Algebraic curves of maximal cyclicity

Math. Proc. Camb. Phil. Soc. (2006), 140, 47 c© 2006 Cambridge Philosophical Society

doi:10.1017/S0305004105008807 Printed in the United Kingdom

47

Algebraic curves of maximal cyclicity

By MAGDALENA CAUBERGH and FREDDY DUMORTIER

Limburgs Universitair Centrum, Universitaire Campus, 3590 Diepenbeek, Belgium.

e-mail: [email protected] and [email protected]

(Received 26 April 2004; revised 7 September 2004)

Abstract

The paper deals with analytic families of planar vector fields, studying methods to

detect the cyclicity of a non-isolated closed orbit, i.e. the maximum number of limit

cycles that can locally bifurcate from it. It is known that this multi-parameter prob-

lem can be reduced to a single-parameter one, in the sense that there exist analytic

curves in parameter space along which the maximal cyclicity can be attained. In

that case one speaks about a maximal cyclicity curve (mcc) in case only the number

is considered and of a maximal multiplicity curve (mmc) in case the multiplicity is

also taken into account. In view of obtaining efficient algorithms for detecting the

cyclicity, we investigate whether such mcc or mmc can be algebraic or even linear

depending on certain general properties of the families or of their associated Bautin

ideal. In any case by well chosen examples we show that prudence is appropriate.

1. Introduction

1·1. Motivation and description of the problem

Hilbert’s sixteenth problem essentially asks for the maximum number of limit

cycles (isolated periodic orbits) in a polynomial vector field, depending uniformly on

the degree. The finiteness part of Hilbert’s 16th problem consists in proving that such

a finite global bound exists. One way to handle this problem is to prove the existence of

local upperbounds in analytic systems [9]. In this way, a global problem is transferred

into local problems: the problem of bounding the maximum number of limit cycles

γ, that can arise after small perturbations of Xλ0 , in small neighbourhoods of a limitperiodic set Γ, inside a given analytic p-parameter family of planar vector fields(Xλ )λ , λ∼λ0 [9]. This number is called the cyclicity of (Xλ )λ along Γ for λ=λ0, andit is denoted by Cycl(Xλ , (Γ, λ0)) or Cycl. Observe that in this paper, we essentiallyuse the common notion of cyclicity, in which one does not count the multiplicity of

the limit cycles. When the multiplicity is taken into account, we will refer to this

number by multiplicity and it will shortly be denoted by Mult(Xλ , (Γ, λ0)) or Mult.A more precise definition of this notion is given below in Section 1·2.An effective tool in this approach is the Poincare, or first-return mapping Pλ ,

and the associated displacement map δλ = Pλ − Id. In this way, limit cycles of Xλ

correspond to isolated zeroes of δλ . In this paper, as in [2], the limit periodic setΓ is supposed to be a regular non-isolated periodic orbit of Xλ0 . We say that Xλ0

www.DownloadPaper.irwww.DownloadPaper.ir

http://www.DownloadPaper.ir

https://www.researchgate.net/publication/251599023_Configurations_of_zeroes_of_analytic_functions?el=1_x_8&enrichId=rgreq-ae07d5b0-12d9-40aa-8b4c-fca56f4225f5&enrichSource=Y292ZXJQYWdlOzIzMTgyMDQ0NztBUzoxMDIxNjk2MzQxNDgzNjVAMTQwMTM3MDU0ODQzNw==

48 M. Caubergh and F. Dumortier

is of center type. In this case, the family of maps (δλ )λ is analytic with δλ0 ≡ 0 andchecking properties on the derivatives of this map leads to a good understanding

of the bifurcation diagram of the local (i.e. close to Γ) limit cycles of Xλ in terms

of λ (λ∼λ0). However, in most concrete situations, finding an explicit expressionor even computing derivatives of the displacement map, remains a tough problem,

especially when a multi-dimensional parameter is involved. In 1-dimensional para-

meter families, there is the well-known technique of computing Melnikov functions

[5, 8]. The aim of this paper is to study in which way multi-parameter bifurcation

problems from a non-isolated periodic orbit, can be reduced to a study of specific

1-parameter subfamilies. Can such 1-parameter subfamilies be taken algebraic or

even linear, depending on the kind of problem one treats and the kind of family one

works with? We describe a number of results either confirming existence or provid-

ing counterexamples. In [2], it was already proven that there exist analytic curves

