) and R(φ,ρ)=Fr(φ)ρ+O(ρ2). Moreover F(x,y)=Fs(x,y)+⋯ and K(x,y)=Kˉr(x,y)+⋯ are the (p,q)-quasihomogeneous expansions of the cofactor K associated to the analytic invariant curve F=0. Here Fs and Kˉr are the leading (p,q)-quasihomogeneous polynomials of weighted degree s and ˉr, respectively, and the dots denote higher (p,q)-quasihomogeneous terms.
Proposition 1.4. Assume that both Ipq and ζpq can be extended with continuity to the origin. Then
log(ηs1)=Ipq(0)=sζpq(0), |
that is,
log(ηs1)=PV∫2π0Kˉr(cosφ,sinφ)Gr(φ)dφ=sPV∫2π0Fr(φ)Gr(φ)dφ, |
provided that both principal values exist.
Remark 1.5. It is worth emphasizing that, in general, Ipq cannot be extended by continuity at ρ0=0. In [10], it is shown that expression (3.4) of η1 is wrong in several examples.
Clearly, if Φ(φ;ρ0) is analytic at ρ0=0 for all φ∈S1 then Π is analytic at ρ0=0. A sufficient condition for that to happen is that Ωpq=∅. Instead, a weaker condition to ensure the analyticity of Π is the following one.
Theorem 1.6. If Ipq is analytic at the origin, then Π is too.
The classical center-focus problem has been studied for decades. In [14,15], some particular degenerate systems with a monodromic singularity were studied. In [16,17], it is proved that some degenerate systems with a monodromic singularity are limit of differential systems with monodromic linear part. In [18,19,20] some sufficient conditions to have a center at a completely degenerate critical point are given. In [21], the relation between the reversivility and the center problem is studied. The textbook [22] is a good summary about the relations between the the center and cyclicity problems. In [23], a geometrical criteria to determine the existence of a center for certain differential systems is given. In [24], the Hopf-bifurcation formulas for some differential systems are established. Final, in [25], the authors solve the center problem for monodromic sigularities with characteristic directions and with inverse integrating factor and [26] the linear term of all the monodromic families known is obtained. However, no general characterization was known until the work [10], where it is proved the following theorem.
Theorem 1.7 ([10]). Let X be a family of analytic planar vector fields having a monodromic singular point at the origin and K the cofactor associated to a real analytic invariant curve through the origin. Then Ipq(ρ0) exists for any initial condition ρ0>0 that is sufficiently small, and the origin is a center if and only if Ipq(ρ0)≡0.
We show other necessary center condition in Λ∖Λpq.
Proposition 1.8. If the origin is a center of X and we restrict to the parameter space Λ∖Λpq then αpq(ρ0)=ζpq(ρ0)≡0.
Notice that from Proposition 1.8 and relation αpq(ρ0)=ζpq(ρ0)−Ipq(ρ0), we obtain a new proof of the necessary part in Theorem 1.7.
Example 1.9. We emphasize that there are focus with parameters in Λ∖Λpq≠∅ and αpq(ρ0)≢0. As an example, we consider the family
˙x=λ1(x6+3y2)(−y+μx)+λ2(x2+y2)(y+Ax3),˙y=λ1(x6+3y2)(x+μy)+λ2(x2+y2)(−x5+3Ax2y). | (1.11) |
In [27], it is proved that
Λ={(λ1,λ2,μ,A)∈R4:3λ1−λ2>0, λ1−λ2>0} |
and the Poincaré map is the linear map Π(ρ0)=η1ρ0. Moreover, F(x,y)=(x2+y2)(x6+3y2) is an inverse integrating factor of the full family (1.11).Using the weights (p,q)=(1,1)∈W(N(X)) we have X=X2+⋯, hence r=2, and the forthcoming formula (2.3) yields α11(ρ0)=−log(η71). The value η1=exp(2πλ1μ+2√3πAλ2/3) is a consequence of the works [6,27]. Notice that in this example 0∈Ω11.
