The second term of the asymptotics of the monodromy map in case of two even edges of Newton diagram

N.Medvedeva, E.Batcheva

Chelyabinsk State University, medv@cgu.chel.su

Abstract

The second term of the asymptotics of the monodromy map of monodromic singular point for some class of vector fields, Newton diagram of which consists of two even edges is computed; in that case the principal term of the asymptotics is an identity mapping. The obtained result allows to formulate the sufficient condition of focus for the singular point from the class under consideration. AMS subject classification: 34C05, 34C20, 34C35, 58F21 Key words and phrazes: monodromic singular point, center, focus, monodromy

map, resolution of singularity, Newton diagram ^{1}

Introduction.

In this paper we calculate the second term of the asymptotics of the monodromy map of monodromic singular point in case when the principal term of the asymptotics is an identity mapping.

It is known ([1],[2]), that the monodromy map (return map) of the monodromic singular point of the analytic vector field on the plane has a linear principal term of the asymptotics

Δ(ρ) = Cρ + o(ρ).

The logarithm of the coefficient of this principal term is computed in ([3]) for a so called Γ-nondegenerate vector field. It is expressed via the Taylor coefficients of the principal part of the vector field defined by Newton diagram Γ. If Δ(ρ) ≡ ρ, then the the singular point is a center. The inequality ln C ≠ 0 is the suffucient condition for the singular point to be a focus.

It was found ([3]) if all the edges of the Newton diagram Γ are even, then ln C is identically equal to zero in the class of all the Γ -nondegenerate vector fields having a monodromic singular point. That is impossible to obtain the sufficient condition of focus with help of the principal term of asymptotics.

1Partially supported by grants d99-411 ISSEP, 9901-00821 and 0001-00745 RFBR.

This paper is in final form and no version of it will be submitted for publication elsewhere.

In this work we consider Γ-nondegenerate vector fields with a monodromic singular point Newton diagram of which consists of two even edges. Under some additional conditions we calculate the second term of asymptotics of the monodromy map.

Let us recall some notions connected with the Newton diagram.

We write an analytic vector field (germ) in the neighbourhood of a singular point (x, y) = (0, 0) in the form

\frac{X(x, y)}{y} \frac{∂}{∂x} + \frac{Y (x, y)}{x} \frac{∂}{∂y} .  (0.1)

Here the functions X and Y are divisible by y and x respectively. The vector field (0.1) defines a dynamical system that it will be convenient to write in the form

yẋ = X(x, y) , xẏ = Y (x, y).  (0.2)

Definitions 1. Let

(∑ a_{ij}x^{i}y^{j}, ∑ b_{ij}x^{i}y^{j})

be a Taylor expansion of the right hand-side of the system (0.2). The support of the vector field (0.1) and of the system (0.2) is the set {(i, j) : (a_{ij}, b_{ij}) ≠ (0, 0)}. The pair (a_{ij}, b_{ij}) is called the vector coefficient of the point (i, j) of the support. The index of the point (i, j) of the support is the quantity

\{ \begin{matrix}\frac{b_{ij}}{a_{ij}} , & if a_{ij} ≠ 0 \\ ∞, & if a_{ij} = 0. \end{matrix}

The vector coefficient of any other integer-valued point we define as (0, 0).

2. Consider the set

⋃_{(i,j)}{(i, j) + R_{+}^{2}},

where R_{+}^{2} is the positive quadrant, the points (i, j) belong to the support. The boundary of the convex hull of this set consists of two open rays and one broken line, which can consist of one point. This broken line is called the Newton diagram of the vector field (0.1). The links of this broken line are called the edges of a Newton diagram, and their end-points are called the vertices of a Newton diagram.

3. The index of an edge of a Newton Diagram is the rational number that is equal to the tangent of the angle between the negative direction of the j− axes, and the edge.

Consider the edge of the Newton diagram of the system (0.2) with index α = \frac{m}{n}, where \frac{m}{n} is irreducible fraction. We can group the terms of the Taylor series of the system (0.2) so that

yẋ = ∑_{d=0}^{∞} X_{d}(x, y),  xẏ = ∑_{d=0}^{∞} Y_{d}(x, y),  (0.3)

where

X_{d}(x, y) = ∑_{ni+mj=d+d_{0}} a_{ij}x^{i}y^{j}, Y_{d}(x, y) = ∑_{ni+mj=d+d_{0}} b_{ij}x^{i}y^{j}  (0.4)

are quasihomogeneous polynomials of degree d + d_{0} with weights n and m in the variables x and y respectively, d_{0} ∈ N.

We set F_{d}(x, y) = Y_{d}(x, y) − αX_{d}(x, y).

Proposition 0.1 ([3]) Let \frac{m}{n} be an irreducible fraction. For any quasihomogeneous polynomial with weights n and m in the variables x and y respectively the decomposition

R(x, y) = Ax^{s_{1}}y^{s_{2}} ∏(y^{n} − b_{i}x^{m})^{k_{i}}

holds, where b_{i} are distinct nonzero complex numbers and k_{i} ≥ 0.