ζc , ζm : I →Rp through λ0 (i.e. ζc(0) = ζm (0) =λ0) such that

Cycl(Xλ , (Γ, λ0)) = Cycl(Xζc (ε), (Γ, 0)) (1·1)

Mult(Xλ , (Γ, λ0)) = Mult(Xζm (ε), (Γ, 0)). (1·2)

An analytic curve ζ(ε) having property (1·1) (respectively (1·2)) is called a curve ofmaximal cyclicity (mcc) (respectively a curve of maximal multiplicity (mmc)).

Throughout the paper we will not really make an explicit distinction between the

parametrization γ: I →Rn and the image γ(I). Neither will we always specify I sincewe are essentially interested in germs. In any case “analytic curve” will mean that

γ: I →Rn is analytic, and the same holds for “algebraic curve” and “linear curve”,

in the sense that the components of γ are respectively polynomial and affine.The problem is interesting, since, in studying bifurcations of vector fields of center

type (e.g. Hamiltonian ones), the technique of Melnikov functions is generally used.

Often one only computes Melnikov functions for 1-parameter subfamilies, sometimes

merely induced by straight lines in parameter space. In this paper, it becomes clear

that one needs more information than only these computations to derive the right

conclusions about the cyclicity of the whole family.

Besides general analytic families of planar vector fields (Xλ )λ , our attention goes

to the following special cases: algebraic systems, i.e.

X0 +

i1+...+ip =N∑

i1+...+ip =1

λi11 . . . λip

p Xi1...ip, (1·3)

and linear systems, i.e.

Xλ = X0 + λ1X1 + · · · + λpXp , (1·4)

where X0 is a vector field of center type; attention also goes to these kind of families

with a regular or principal Bautin ideal.

1·2. Displacement map, Bautin ideal and Melnikov functions

Let us recall some notions and properties of the displacement map, essentially

based on [9]. As mentioned in Section 1·1, we suppose (Xλ )λ to be an analytic

family of planar vector fields that unfolds a vector field of center type Xλ0 , andΓ is a regular non-isolated periodic orbit of Xλ0 . We suppose that Σ is a section




https://www.researchgate.net/publication/231900533_Successive_derivatives_of_a_first_return_map_application_to_the_study_of_quadratic_vector_fields?el=1_x_8&enrichId=rgreq-ae07d5b0-12d9-40aa-8b4c-fca56f4225f5&enrichSource=Y292ZXJQYWdlOzIzMTgyMDQ0NztBUzoxMDIxNjk2MzQxNDgzNjVAMTQwMTM3MDU0ODQzNw==

Algebraic curves of maximal cyclicity 49

transverse to Γ, and let s be a regular analytic parameter on Σ. The periodic or-bit Γ will be represented by s0, the point of intersection of Σ and Γ. Suppose thatP : Σ×W ⊂R×R

p →Σ: (s, λ)→Pλ (s) is the analytic family of first-returnmaps (alsocalled Poincare maps) Pλ : Σ→Σ, for λ ∈ W, where W is an open neighbourhood of

λ0 in Rp . The analytic family of displacement maps is then defined by δ: Σ×W ⊂R×R

p →R: (s, λ)→ δλ (s) =Pλ (s)− s. Obviously, periodic orbits of Xλ are represen-

ted by roots of δλ . Now, we can express the cyclicity and multiplicity of the familyof planar vector fields in terms of the family of displacement maps:

Cycl(Xλ , (Γ, λ0)) = lim supλ→λ0,s→s0

{number of isolated zeros s of δλ}

and

Mult(Xλ , (Γ, λ0)) = lim supλ→λ0,s→s0

{number of isolated zeros s of δλ}counted with their multiplicity

.

Expanding δλ in a Taylor series around s0, defines a sequence of analytic functionsα0, α1, . . . , αn , . . . in a neighbourhood of λ0:

δλ (s) =

∞∑

i=0

αi (λ) (s − s0)i , s ∼ s0.