We consider analytic degenerate infinity vector fields
X=Xn+AXE | (1.12) |
with Xn=Pp+n(x,y)∂x+Qq+n(x,y)∂y a (p,q)-quasihomogeneous polynomial vector field of degree n, XE=px∂x+qy∂y is the radial Euler field, and A(x,y) a real analytic function in R2 whose Taylor expansion at the origin starts with (p,q)-quasihomogeneous terms of degree m−1 with m>n+1. In the work [28] the particular homogeneous case (p,q)=(1,1) is analyzed, and here we generalize it.
Proposition 1.10. Any degenerate infinity vector field (1.12) has the homogeneous algebraic invariant curve
F(x,y)=pxQq+n(x,y)−qyPp+n(x,y)=0 |
with cofactor
K(x,y)=div(Xn)+(n+p+q)A(x,y). |
Moreover, it also has the inverse integrating factor V=FHm−n−1d provided that A is (p,q)-quasihomogeneous of degree m−1, where H is a (p,q)-quasihomogeneous first integral of degree d of Xn.
Proposition 1.11. Any degenerate infinity vector field (1.12) with a monodromic singularity at the origin becomes a focus provided that div(Xn)+(n+p+q)A(x,y) is a positive or negative defined function in a neighborhood of the origin. If additionally the origin is a center of Xn and A is (p,q)-quasihomogeneous then the Poincaré map associated with the origin is analytic and has the form Π(ρ)=ρ+ηm−nρm−n(1+O(ρ)) with
ηm−n=∫2π0(pcos2φ+qsin2φ)A(cosφ,sinφ)F(cosφ,sinφ)(H(cosφ,sinφ))(m−n−1)/ddφ. |
In particular, the origin is a center if and only if ηm−n=0 and the cyclicity of the origin in family (1.12) is zero.
Remark 1.12. We note that the (p,q)-quasihomogeneous first integral H of the monodromic vector field Xn appearing in Proposition 1.10 only exists in case the origin be a center of Xn. This statement can be inferred from statement (ⅱ) of Lemma 4 in [29] because the transformed first integral to (p,q)-weighted polar coordinates becomes a Laurent first integral of Xn. In particular, Π(ρ)=ρ+o(ρ) in agreement with Proposition 1.11.
In this section we restrict our attention to the monodromic subset Λ∖Λpq so that Θ−1(0)∖{ρ=0}=∅. In particular Ωpq can be empty or not.
Let us assume the particular case that F(x,y) is an analytic inverse integrating factor of X; that is, div(X/F)≡0 holds. Then the function
V(φ,ρ)=ˆF(φ,ρ)ρrJ(φ,ρ)Θ(φ,ρ) | (2.1) |
is an inverse integrating factor of ˆX in C∖{Θ−1(0)∪{ρ=0}}, that is, V satisfies the equation ˆX(V)=∂ρ(F)V. Applying the differential operator ˆX on both sides of (2.1) and taking into account (1.4), we obtain the relation between ˆK and ∂ρF given by
∂ρF=ˆK−ˆX(log|ρrJΘ|). | (2.2) |
On the other hand, using (2.2) and recalling that αpq(ρ0)=ζpq(ρ0)−Ipq(ρ0), we obtain
αpq(ρ0)=−∫2π0ˆX(log|ρrJΘ|)∘(φ,Φ(φ;ρ0))dφ=−∫2π0ddφ(log|ρrJΘ|∘(φ,Φ(φ;ρ0)))dφ,=−log|Πr(ρ0)J(0,Π(ρ0))Θ(0,Π(ρ0))ρrJ(0,ρ0)Θ(0,ρ0)|, | (2.3) |
where we have used the 2π-periodicity of J and Θ in the variable φ. Using this expression of αpq and taking into account (2.1), Eq (1.10) in Λ∖Λpq is written in the simpler form V(0,Π(ρ0))=V(0,ρ0)Π′(ρ0). In this way we rediscover the formula obtained in [6] by other methods. This formula for the special case of degenerate differential systems without characteristic directions was given in [30]; see also [31].