Definition. A factor of the form y^{n} − b_{i}x^{m}, b_{i} ≠ 0, is called the prime factor of the polynomial R(x, y), the number k_{i} is called the multiplicity of this prime factor.

Definition. A vector field (germ) with Newton diagram Γ is Γ-nondegenerate, if 1) none of the polynomials F_{0}(x, y) corresponding to edges of the Newton diagram Γ has a prime factor of multiplicity larger than one; 2) the index of any vertex not lying on a coordinate axis is different from the indices of the edges adjacent to it.

The set of Γ-nondegenerate vector fields having zero as a monodromic singular point will be denoted by M_{Γ}.

Definition. We call the Newton diagram Γ monodromic, if the set M_{Γ} is nonempty. A Newton diagram is monodromic, if and only if it has one vertex on each coordinate axis and the lengths of the projections of the edges on the coordinate axes are all even numbers ([3]).

Definition. Let α = \frac{m}{n} be an irreducible fraction. The edge of the Newton diagram with index α will be called even, if one of the numbers m and n and odd otherwise.

Theorem 1 ([3]) Let Γ be a monodromic Newton diagram. ln c = 0 is identically zero on M_{Γ} if and only if all the edges of the Newton diagram Γ are even.

Let the Newton diagram of the vector field V consist of two edges with indices α = \frac{m}{n}, α̃ = \frac{m̃}{ñ}, where α̃ > α. For each edge we can consider the expansion (0.3) - (0.4). The polynomials analogous to X_{d}, Y_{d}, F_{d} for the edge with index α̃ we denote X̃_{d}, Ỹ_{d}, F̃_{d} respectively.

In this paper we prove the following theorem.

Theorem 2 Let Γ be a monodromic Newton diagram consisting of two even edges with indices α = \frac{m}{n} and α̃ = \frac{m̃}{ñ} (α̃ > α) and V be a Γ-nondegenerate vector field having (0, 0) as a monodromic singular point, (A_{0}, B_{0}) be a vector coefficient of the vertex of the Newton diagram between its edges. Let in addition the following conditions hold 1)λ = \frac{ñ(B_{0}−α̃A_{0})}{B_{0}−αA_{0}} > 1; λ is irrational number;

2) \frac{A_{0}}{(B_{0}−α̃A_{0})} < 0.

Then the monodromy map associated to the origin (taking the axis of abscissa as transversal with a suitable chosen parameter) has possibly after a time reversal the form

Δ(ρ) = ρ(1 + F_{2}ρ^{\frac{1}{n}} + o(ρ^{\frac{1}{n}})), ρ → 0,

where in case n > 1 F_{2} = 0, in case n = 1, m̃ – even number, r = m̃n − ñm > 1,

F_{2} = 2 ∫_{−∞}^{+∞}Φ̃_{1}(1, ξ)e^{∫_{0}^{ξ} Φ̃_{0}(1,τ)dτ}dξ

Φ̃_{0}(x, y) = \frac{X̃_{0}(x, y)}{ñyF̃_{0}(x, y)}, Φ̃_{1}(x, y) = \frac{Ỹ_{0}(x, y)X̃_{1}(x, y) − Ỹ_{1}(x, y)X̃_{0}(x, y)}{ñyF̃_{0}^{2}(x, y)}.

If F_{2} ≠ 0, then the origin is a focus.

1 Resolution of the singularity.

Let the Newton diagram of the vector field V consist of two edges with indices α = \frac{m}{n}, α̃ = \frac{m̃}{ñ}, where α̃ > α. In according to [3] in such a case the resolution of the singularity connected with a Newton diagram consists of the following: the first quadrant of the plain (x, y) is broken up into sectors S_{α}, Sα̃ and S_{αα̃}, corresponding to the edges and the vertex between them of the Newton diagram.

Let ε > 0, δ > 0 be small enough. The change of coordinates

x = wz^{n}, y = z^{m}.  (1.5)

turns the sector S_{α} = {εx^{α} ≤ y ≤ δ} into the rectangle

P_{α} = \{0 ≤ w ≤ ε^{− \frac{1}{α}} , 0 ≤ z ≤ δ^{\frac{1}{m}} \}.

The change of coordinates

x = u^{nñ}v^{n}, y = u^{nm̃}v^{m},  (1.6)

turns the sector S_{αα̃} = \{\frac{1}{ε}xα̃ ≤ y ≤ εx^{α}\} into the rectangle

P_{αα̃} = {0 ≤ u ≤ ε_{1}, 0 ≤ v ≤ ε_{2}},

where ε_{1} = ε^{\frac{1}{nñ(α̃−α)}} , ε_{2} = ε^{\frac{1}{n(α̃−α)}} .

Finally the change of coordinates

x = z̃ñ, y = z̃m̃w̃,  (1.7)

turns the sector Sα̃ = \{0 ≤ y ≤ \frac{1}{ε}xα̃\} into the rectangle

Pα̃ = \{0 ≤ w̃ ≤ \frac{1}{ϵ}, 0 ≤ z̃ ≤ δ^{\frac{1}{ñ}} \}.

(See fig.1.)