The ideal I, generated by the germs of the analytic functions αi at λ0, is called theBautin ideal. Since the local ring Oλ0 of analytic function germs at λ0 is Noetherian[7], this ideal is finitely generated. Then, by Nakayama’s lemma [7], there exists

a minimal set of generators that is adapted at s0, say ϕ1, . . . , ϕl . This means that{ϕ1, . . . , ϕl}modMI is a basis of the real vector space I/MI, whereM denotes the

maximal ideal of Oλ0 ; in this case, we define the dimension of the Bautin ideal as:

dim I = l. Moreover, δ can locally be divided in the Bautin ideal, as follows:

δ(s, λ) =

l∑

i=1

hi(s, λ)ϕi(λ), s ∼ s0, λ ∼ λ0

for certain analytic functions hi(s, λ), such that the “factor functions” Hi are

uniquely and globally defined by Hi(s) =hi(s, λ0), i=1, . . . , l with strictly increas-

ing order at s0: orderH1(s0)< orderH2(s0)< · · · < orderHl(s0). Now it follows by adivision-derivation algorithm, based on Rolle’s theorem, that

Cycl(Xλ , (Γ, λ0)) � Mult(Xλ , (Γ, λ0)) � orderHl(s0) = sδ (s0). (1·5)

Let us remark that the numbers in (1·5) are possibly all distinct, as shown e.g., bythe family in (2·2) with specification (2·3). In [9], orderHl(s0) is called the index ofthe Bautin ideal at s0 and it is denoted by sδ (s0) or Index(Xλ , (Γ, λ0)). This indexonly depends on the displacement map δ and s0. Let us recall that an ideal is said tobe regular if it has a regular set of generators; an ideal is said to be principal if it can

be generated by only one function. If the Bautin ideal is regular, we also have:

dim I − 1 � Cycl(Xλ , (Γ, λ0)) � Mult(Xλ , (Γ, λ0)) � Index(Xλ , (Γ, λ0)). (1·6)



https://www.researchgate.net/publication/50333927_Ideals_of_Differentiable_Functions?el=1_x_8&enrichId=rgreq-ae07d5b0-12d9-40aa-8b4c-fca56f4225f5&enrichSource=Y292ZXJQYWdlOzIzMTgyMDQ0NztBUzoxMDIxNjk2MzQxNDgzNjVAMTQwMTM3MDU0ODQzNw==



If we deal with a 1-dimensional parameter family (Xε)ε∈R of vector fields (ε∼ 0),then the displacement map can be expanded in terms of ε:

δ(s, ε) =

∞∑

i=1

Mi(s)εi , s ∼ s0, ε ∼ 0,

where the coefficients Mi are analytic in s, i∈N�. The function Mi is called the ithMelnikov function (i∈N�). If Mk is the first non-identical zero Melnikov function,

i.e.

Mi ≡ 0, ∀1 � i < k and Mk � 0

then the Bautin ideal is principal; more precisely, in this case the Bautin ideal is

generated by the germ of the map ε �→ εk at ε=0. In particular, the correspondingfactor function is given by Mk ; as a consequence, Index(Xε , (Γ, 0)) = orderMk (s0).Algorithms to compute the first non-vanishing Melnikov function can be found in

[5, 8]. Using these algorithms, the index of 1-parameter subfamilies can be computed;

therefore, we also investigate the existence of curves in parameter space such that

the index of the induced 1-dimensional parameter family is equal to the index of the

p-parameter family. In [10], using the desingularisation theory of Hironaka, it wasproven by R. Roussarie, that there exists an analytic curve ζ with

Index(Xλ , (Γ, λ0)) = Index(

Xζ (ε), (Γ, 0))

. (1·7)

An analytic curve ζ(ε) having property (1·7) is called a curve of maximal index (mic).

1·3. Organisation of the paper and summary of the results

First, we investigate the existence of a linear, algebraic mcc, mmc and mic, in

analytic, algebraic and linear families. Let us give a summary of the most important

results. In a general analytic system, there always exists an algebraic mic. In case of

a regular or principal Bautin Ideal, there even exists a linear mic. Furthermore,

Linear system Algebraic system Analytic system

Linear mic

Linear mmc

Linear mcc

Algebraic mmc

Algebraic mcc

I II III

yes yes

no no

I II III

no yes yes

no no no

no no no

I II III

no yes yes

no no no

no no no

no no no

no no no

In this table, I, II and III correspond to respectively the general case, the case of a

regular Bautin ideal and the case of a principal Bautin ideal. The word ‘yes’ means

‘always exists’, while ‘no’ means ‘does not always exist’. In case of a blank, the

question of existence still remains open. Proofs and counterexamples are provided in

Sections 2 to 5. Important to observe is that even in a linear system there does not

need to exist a linear mcc. Counterexamples even exist if the Bautin ideal is regular.