Proof. To prove the first statement, we evaluate (1.4) along the flow Φ(φ;ρ0), and we obtain
ddφˆF(φ,Φ(φ;ρ0))=ˆK(φ,Φ(φ;ρ0))ˆF(φ,Φ(φ;ρ0)), |
hence
PV∫2π0ddφˆF(φ,Φ(φ;ρ0))ˆF(φ,Φ(φ;ρ0))dφ=PV∫2π0ˆK(φ,Φ(φ;ρ0))dφ, | (3.1) |
where the last principal value exist for any initial condition ρ0>0 sufficiently small as it was proved in [10]. Therefore, (3.1) takes the form
ˆF(2π,Φ(2π;ρ0))=ˆF(0,ρ0)exp(Ipq(ρ0)). |
Using the 2π-periodicity of ˆF in the variable φ and the definition of Π, the former equation becomes (1.8).
To prove the second part, we use the definition of Φ(φ;ρ0), that is,
∂Φ∂φ(φ;ρ0)=F(φ,Φ(φ;ρ0)), Φ(0;ρ0)=ρ0>0. | (3.2) |
Working in Λ∖Λpq we know that F is analytic in C∖{ρ=0}; hence, Φ is also analytic there, and differentiating both expressions in (3.2) with respect to ρ0 yields
∂∂φ(∂Φ∂ρ0(φ;ρ0))=∂F∂ρ(φ,Φ(φ;ρ0))∂Φ∂ρ0(φ;ρ0), ∂Φ∂ρ0(0;ρ0)=1. | (3.3) |
Clearly (φ,Φ(φ;ρ0))∈C∖{ρ=0} for all φ∈[0,2π], and therefore the function ∂ρF(φ,Φ(φ;ρ0)) is continuous in S1×{0<ρ0≪1}. Thus, we may integrate the first equality in (3.3), yielding
∫2π0∂F∂ρ(φ,Φ(φ;ρ0))dφ=∫2π0∂∂φ(∂Φ∂ρ0(φ;ρ0))∂Φ∂ρ0(φ;ρ0)dφ. |
So we obtain
∫2π0∂F∂ρ(φ,Φ(φ;ρ0))dφ=[log(∂Φ∂ρ0(φ;ρ0))]φ=2πφ=0=log(∂Φ∂ρ0(2π;ρ0)) |
that can be written as Eq (1.9). Finally, Eq (1.8) is rewritten as the fundamental ordinary differential equation (1.10).
Proof. Using the definition (1.5) of ˆK together with the (p,q)-quasihomogeneous expansion K(x,y)=Kˉr(x,y)+⋯ and the Taylor expansion Θ(φ,ρ)=Gr(φ)+O(ρ) with ˉr≥r we obtain
ˆK(φ,ρ)=ρˉr−rKˉr(cosφ,sinφ)Gr(φ)+O(ρˉr−r+1). |
From [10], we know that it is proved that the principal value Ipq(ρ0) defined in (1.6) exists for any ρ0>0 and small. If Ipq can be extended with continuity to the origin, then, using that Φ(φ;0)=0, we would have
Ipq(0)={0ifˉr>r,PV∫2π0Kˉr(cosφ,sinφ)Gr(φ)dφifˉr=r, |
in case that this principal value exists. Moreover, from the (p,q)-quasihomogeneous expansion F(x,y)=Fs(x,y)+⋯, we could use Eq (1.8) to express
Ipq(ρ0)=log|ˆF(0,Π(ρ0))ˆF(0,ρ0)|=log|Πs(ρ0)Fs(cosφ,sinφ)+O(Πs+1(ρ0))ρs0(Fs(cosφ,sinφ)+O(ρ0))| |
whose extension to ρ0=0 gives Ipq(0)=log(ηs1), taking into account that Π(ρ0)=η1ρ0+o(ρ0). Comparing both expressions of Ipq(0), we have that if ˉr>r, then η1=1, whereas
log(ηs1)=PV∫2π0Kˉr(cosφ,sinφ)Gr(φ)dφ, | (3.4) |
when ˉr=r under the restriction that the former principal value exists.