[figure: x; y; ρ; Δ(ρ); S_{α}; S_{αα̃}; Sα̃; z; w; Γ_{0}; f_{α}; P_{α}; Γ′_{1}; ε^{− \frac{n}{m}}; (v); Γ_{1}; (v); g; P_{αα̃}; Γ_{2}; (y); (y); Γ′_{2}; Γ̃_{0}; Pα̃; v; w̃; z̃; u; Δ_{1}; Δ_{2}]

Figure 1: Resolution of singularity.

2 Change of coordinates in the sector corresponding to the edge of a Newton diagram.

Take the change (1.5) in the system (0.2), we obtain

\frac{dz}{dw} = z(Φ_{0}(w, 1) + zΦ_{1}(w, 1) + ...),  (2.8)

where

Φ_{0}(x, y) = −\frac{Y_{0}(x, y)}{nxF_{0}(x, y)},

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 19, p. 5

Φ_{1}(x, y) = −\frac{m(Y_{1}(x, y)X_{0}(x, y) − X_{1}(x, y)Y_{0}(x, y))}{n^{2}xF_{0}^{2}(x, y)}.

Analogously take the change (1.7), we obtain

\frac{dz̃}{dw̃} = z̃(Φ̃_{0}(1, w̃) + z̃Φ̃_{1}(1, w̃) + ...),  (2.9)

where Φ̃_{0}(x, y),*** Φ̃_{1}(x, y) are defined at the statement of the theorem 2.

3 Reflected vector fields.

Let S^{x} and S^{y} be the reflections of the (x, y)-plane about the x- and y-axes respectively, and let S^{xy} = S^{x} ∘ S^{y}. The images of the vector field V after reflections S^{x}, S^{y}, S^{xy} we denote V^{x}, V^{y}, V^{xy} respectively. Consider in the first quadrant four vector fields V, V^{x}, V^{y}, V^{xy} and apply the described resolution of singularity to them. Corresponding formulas for the vector field V are given at the previous section. The polinomials analogous to X_{d}, Y_{d}, F_{d}, Φ_{d} for the reflected vector fields we denote by the same letters with the corresponding index above.

From ([3]) we obtain

Lemma 3.1

Φ_{d}^{x}(x, y) = Φ_{d}(x, −y), Φ̃_{d}^{x}(x, y) = −Φ̃_{d}(x, −y),
Φ_{d}^{y}(x, y) = −Φ_{d}(−x, y), Φ̃_{d}^{y}(x, y) = Φ̃_{d}(−x, y),
Φ_{d}^{xy}(x, y) = −Φ_{d}(−x, −y), Φ̃_{d}^{xy}(x, y) = −Φ̃_{d}(−x, −y).

where d = 0, 1.

4 Transition map in the rectangle P_{αα̃}.

In this section we denote F (u, v) be any analytic at the point (0, 0) function of variables u, v.

Lemma 4.1 Let (A_{0}, B_{0}) be the vector coefficient of the vertex of the Newton diagram Γ joining its edges, (A_{1}, B_{1}) be the vector coefficient of the nearest to this vertex integervalued point on the edge with the index α̃; (A_{2}, B_{2}) be the vector coefficient of the nearest to this vertex integer-valued point on the edge with the index α. The change of variables (1.6) in the sector S_{αα̃} followed by division by a power function converts fector field V into a vector field

\begin{matrix}u̇ = u(Ã_{0} + v^{r}(Ã_{1} + f_{1}(v^{r})) + u^{nr}(Ã_{2} + f_{2}(u^{nr})) + uvF (u, v)), \\ v̇ = v(B̃_{0} + v^{r}(B̃_{1} + g_{1}(v^{r})) + u^{nr}(Ã_{2} + g_{2}(u^{nr})) + uvF (u, v)), \end{matrix}  (4.10)

where (Ã_{i}, B̃_{i}) = (\frac{1}{r}(B_{i} − αA_{i}); −\frac{ñ}{r}(B_{i} − α̃A_{i})), i = 0, 1, 2, r = m̃n − ñm, f_{j}, g_{j}− polynomials, f_{j}(0) = g_{j}(0) = 0, j = 1, 2.

Proof. The matrix of exponents corresponding to the change (1.6) has the form

C = ( \begin{matrix}nñ & n \\ nm̃ & m \end{matrix} ) .

In according to ([3]) the support of the new vector field is the image of the support of the vector field V by means of the map C^{T}; vector coefficients are transformed by means of matrix C^{−1}:

C^{T} = ( \begin{matrix}nñ & nm̃ \\ n & m \end{matrix} ) , C^{−1} = −\frac{1}{nr} ( \begin{matrix}m & −n \\ −nm̃ & nñ \end{matrix} ) .

The transformation of the support of the vector field V is shown on the fig.2. From the form of the support of the transformed vector field and from the equality

[figure: C^{T}; i; j; j(v); i(u); r; nr; 1; 1; n; m; ñ; m̃; α; α̃; (A_{2}, B_{2}); (A_{0}, B_{0}); (A_{1}, B_{1}); (Ã_{1}, B̃_{1}); (Ã_{0}, B̃_{0}); (Ã_{2}, B̃_{2})]

Figure 2: Transformation of the support.