In Section 6, we determine a condition that guarantees the existence of algebraic

mcc’s and mmc’s. In Section 6·1, two useful specifications of the curve selectionlemma for open subanalytic sets are derived. Recall that the curve selection lemma

ensures the existence of an analytic curve ζ entering the considered subanalytic set V.By use of the �Lojasiewicz inequality (to ensure that certain closed subanalytic stay



https://www.researchgate.net/publication/231900533_Successive_derivatives_of_a_first_return_map_application_to_the_study_of_quadratic_vector_fields?el=1_x_8&enrichId=rgreq-ae07d5b0-12d9-40aa-8b4c-fca56f4225f5&enrichSource=Y292ZXJQYWdlOzIzMTgyMDQ0NztBUzoxMDIxNjk2MzQxNDgzNjVAMTQwMTM3MDU0ODQzNw==

https://www.researchgate.net/publication/226610219_Melnikov_functions_and_Bautin_ideal?el=1_x_8&enrichId=rgreq-ae07d5b0-12d9-40aa-8b4c-fca56f4225f5&enrichSource=Y292ZXJQYWdlOzIzMTgyMDQ0NztBUzoxMDIxNjk2MzQxNDgzNjVAMTQwMTM3MDU0ODQzNw==


at some distance from each other), we prove that if the subanalytic set V is open,

there exists an (analytic) ‘cone of curves surrounding ζ’ entering V ; for a precisedefinition see Section 6·1. If the family (Xλ )λ has an mc-stratum (respectively mm-

stratum) with a non-empty interior at λ0, then we can construct an open subanalyticsetW , such that curves enteringW at λ0 are mcc’s (respectively mmc’s). The precisedefinition of this condition can be found in Section 6·2.Under this condition, the specifications of the curve selection lemma for open

subanalytic sets, described above, imply the existence of algebraic mcc’s (respectively

mmc’s). This is the main result of Section 6·3 (respectively Section 6·4).If the family (Xλ )λ does not satisfy this condition, then we do not yet know

whether an algebraic mcc (respectively mmc) always exists. In the algebraic example

(2·2) with specification (2·10), there is an algebraic mcc (respectively mmc) present.Moreover, for the moment we don’t have an example of a linear family that does not

satisfy the condition.

In Section 7 we discuss some extra problems, such as the problem of the minimal

degree of an algebraic mcc (resp. mmc) for certain specific families. However, we do

not know whether a uniform bound exists only depending on the degree of algeb-

raicity in λ. Yet we still have one result for analytic families of planar vector fieldswith an l-dimensional regular Bautin ideal and Index equal to l− 1. In that case, weprove the existence of an algebraic mcc (resp. mmc) of degree � [(l+2)/2], and everyanalytic curve with the same [(l+2)/2]-jet enters the mc-stratum (resp. mm-stratum)at λ0 (precise conditions on the Bautin ideal are formulated in Theorem 7·7). As aconsequence, for a family with a 2-dimensional regular Bautin ideal of index 1, thereexists a linear mcc (mmc).

2. Elaboration of examples

In this section we elaborate a few cases that will serve as counterexamples needed

to prove a number of statements throughout the rest of the paper. We first give a

proposition that enables to generate a lot of examples with or without a linear (or

algebraic) mic, mcc or mmc in algebraic families (Xλ )λ . The proof is a straightforwardapplication of the division-derivation algorithm; we leave it to the reader.