On the other hand, if ζpq can be extended with continuity to the origin, then, by (1.9),
log(η1)=ζpq(0)=PV∫2π0Fr(φ)Gr(φ)dφ, |
assuming this last principal value exists. Then the proposition follows.
Proof. We define
f(ρ0,Π)=ˆF(0,Π)−ˆF(0,ρ0)exp(Ipq(ρ0)). |
The analyticity of Ipq at the origin implies that (1.8) can be written as f(ρ0,Π)=0, where f is an analytic function in a neighborhood of (ρ0,Π)=(0,0). Therefore, the Poincaré map Π(ρ0)=η1ρ0+o(ρ0) must be a branch of f at the origin and, consequently, admits the convergent Puiseux expansion
Π(ρ0)=∑i≥0ηi+1ρ1+in0 | (3.5) |
with some index n∈N∗. Now we are going to compute n.
By Proposition 1.4 we know that Ipq(0)=log(ηs1). Then exp(Ipq(ρ0))=ηs1+O(ρ0) with η1>0. Using that ˆF(0,ρ0)=ρs0(Fs(1,0)+O(ρ0)), we obtain that f(ρ0,Π)=Fs(1,0)(Πs−ηs1ρs0)+⋯ from where we deduce that the Newton diagram of f only contains the edge L joining the endpoints (s,0) and (0,s) provided that Fs(1,0)≠0. We can take Fs(1,0)≠0 because, without loss of generality, we take 0∉Ωpq after a rotation (if necessary), and moreover we claim that the eventual real roots of Fs(cosφ,sinφ) must belong to Ωpq. To prove the claim, we observe that
ˆF(φ,ρ)=F(ρpcosφ,ρqsinφ)=ρs[Fs(cosφ,sinφ)+o(ρ)], |
hence ˆF(φ,ρ)/ρs=0 is an invariant curve of the differential equation (1.2). Clearly that invariant curve can only intersect the monodromic polycycle ρ=0 at its singularities (φ,ρ)=(φ∗,0) with φ∗∈Ωpq.
Once we know that Fs(1,0)≠0 so that the edge L has slope −1, then we compute
f(ρ0,ρ0μ)=ρs0((μs−ηs1)Fs(1,0)+O(ρ0)). |
From this expression we deduce that the determining polynomial P(μ) associated to L is P(μ)=Fs(1,0)(μs−ηs1). Therefore μ=η1 is a simple root of P, and it follows that n=1 in the Puiseux expansion (3.5); this is a classical result that is proved, for instance, in [32]. Therefore, Π admits a convergent power series expansion at the origin finishing the proof.
Proof. Let the origin be a center. Then Eq (1.10) must have the solution Π(ρ0)=ρ0 and therefore ˆF(0,ρ0)=ˆF(0,ρ0)exp(−αpq(ρ0)) holds. We claim that ˆF(0,ρ0)≢0 so it must occur αpq(ρ0)≡0 proving the proposition.
To prove the claim, we use the (p,q)-quasihomogeneous expansion F(x,y)=Fs(x,y)+⋯ with Fs(x,y)≢0 a (p,q)-quasihomogeneous polynomial of degree s and the dots are (p,q)-quasihomogeneous terms of higher degree. Notice that s≥1 because (x,y)=(0,0) is an isolated real zero of F. Now we consider the expression
ˆF(φ,ρ)=F(ρpcosφ,ρqsinφ)=ρs[Fs(cosφ,sinφ)+O(ρ)], | (3.6) |
and we observe that ˆG(φ,ρ)=ˆF(φ,ρ)/ρs=0 is an invariant curve ˆX. Thus either
ˆG−1(0)∩{ρ=0}=∅ |
or
ˆG−1(0)∩{ρ=0}⊂{(φ,ρ)=(φ∗,0):φ∗∈Ωpq} |
by uniqueness of solutions of ˆX. But ˆG−1(0)∖{ρ=0}=∅ by the monodromy of the polycycle {ρ=0}. Therefore ˆF(φ,ρ) has an isolated zero at ρ=0 for any φ∈S1, and, in particular, the claim follows.