( Ã_{i} ; B̃_{i} ) = C^{−1} ( \begin{matrix}A_{i} \\ B_{i} \end{matrix} )

the conclusion of lemma follows.

From conditions of Γ - nondegegeneracy Ã_{0} ≠ 0, B̃_{0} ≠ 0. After division the system (4.10) on the expression in brackets from the second equation we obtain the system

u̇ = u(−\frac{1}{λ} + v^{r}(a + h_{1}(v^{r})) + u^{nr}(b + h_{2}(u^{nr})) + uvF (u, v)), v̇ = v,  (4.11)

where

λ = \frac{ñ(B_{0} − α̃A_{0})}{B_{0} − αA_{0}}, a = \frac{A_{0}B_{1} − B_{0}A_{1}}{ñ(B_{0} − α̃A_{0})^{2}} (α̃ − α), b = \frac{A_{0}B_{2} − B_{0}A_{2}}{ñ(B_{0} − α̃A_{0})^{2}} (α̃ − α),

h_{1}, h_{2}− polynomials, h_{1}(0) = h_{2}(0) = 0.

Lemma 4.2 The system (4.11) is reduced to the linear normal form

v̇ = v, ẏ = −\frac{1}{λ}y

with help of the C^{∞}-change of variables of the form

u = y(1 + v^{r}(ã + h̃_{1}(v^{r})) + u^{nr}(b̃ + h̃_{2}(u^{nr})) + uvF (u, v)),  (4.12)

where ã = −\frac{a}{r}, b̃ = \frac{bλ}{nr}, h̃_{1}, h̃_{2}− polynomials, h̃_{1}(0) = h̃_{2}(0) = 0.

Proof. In according to ([4],[5]) there exists C^{∞}-change of variabels which linearized the system (4.11). We shall look for it in the form (4.12).

From (4.12) and (4.11) we obtain

ẏ = y(−\frac{1}{λ} + (a + ãr)v^{r} + (b − \frac{b̃nr}{λ})y^{nr} + …).

Setting equal ẏ to −\frac{1}{λ}y) we obtain: ã = −\frac{a}{r}, b̃ = \frac{bλ}{nr}. Lemma is proved.

Because the vector field is Γ - nondegenerate and the singular point is monodromic we have λ > 0.

On the plain (y, v) we consider the rectangle 0 ≤ y ≤ ε_{1}, 0 ≤ v ≤ ε_{2}. Let L_{1} and L_{2} be the sides of the rectangle not lying on the coordinate axes: L_{1} = {y = ε_{1}}, Ł_{2} = {v = ε_{2}}. Let g : L_{2} → L_{1} be a transition map along the trajectories of the linear system

ẏ = y, v̇ = −λv.  (4.13)

On L_{1} we consider the parameter v, on L_{2} - parameter y.

Lemma 4.3 v = g(y) = ε_{2} (\frac{y}{ε_{1}} )^{λ} .

Proof. The trajectory of the system (4.13) goes from the point (y, ε_{2}) to the point (ε_{1}, v) during the time

t = ∫_{y}^{ε_{1}} \frac{dy}{y} = − ∫_{ε_{2}}^{v} \frac{dv}{λv} .

From here ln \frac{ε_{1}}{y} = −\frac{1}{λ} ln \frac{v}{ε_{2}} , v = ε_{2} (\frac{y}{ε_{1}} )^{λ} .

Lemma is proved.

5 Parametrisation of transversals.

The change (4.12) converts the segments L_{1} and L_{2} into the curves Γ_{1} and Γ_{2}. On Γ_{1} we consider the parameter v, on Γ_{2} − y. Then Γ_{1} and Γ_{2} accoding to (4.12) have the following form (r > 1):

Γ_{1} : u = ε_{1}(1 + o(ε_{1}))(1 + O(ε_{1})v + o(v)), v = v,  (5.14)
Γ_{2} : \begin{matrix}u = y(1 + o(ε_{2}))(1 + O(ε_{2})y + o(y)), \\ v = ε_{2}. \end{matrix}  (5.15)

From (1.5) and (1.6) we find that the connection between coordinates (z, w) in the rectangle P_{α} and (u, v) in the rectangle P_{αα̃} is the following

w = u^{\frac{nñ(α−α̃)}{α}} , z = u^{\frac{m̃}{α}} v.  (5.16)

Analogously from (1.6) and (1.7) we obtain that the coordinates (z̃, w̃) in the rectangle Pα̃ and (u, v) in the rectangle P_{αα̃} are connected by following formulas

w̃ = v^{n(α−α̃)}, z̃ = u^{n}v^{\frac{n}{ñ}} .  (5.17)

Substituting (5.14) in (5.16) we obtain that Γ_{1} in the coordinates (z, w) has the form

Γ′_{1} : \begin{matrix}w = ε_{1}^{∗}(1 + o(v)) \\ z = ε̃_{1}v(1 + O(ε_{1})v + o(v)), \end{matrix}  (5.18)

where ε_{1}^{∗} = ε^{− \frac{n}{m}} (1 + o(ε_{1})), ε̃_{1} = ε1^{\frac{m̃}{α}} (1 + o(ε_{1})).