Proposition 2·1. Suppose that (δλ : I →R)λ∈W is an analytic family of displacement

maps associated to the family (Xλ )λ with δλ0 ≡ 0, where I is a neighbourhood of s0in R and W is a neighbourhood of λ0 in Rp . Suppose that h1, . . . , hl : I ×W →R,

ϕ= (ϕ1, . . . , ϕl) : W →Rl are analytic functions such that

δ(s, λ) =

l∑

i=1

ϕi(λ)hi(s, λ), ∀(s, λ) ∈ I × W

with orderH1(s0)< orderH2(s0)< . . . < orderHl(s0), for Hi ≡hi(·, λ0), 1� i� l. Define

V i = {λ ∈ W : |ϕj (λ)|� |ϕi(λ)|, ∀1� j � l}, ∀1� i� l. Then:(i) Index((Xλ )λ∈V l , (Γ, λ0)) = orderHi(s0), ∀1� i� l. Furthermore, if V i (1� i� l)contains the germ of an analytic curve ζ at λ0, then

Index((

Xζ (ε)

)

ε, (Γ, 0)

)

= orderHi(s0).

In particular, if V l contains the germ of an analytic curve ζ at λ0, then ζ is anmic;




(ii) let k ∈ N \ {0} . If V l contains the germ of an algebraic curve of degree at most kat λ0, then there exists an algebraic mic of degree � k;

(iii) let k∈N \ {0}. Suppose that 1� l0 < l and that V i does not contain the germ of

an algebraic curve of degree � k at λ0, ∀i > l0, then(a) sup ζ Index((Xζ (ε))ε , (Γ, 0))� orderHl0(s0), where the sup is taken over all

algebraic curves ζ through λ0 of degree � k.(b) If k=1, then there does not exist a linear mic.(c) If moreover ϕ is a submersion at λ0 and orderHl0(s0)< l− 1, then there does

not exist an algebraic mcc or mmc of degree � k. In particular, for everyalgebraic curve ζ of degree � k through λ0,

Index((Xζ (ε))ε , (Γ, 0)) < Cycl(Xλ , (Γ, λ0)) � Mult(Xλ , (Γ, λ0)).

By Proposition 2·1, we can easily give examples of algebraic families without alinear mcc. Next proposition even enables us to construct linear families without a

linear mcc.

Proposition 2·2. Suppose that (δλ : I →R)λ∈W is an analytic family of displacement

maps associated to the family (Xλ )λ with δλ0 ≡ 0, where I is a neighbourhood of s0 in R

and W is a neighbourhood of λ0 in Rp . Suppose that

δ(s, λ) =

l∑

i=1

ϕi(λ)hi(s, λ), ∀(s, λ) ∈ I × W,

with l � 3, and up to reordering of the parameters λ1, . . . , λl:

ϕi(λ) = λicii(λ) + · · · + λ1c

i1(λ) for i = 1, . . . , l, (2·1)

where hi : I ×W →R and ci :W →R are analytic functions with

lim s→s0 hi(s, λ)/(s − s0)n i =1, for non-negative integers n1 < · · · < nl and ci

i(0)�0,

∀i=1, . . . , l. Then there does not exist a linear mcc.

Remark 2·3. Under the conditions of Proposition 2·2, the Bautin ideal is regular.

Proof. From (2·1) it follows that Cycl(Xλ , (Γ, λ0))� l− 1� 2. Now take any linear

curve in parameter space expressed by λ0+ελ for λ� 0 (λ0 = 0). Choose the smallestinteger j such that λj � 0, then we can write

δ(s, ελ) = ε(s − s0)n j

(

λjcjj (0) +O(s − s0)

)

, s → s0.

We clearly see that Cycl(Xελ , (Γ, 0))� 1< Cycl(Xλ , (Γ, 0)).

The rest of this section deals with analytic families of planar vector fields (Xλ )λ of

the form

Xλ =

(

y∂

∂x− x

∂

∂y

)

+ δ(s, λ)

(

x∂

∂x+ y

∂

∂y

)

, (2·2)

where the family δ : (R×Rp , (s0, λ0))→ R is analytic with δλ0 ≡ 0, s

2 = x2+y2, s ∼ 0,for a fixed s0 > 0, such that Γ = {s = s0}. By definition, the family (Xλ )λ is algebraic

(respectively linear) if and only if the family δ depends in a polynomial (respectivelylinear) way on λ. Up to a non-zero factor U , the map δ is the family of displacementmaps associated to (Xλ )λ :

δ(s, λ) = U (s, λ) · δ(s, λ) with U (s0, λ0)� 0.