The second part, that ζpq(ρ0)≡0, is a trivial consequence of the relation (1.9).
Proof. The first part is straightforward since X(F)=KF holds because Xn is a (p,q)-quasihomogeneous polynomial vector field of degree n; hence its components Pp+n and Qq+n satisfy Euler relations
XE(Pp+n)=(p+n)Pn+p, XE(Qq+n)=(q+n)Qq+n. | (3.7) |
The second part is also straightforward since X(V)=div(X)V holds, taking into account Euler relations (3.7) and XE(A)=(m−1)A, together with the fact that system Xn(H)=0 and XE(H)=dH can be solved as ∂xH=dQq+nH/F and ∂yH=−dPp+nH/F.
Proof. In (p,q)-weighted polar coordinates, the angular speed of vector field (1.12) is ˙φ=Gr(φ)=F(cosφ,sinφ)/(pcos2φ+qsin2φ). Therefore, a monodromic necessary condition for the origin of X is that F has no real factors in R[x,y] so that Gr≠0 in S1. In particular, F has an isolated singularity at the origin, X∈Mo(p,q) so Π is analytic at the origin, Λpq=∅ and the origin is the unique real finite singularity of X. We observe that Xn has a monodromic singular point at the origin when X has it because both vector fields share the same angular speed ˙φ. Indeed, the origin is a center of Xn under our hypothesis that implies the existence of the first integral H of Proposition 1.10, see Remark 1.12.
The monodromy of the origin implies that N(X)=N(Xn) has only one edge and that when we only vary the parameters of X in the monodromic space Λ, then the Newton diagram N(X|Λ) of the restricted vector field X|Λ is fixed. The inverse integrating factor
V(φ,ρ)=ρm−n(H(cosφ,sinφ))(m−n−1)/d. |
The explicit Taylor expansion of Π at the origin is just a consequence of statement (ⅱ) of Theorem 4 in [6]. Finally, we can use Theorem 7 in [6] to conclude that
ηm−n=∫2π0F(φ,r)V(φ,r)dφ=I1+rn−m+1I2, |
where Ii are integrals independents of r and
I1=∫2π0(pcos2φ+qsin2φ)A(cosφ,sinφ)F(cosφ,sinφ)(H(cosφ,sinφ))(m−n−1)/ddφ. |
Since the expression of ηm−n must be independent of r>0 and sufficiently small, by Theorem 7 in [6], we deduce that I2=0, and the proposition follows.
In this paper we have considered planar analytic vector fields X having a monodromic singular point with Poincaré map Π. Using the fact that always exists a real analytic invariant curve F=0 of X in a neighborhood of that singularity in the paper are given the relations between Π and F that can be used to determine new conditions in order to guarantee the analyticity of Π at the singularity.
The special case when F is inverse integrating factor of X we rediscover the formula obtained previously in [25] by an other method. Finally some applications to the center-focus problem and also to vector fields with degenerate infinity are given.
All authors carried out the main results of this article, drafted the manuscript, and read and approved the final manuscript. All authors have read and approved the final version of the manuscript for publication.
The authors declare they have used Artificial Intelligence (AI) tools in the creation of this article.
The authors are partially supported by the Agencia Estatal de Investigación grant PID2020-113758GB-I00 and an AGAUR (Generalitat de Catalunya) grant number 2021SGR 01618.
The author is partially supported by a MICIN grant number PID2020-113758GB-I00 and an AGAUR grant number 2017SGR-1276.
Prof. Jaume Giné is the Guest Editor of special issue "Advances in Qualitative Theory of Differential Equations" for AIMS Mathematics. Prof. Jaume Giné was not involved in the editorial review and the decision to publish this article. The authors declare no conflicts of interest.
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