Analogously substituting (5.15) in (5.17) we obtain that Γ_{2} in coordinates (z̃, w̃) has the form

Γ′_{2} : \begin{matrix}w̃ = \frac{1}{ε} \\ z̃ = ε̃_{2}y^{n}(1 + O(ε_{2})y + o(y)), \end{matrix}  (5.19)

where ε̃_{2} = ε2^{\frac{n}{ñ}} (1 + o(ε_{2})).

6 Transition maps in the rectangels corresponding to edges.

Consider in coordinates (z̃, w̃) two transversals Γ̃_{0} = {w̃ = 0} with parameter ρ = z̃ and Γ′_{2} (see (5.19)) with parameter y. Calculate the coefficients of the transition map fα̃ : Γ′_{2} → Γ̃_{0}.

Lemma 6.1 The map fα̃ has the asymptotics

ρ = d̃y^{n}(1 + d̃_{1}y + o(y)),  (6.20)

d̃= ε̃_{2}e^{− ∫_{0}^{\frac{1}{ε}} Φ̃_{0}(1,ξ)dξ},  (6.21)
\frac{d̃_{1}}{d̃} = − ∫_{0}^{\frac{1}{ε}} Φ̃_{1}(1, ξ)e^{∫_{0}^{ξ} Φ̃_{0}(1,τ)dτ}dξ + \frac{O(ε_{2})}{d̃}, if n = 1, d̃_{1} = O(ε_{2}) if n > 1. (6.22)

Proof. We look for the solution z̃(w̃, ρ) of the equation (2.9) with the initial condition z̃(0, ρ) in the form

z̃(w̃, ρ) = ρ(C̃_{0}(w̃) + C̃_{1}(w̃)ρ + ...),  (6.23)

where C̃_{0}(0) = 1, C̃_{i}(0) = 0 as i ≥ 0.

Solving corresponding equations in variations we obtain

C̃_{0}(w̃) = e^{∫_{0}^{w̃} Φ̃_{0}(1,ξ)dξ}, C̃_{1}(w̃) = C̃_{0}(w̃) ∫_{0}^{w̃} C̃_{0}(ξ)Φ̃_{1}(1, ξ)dξ.  (6.24)

For the map ρ = fα̃(y) from (6.23) and (5.19) we obtain the equation

C̃_{0}ρ + C̃_{1}ρ^{2} + ... = ε̃_{2}y^{n}(1 + O(ε_{2})y + o(y)),  (6.25)

where C̃_{i} = C̃_{i}(\frac{1}{ε}), i = 0, 1.

We shall look for ρ = fα̃ in the form (6.20). Substituting (6.20) in (6.25) we obtain

C̃_{0}d̃y^{n}(1 + d̃_{1}y + o(y)) + C̃_{1}d̃^{2}y^{2n}(1 + d̃_{1}y + o(y))^{2} + o(y^{2}) = ε̃_{2}y^{n}(1 + O(ε_{2})y + o(y)).

Let n = 1. Then

C̃_{0}d̃y(1 + d̃_{1}y + o(y)) + C̃_{1}d̃^{2}y^{2}(1 + 2d̃_{1}y + o(y)) + o(y^{2}) = ε̃_{2}y(1 + O(ε_{2})y + o(y))

or C̃_{0}d̃y + (C̃_{0}d̃d̃_{1} + C̃_{1}d̃^{2})y^{2} + o(y^{2}) = ε̃_{2}y + ε̃_{2}O(ε_{2})y^{2} + o(y^{2})

Setting equal coefficients in the equal powers of y, we obtain the conclusion of lemma. In the case that n > 1 we obtain the same expression for d̃, and d̃_{1} = O(ε_{2}). Lemma

is proved.

Consider in coordinates (z, w) two curves: Γ_{0} = {w = 0} with parameter ρ = z and Γ1^{′} (see (5.18)) with parameter v. It is evident, that the transition map f_{α} : Γ′_{1} → Γ_{0} has the asymptotics

ρ = dv(1 + d_{1}v + ...)  (6.26)

As in the proof of Lemma 6.1 we obtain that

d = ε̃_{1}e^{− ∫_{0}^{ε^{∗}_{1}}Φ_{0}(ξ,1)dξ}.  (6.27)

7 Coefficients of the composition of the transition maps.

The monodromy map Δ is the composition of the following maps:

Δ = Δ_{2} ∘ Δ_{1},

where Δ_{1} is a transition map for the upper half-plane which transforms the positive xhalf-axis near the origin into the negative x-half-axis in the positive direction along the trajectories of the vector field, and Δ_{2} is the analogous map for the lower half-plane (see fig.1).

The maps analogous to f_{α}, fα̃, g for the reflected vector fields we denote by the same letters with corresponding index above. Then

Δ_{1} = f y ; α̃ ∘ (g^{y})^{−1} ∘ (f_{α}^{y})^{−1} ∘ f_{α} ∘ g ∘ (fα̃)^{−1}, Δ_{2} = (Δ_{1}^{x})^{−1},

where

Δ_{1}^{x} = f xy ; α̃ ∘ (g^{xy})^{−1} ∘ (f_{α}^{xy})^{−1} ∘ f_{α}^{x} ∘ g^{x} ∘ (f x ; α̃)^{−1}.