Remark that the conditions of Propositions 2·1 and 2·2 can be checked by theanalogues on δ by the following observations. Multiplication of an analytic functionby a non-zero analytic function, respects the number of zeroes and the order of the

analytic function at s0. Furthermore, if we have a local division for δ, say

δ(s, λ) =

l∑

i=1

ϕi(λ)hi(s, λ),

that satisfies the conditions of Proposition 2·1 or 2·2, then the following local divisionfor δ also satisfies these conditions:

δ(s, λ) =

l∑

i=1

ϕi(λ)hi(s, λ)

with ϕi(λ) = ϕi(λ)/U (s0, λ) and hi(s, λ) =U (s0, λ)/U (s, λ)hi(s, λ), ∀1� i� l.Let us start by showing that the notions Cyclicity, Multiplicity and Index are not

equal. Let λ= ε∈R, λ0 =0 and

δ(s, ε) = εh(s, ε) = ε(s − (s0 + ε))2((s − s0)2 + ε2). (2·3)

Clearly, Cycl(Xε , (Γ, 0)) = 1 < Mult(Xε , (Γ, 0)) = 2 < Index(Xε , (Γ, 0)) = 4.Next, let us give a concrete example without a linear mic, mcc or mmc. Let λ0 = 0,

λ = (λ1, λ2) ∈ R2 and

δ(s, λ) = ϕ1(λ) + ϕ2(λ)(s − s0) =(

λ22 + λ41 − 2λ21λ2

)

+ λ61(s − s0). (2·4)

One can check easily that no linear curve through the origin enters the semi-algebraic

set V 2. By Proposition 2·1, there does not exist a linear mic. Furthermore,

supζ

Index(

(Xζ (ε))ε , (Γ, 0))

= 0 < 1 = Index(Xλ , (Γ, λ0)),

where the sup is taken over all linear curves through λ0. As a consequence, for everylinear curve ζ through λ0, we have

Cycl((

Xζ (ε)

)

ε, (Γ, 0)

)

= Mult((

Xζ (ε)

)

ε, (Γ, 0)

)

= Index((

Xζ (ε)

)

ε, (Γ, 0)

)

= 0.

Since Cycl((Xλ )λ , (Γ, λ0)) � 1 and Cycl((X(ε,ε2))ε , (Γ, λ0)) = 1, it follows that

Cycl(Xλ , (Γ, λ0)) = Mult(Xλ , (Γ, λ0)) = 1 > supζ

Index((

Xζ (ε)

)

ε, (Γ, 0)

)

= 0,

where sup is taken over all linear curves through λ0. In particular, there does notexist a linear mmc or mcc.

The following is a concrete example of a linear family without a linear mcc:

δ(s, λ) = λ1 + λ2s2 + λ3s

4, where λ = (λ1, λ2, λ3). (2·5)

By means of Taylor’s theorem one finds that δ(s, λ) =∑4

i=0 ϕi+1(λ)(s − s0)i with

ϕ1 (λ) = λ1 + λ2s20 + λ3s

40

ϕ2 (λ) = 2λ2s0 + 4λ3s30

ϕ3 (λ) = λ2 + 6λ3s20

ϕ4 (λ) = 4λ3s0ϕ5 (λ) = λ3.

(2·6)




From (2·6), it is seen that λ �→ (ϕ1(λ), ϕ2(λ), ϕ3(λ)) defines a linear coordinate trans-formation. As a consequence, the set {ϕ1, ϕ2, ϕ3} defines a minimal set of generatorsfor the associated Bautin ideal. By using the inverse transformation defined in (2·6),one can divide the map δ in ϕ1, ϕ2, ϕ3 as follows:

δ(s, λ) =3

∑

i=1

ϕi(λ)hi(s), (2·7)

with

h1 (s) = 1

h2 (s) = (s − s0)−12s20(s − s0)

3 − 18s20(s − s0)

4

h3 (s) = (s − s0)2 + 1

s0(s − s0)

3 + 14s20(s − s0)

4 .

(2·8)

Hence, (2·7) displays a division of the map δ as in Proposition 2·2, implying thatCycl(Xλ , (Γ, λ0)) = 2 and that there does not exist a linear mcc (ϕ1(ε), ϕ2(ε), ϕ3(ε)).Furthermore, since λ �→ (ϕ1(λ), ϕ2(λ), ϕ3(λ)) is a linear transformation, there neitherdoes exist a linear mcc (λ1(ε), λ2(ε), λ3(ε)).However, the curve λc(ε) = (− 5

8s0ε

2 + 14s20ε,

34s0

ε2 − 12ε,− 1

8s30ε2 + 1

4s20ε) is a quadratic

mcc. Indeed, λc(ε) corresponds to the quadratic curve (ϕc1(ε), ϕ

c2(ε), ϕ

c3(ε)) = (0, ε

2, ε),by (2·6). Then, by (2·7) and (2·8), the map δ(·, λc(ε)) reads for s→ s0 as: δ(s, λ

c(ε)) =ε(s − s0)[ε(1 +O(s − s0)

2) + ((s − s0) +O(s − s0)2)] ≡ ε(s − s0)f (s, ε). Now, it is clear

that, for ε � 0, zeroes of δ(·, λc(ε)) correspond to zeroes of f (·, ε) and s = s0. Sincef (s0, 0) = 0 and (∂f/∂ε)(s0, 0) = 1� 0, the implicit function theorem guarantees theexistence of an analytic curve s(ε), such that f (s(ε), ε) = 0, for all ε sufficiently smallbut positive, and s(0) = s0. In particular, f (s0, ε) � 0 if ε � 0; hence, s(ε) � s0 ifε� 0. As a consequence, the curve λc is an mcc.

Notice, however, that there exist a linear mmc and mic; for instance, we claim

that the linear curve defined by ζ(ε) = (λ1(ε), λ2(ε), λ3(ε)) = ε( 14s20,−

12, 14s20) is an mmc

and mic. By (2·6), the curve ζ corresponds to the linear curve (ϕ1(ε), ϕ2(ε), ϕ3(ε)) =(0, 0, ε). Now, our claim follows immediately since

δ(s, ζ(ε)) = ε(s − s0)2(1 +O(s − s0)), s −→ s0.

In fact, the existence of a linear mic is not surprising, since in this example the

Bautin ideal is regular. Although in this example there exists a linear mmc, there

exist algebraic families having a regular Bautin ideal without a linear mmc. Consider

for example, the 3-parameter family with, for λ = (λ1, λ2, λ3),

δ(s, λ) = (λ1 + λ2(s − s0) + λ3(s − s0)2)

(

(s − s0)2 +

(

λ1 − λ23)2)

. (2·9)

Then, the Bautin ideal is regular since it is generated by the germs at λ0 = 0 of themaps

ϕi : R3 −→ R : λ = (λ1, λ2, λ3) �−→ λi , i = 1, 2, 3.

Furthermore, it is clear thatMult(Xλ , (Γ, λ0)) � Index(Xλ , (Γ, λ0)) = 4.On the otherhand, the 1-parameter subfamily induced by ζ(ε) = (ε4, 2ε3, ε2), has multiplicity 4,since the map δ for this 1-parameter subfamily reduces to δ(s, ζ(ε)) = ε2(s − (s0 −ε))2(s− s0)

2.Hence,Mult(Xλ , (Γ, λ0)) = 4.However, for any linear curve R(ε) = λε =(λ1ε, λ2ε, λ3ε),

Mult(XR (ε), (Γ, 0)) < 4 = Mult(Xλ , (Γ, λ0)).




Indeed, if λ1 � 0, then Mult(XR (ε), (Γ, 0)) = 0. If λ1 = 0, then the map δ writesas: δ(s, R(ε)) = ε(s − s0)(λ2 + λ3(s − s0)) · ((s − s0)

2 + λ43ε4). In case λ3 � 0, then

Mult(XR (ε), (Γ, 0)) � 2. In case λ3 = 0 (to have isolated zeroes, it is then necessarythat λ2 � 0), the map δ reduces to: δ(s, R(ε)) = ελ2(s − s0)

3; hence, in this case,

Mult(XR (ε), (Γ, 0)) = 3.Let us now consider a family in which parameter values corresponding to maximal

cyclicity (as well as multiplicity) belong to a curve:

δ (s, λ) = λ2((s − s0)2 + (λ1 − g (λ2))

2) where λ = (λ1, λ2) ∈ R2, (2·10)

where g:R→R is an analytic function with g(0) = 0 and suppose that λ0 = 0. Noticethat the associated Bautin ideal is principal. Clearly,

Cycl(Xλ , (Γ, λ0)) = 1 and Mult(Xλ , (Γ, λ0)) = 2,

and the ‘stratum of maximal cyclicity’ is given by the graph λ1 = g(λ2). Moreover, ifζ is an mcc (mmc), then ζ necessarily is a parametrisation of the graph λ1 = g(λ2),by the definition of cyclicity (multiplicity) and the property that analytic functions

can only have isolated zeroes unless they are identically zero.