According to Lemma 4.3 v = g(y) = ε^{γ}y^{λ}, where ε^{γ} = ε_{2}ε_{1}^{−λ}. Observe that all the analogous to g transition maps for the reflected vector fields have the same formula, because the number λ is the same for all of them ([3]).

The maps f_{α} and fα̃ are defined by formulas (6.26) and (6.20) respectively. Inverse maps have respectively forms

v = (f_{α}^{y})^{−1}(ρ) = \frac{1}{d^{y}} ρ(1 − \frac{d_{1}^{y}}{d^{y}} ρ + ...),

y = f −1 ; α̃(ρ) = (\frac{1}{d̃})^{\frac{1}{n}} ρ^{\frac{1}{n}} (1 − \frac{d̃_{1}}{nd̃^{\frac{1}{n}}} ρ^{\frac{1}{n}} + o(ρ^{\frac{1}{n}})).

It is easy to compute, that

y = g^{−1}(v) = ε^{\frac{−γ}{λ}} v^{\frac{1}{λ}} .

Taking into account that λ > 1 we obtain in consecutive order:

(f_{α}^{y})^{−1} ∘ f_{α} = \frac{d}{d^{y}} v(1 + \frac{d^{y}d_{1} − dd_{1}^{y}}{d^{y}} v + o(v)),

(g^{y})^{−1} ∘ (f_{α}^{y})^{−1} ∘ f_{α}) ∘ g = (\frac{d}{d^{y}} )^{\frac{1}{λ}} y(1 + o(y)).

Finally

Δ_{1} = f y ; α̃ ∘ (g^{y})^{−1} ∘ (f_{α}^{y})^{−1} ∘ f_{α} ∘ g ∘ f −1 ; α̃ = cρ(1 + c_{1}ρ^{\frac{1}{n}} + o(ρ^{\frac{1}{n}})),

where

∗ ∗ ∗ = \frac{d̃^{y}}{d̃} (\frac{d}{d^{y}} )^{\frac{n}{λ}} , c_{1} = \frac{d̃_{1}^{y}}{d̃^{\frac{1}{n}}} (\frac{d}{d^{y}} )^{\frac{1}{λ}} − \frac{d̃_{1}}{d̃^{\frac{1}{n}}} ,  (7.28)

Analogously Δ_{1}^{x} = c^{x}ρ(1 + c_{1}^{x}ρ^{\frac{1}{n}} + o(ρ^{\frac{1}{n}})), where

c^{x} = \frac{d̃^{xy}}{d̃^{x}} (\frac{d^{x}}{d^{xy}} )^{\frac{n}{λ}} , c_{1}^{x} \frac{d̃_{1}^{xy}}{(d̃^{x})^{\frac{1}{n}}} (\frac{d^{x}}{d^{xy}} )^{\frac{1}{λ}} − \frac{d̃_{1}^{x}}{(d̃^{x})^{\frac{1}{n}}} ,  (7.29)

From here Δ_{2} = (Δ_{1}^{x})^{−1} = \frac{1}{c^{x}} ρ(1 − \frac{c_{1}^{x}}{(c^{x})^{\frac{1}{n}}} ρ^{\frac{1}{n}} + o(ρ^{\frac{1}{n}})). Finally

Δ = Δ_{2} ∘ Δ_{1} = \frac{c}{c^{x}} ρ(1 + F_{2}ρ^{\frac{1}{n}} + o(ρ^{\frac{1}{n}})),

where

F_{2} = \frac{c_{1}(c^{x})^{\frac{1}{n}} − c_{1}^{x}c^{\frac{1}{n}}}{(c^{x})^{\frac{1}{n}}} .

Since \frac{c}{c^{x}} is a coefficient of the principal term of the asymptotics of the monodromy map Δ, then in our case of even edges it is equal to 1, hence

F_{2} = c_{1} − c_{1}^{x}.  (7.30)

The formula (6.22) implies that in case n > 1 d̃_{1} = O(ε_{2}), d̃_{1}^{y} = O(ε_{2}). Consider the coefficient c_{1}. Notice that as ε → 0

e^{\frac{1}{n} ∫_{0}^{ξ} Φ_{0}(ξ,1)dξ} ≈ ∗ ∗ ∗ξ^{b_{0}}, c > 0.  (7.31)

where b_{0} = \frac{A_{0}}{nñ(B_{0}−α̃A_{0})} < 0.

From here

e^{\frac{1}{n} ∫_{0}^{\frac{1}{ε}} Φ_{0}(ξ,1)dξ} ≈ ∗ ∗ ∗ε^{−b_{0}}.

Hence

\frac{d̃_{1}}{d̃^{1/n}} = \frac{O(ε_{2})}{ε̃_{2}^{1/n}}e^{\frac{1}{n} ∫_{0}^{ξ} Φ_{0}(−ξ,1)dξ} = \frac{O(ε_{2})}{ε_{2}^{1/ñ}(1 + o(ε_{2}))}ε^{−b_{0}} = \frac{O(ε_{2})}{ε_{2}^{1/ñ}}ε^{−b_{0}} → 0

as ε → 0. Analogously \frac{d̃_{1}^{y}}{d̃^{1/n}} → 0 as ε → 0.

We show, that the value \frac{d}{d^{y}} is bounded as ε → 0. Really from the formula (6.27) and lemma 3.1

d^{y} = ε̃_{1}e^{− ∫_{0}^{ε^{∗}_{1}}Φ_{0}^{y}(ξ,1)dξ} = ε̃_{1}e^{∫_{0}^{ε^{∗}_{1}}Φ_{0}(−ξ,1)dξ} = ε̃_{1}e^{∫_{−ε^{∗}_{1}}^{0} Φ_{0}(ξ,1)dξ}.

From here

\frac{d}{d^{y}} = (1 + o(ε_{1})e^{− ∫_{−ε^{∗}_{1}}^{ε^{∗}_{1}} Φ_{0}(ξ,1)dξ}

Limits in the integral in the exponent are not symmetric, because the quantity

ε_{1}^{∗} = ε^{− \frac{n}{m}} (1 + o(ε_{1}))  (7.32)

contains o - small, which for the reflected vector field can be differ from the analogous quantity for the initial vector field.

Therefore

\frac{d}{d^{y}} = (1 + o(ε_{1}))e^{− ∫_{−ε^{∗}_{1}}^{ε^{∗}_{1}} Φ_{0}(ξ,1)dξ}e^{∫_{I(ε)} Φ_{0}(ξ,1)dξ},  (7.33)

where the limits in the integral in the first exponent are symmetric, and both of the ends of the segment I(ε) have the asymptotics (7.32). Because from Γ - nondegeneracy conditions

Φ_{0}(ξ, 1) = O(\frac{1}{ξ^{k}}), k ≥ 1,  (7.34)

as ξ → ∞, then the integraal in the first exponent turns to the finite limit as ε → 0. From (7.34) we also obtain, that the integral in the second exponent in the formula (7.33) is O(ε^{\frac{n}{m} k})ε_{1}^{− \frac{n}{m}} o(ε_{1}) = o(ε_{1}). Hence the second exponent turns to 1 as ε → 0. From here the ratio \frac{d}{d^{y}} is bounded. Notice that if m is even the integrand in the first exponent (7.33) is odd and so

\frac{d}{d^{y}} = 1 + o(ε_{1}).  (7.35)

Analogously

\frac{d^{x}}{d^{xy}} = 1 + o(ε_{1}).  (7.36)

Because the ratio \frac{d}{d^{y}} is bounded we have c_{1} → 0 as ε → 0. Analogously c_{1}^{x} → 0 as ε → 0. From here and because F_{2} is undependent on ε we obtain that F_{2} = 0.

Let n = 1.

Because in this case m is even, that from (7.30),(7.28), (7.29), (7.35), (7.36) we obtain that

F_{2} = \frac{d̃_{1}^{y}}{d̃} − \frac{d̃_{1}}{d̃} + o(ε_{1})\frac{d̃_{1}^{y}}{d̃} − (\frac{d̃_{1}^{xy}}{d̃^{x}} − \frac{d̃_{1}^{x}}{d̃^{x}} + o(ε_{1})\frac{d̃_{1}^{xy}}{d̃^{x}} ) .  (7.37)

Because m̃ is even, that from [3]

Φ̃_{0}(−x, y) = Φ̃_{0}(x, y), Φ̃_{1}(−x, y) = −Φ̃_{1}(x, y).  (7.38)

From here and from lemma 3.1 we obtain, that

d̃^{y} = ε̃_{2}e^{− ∫_{0}^{\frac{1}{ε}} Φ̃_{0}^{y}(1,ξ)dξ} = ε̃_{2}e^{− ∫_{0}^{\frac{1}{ε}} Φ̃_{0}(−1,ξ)dξ} = ε̃_{2}e^{− ∫_{0}^{\frac{1}{ε}} Φ̃_{0}(1,ξ)dξ} = d̃(1 + o(ε_{2})).

Analogously

\frac{d̃_{1}^{y}}{d̃} = \frac{d̃_{1}^{y}}{d̃^{y}} \frac{d̃^{y}}{d̃} = −\frac{d̃_{1}}{d̃} (1 + o(ε_{2})) + \frac{O(ε_{2})}{d̃}.  (7.39)

Analogously

\frac{d̃_{1}^{xy}}{d̃^{x}} = −\frac{d̃_{1}^{x}}{d̃^{x}} (1 + o(ε_{2})) + \frac{O(ε_{2})}{d̃^{x}}.

From here, from (7.39) and (7.37) we obtain, that

\begin{matrix}F_{2} = c_{1} − c_{1}^{x} = 2(\frac{d̃_{1}^{x}}{d̃^{x}} − \frac{d̃_{1}}{d̃}) + (o(ε_{1}) + o(ε_{2}))\frac{d̃_{1}}{d̃} + (o(ε_{1}) + o(ε_{2}))\frac{d̃_{1}^{x}}{d̃} \\ +\frac{O(ε_{2})}{d̃} + \frac{O(ε_{2})}{d̃^{x}}. \end{matrix}  (7.40)

Investigate the asymptotics of the integral in the formula (6.22) for the quantity \frac{d̃_{1}}{d̃} . The edge with index α̃ is situated on the line l : ñi + m̃j = d̃_{0}, the edge with index α is situated on the line ni + mj = d_{0}.

Let (i_{0}, j_{0}) be the coordinates of the vertex, joining these edges. Then

ni_{0} + mj_{0} = d_{0}, ñi_{0} + m̃j_{0} = d̃_{0}.

Consider the right line l_{1}, passing throw the point (i_{0} −m, j_{0} +n) in parallel to l. Then l_{1} has the equation ñi + m̃j = d and ñ(i_{0} − m) + m̃(j_{0} + n) = d. From here d − d̃_{0} = r > 1. Therefore the supports of the functions X̃_{1} and Ỹ_{1} do not lie on the right line l_{1}, hence they lie lower. Because n = 1, then the upper points of these supports lie not above the horizontal line j = j_{0}. So the powers of the polynomials X̃_{1}(1, ξ) and Ỹ_{1}(1, ξ) are not greater than the power j_{0} of polynomials X̃_{0}, Ỹ_{0}, F̃_{0}. Therefore Φ̃_{1}(1, ξ) = O(\frac{1}{ξ^{m}}), m ≥ 1 as ξ → ∞.

From here and from the formula (7.36) we obtain, that the integrand in (6.22) is O(\frac{1}{ξ^{l−b_{0}}} ). So from the condition b_{0} < 0 the integral in the formula (6.22) turns to the finite limit as ε → 0.

We proved early, that two last terms in (7.40) turn to 0 as ε → 0. From here and because the quantity F_{2} does not depend on ε we conclude that

F_{2} = lim_{ε→0} 2(\frac{d̃_{1}^{x}}{d̃^{x}} − \frac{d̃_{1}}{d̃}).  (7.41)

Continue the proof of the theorem. Twice changing the variable on the opposite one we obtain

\begin{matrix}\frac{d̃_{1}^{x}}{d̃^{x}} = −(1 + o(ε_{2})) ∫_{0}^{\frac{1}{ε}} Φ̃_{1}^{x}(1, ξ)e^{∫_{0}^{ξ} Φ̃_{0}^{x}(1,τ)dτ}dξ + \frac{O(ε_{2})}{d̃^{x}} = \\ (1 + o(ε_{2})) ∫_{0}^{\frac{1}{ε}} Φ̃_{1}(1, −ξ)e^{− ∫_{0}^{ξ} Φ̃_{0}(1,−τ)dτ}dξ + \frac{O(ε_{2})}{d̃^{x}} = \\ (1 + o(ε_{2})) ∫_{− \frac{1}{ε}}^{0} Φ̃_{1}(1, ξ)e^{∫_{0}^{ξ} Φ̃_{0}(1,τ)dτ}dξ + \frac{O(ε_{2})}{d̃^{x}}. \end{matrix}  (7.42)

From here, from (6.22) and (7.41) we obtain

F_{2} = 2 lim_{ε→0} (\frac{d̃_{1}^{x}}{d̃^{x}} − \frac{d̃_{1}}{d̃} ) =

2 lim_{ε→0} ( ∫_{− \frac{1}{ε}}^{\frac{1}{ε}} Φ̃_{1}(1, ξ)e^{∫_{0}^{ξ} Φ̃_{0}(1,τ)dτ}dξ + \frac{O(ε_{2})}{d̃} + \frac{O(ε_{2})}{d̃^{x}} ) = 2 ∫_{−∞}^{∞} Φ̃_{1}(1, ξ)e^{∫_{0}^{ξ} Φ̃_{0}(1,τ)dτ}dξ.

Theorem 2 is proved.

References

[1] V.I.Arnol’d and Yu.S.Il’yashenko, Ordinary differential equations. 1. in: ”Sovremennye problemy matematiki. Fundamental’nye napravleniya, Vol.1, Itogi Nauki i Tekhniki, VINITI AN SSSR, Moscow, 1985, 7-149 (English: Encyclopaedia of Mathematical Science, Vol.1, Springer, Heidelberg).

[2] N.B.Medvedeva, The principal term of the first return function of a monodromic singular point is linear, Sibirskii matematicheskii zhurnal. 1992, Vol.33, No.2 p.116-124.

[3] F.S.Berezovskaja and N.B.Medvedeva, A complicated Singular point of ”Centerfocus” type and the Newton diagram, Selecta Mathematica Vol.13, No 1, 1994, p.1-15.

[4] Yu.S.Il’yashenko, The memoir of Dulak ”About the limit cycles” and the ajacent questions of the local theory of the differential equations, Uspekhi Mat. Nauk 40(1985), p.41-78.

[5] Chen K.T. Equivalence and decomposition of vector fields about an elementary critical point. - Amer.J. Math., 1963, v. 85, N4, p.693 –722.