To end this section, we consider the family (Xλ )λ defined by (2·2) with

δ(s, λ) =(

λ21 − λ32)

+ λ32s, for λ = (λ1, λ2) ∈ R2. (2·11)

In this case the set of parameter values corresponding to maximal cyclicity (and

similar for maximal multiplicity) has a germ at λ0 which cannot be defined inde-pendently of the chosen domain in s-space. We say that the mc-stratum (respectivelymm-stratum) cannot be defined. The phenomenon will be considered in Section 6.

Let Γ = S1 be the unit circle and λ0 = (0, 0), then Cycl(Xλ , (Γ, λ0)) = 1. It is clear thatδ(·, λ) only has a non-isolated zero if λ2�0. If λ2�0, the zero of δ(·, λ) can be writtenas:

s(λ1, λ2) = 1−λ21λ32

.

As a consequence, ∀M > 0 and for every neighbourhood W of 0 in R2:

ZWM =

{

(λ1, λ2) ∈ W : |λ1| <√

M |λ2|3}

.

Hence, ZW0 =

⋂

M ↓0ZWM = {(λ1, λ2)∈W : λ1 =0}. Clearly, the curve γ(ε) = (0, ε) is an

algebraic (even a linear) mcc. Moreover, for every analytic curve γ : I ⊂R→R2 with

j1(γ − γ)(0) = 0, and for every M > 0, there exists E(M )> 0 such that γ(ε)∈ZM ,∀0< ε < E(M ). Hence, γ also is an mcc.

3. General case

3·1. Linear curves

From example (2·2) with (2·4), we immediately get:

Theorem 3·1.

(i) There are algebraic families without a linear mic, mmc or mcc.

(ii) Moreover, there are algebraic families with

Cycl(Xλ , (Γ, λ0)) > max{Index(Xλ0+λε , (Γ, 0)) : λ� 0}




and

Mult(Xλ , (Γ, λ0)) > max{Index(Xλ0+λε , (Γ, 0)) : λ� 0}.

From example (2·2) with (2·5), we immediately get:

Theorem 3·2. There are linear families without a linear mcc.

However, let us remark that, in the above mentioned example, there is a quadratic

mcc, a linear mic and a linear mmc present. In fact, in Section 4, we will show that

there always exists a linear mic, if the Bautin ideal is regular (as is clearly the case

here).

3·2. Algebraic curves

Theorem 3·3 ([10]). There always exists an algebraic mic.

Theorem 3·4. There are analytic p-parameter families of planar vector fields withoutan algebraic mmc or algebraic mcc.

Proof. Consider example (2·2) with (2·10) and g(λ2) = sin λ2. There is exactly onelimit cycle Γ= {s= s0} with multiplicity 2 for parameter values (λ1, λ2) on the graphof λ1 = sin λ2. For all other parameter values there are no limit cycles. The multi-plicity (and cyclicity) is only attained on the analytic curve ζ(ε) = (ε, sin ε), ε∈R.It is easy to show that there is no algebraic reparametrisation of this curve. There

does also not exist an algebraic mmc ζm or an algebraic mcc ζc . If there was such analgebraic curve ζm (respectively ζc), this would imply that ζm (respectively ζc) had

infinitely many intersections with the analytic curve ζ, accumulating on λ0, whichcan easily be proven to be impossible.

Remark that, in the example given in the proof, the mc-stratum (andmm-stratum)

is nowhere dense, since ∀M > 0, the interior of the subanalytic set ZM

ZM = {λ : ∃s ∈]R − M, R +M [ such that δ(s, λ) = 0}

= {(λ1, λ2) : λ1 = sin λ2}

is empty (i.e. ZM =�). In Section 6, it will be proven that there exists an algebraicmcc (repectively mmc) if the family (Xλ )λ has a ‘stratum of maximal cyclicity

(respectively multiplicity) with non-empty interior at λ0’.

4. Regular Bautin ideal

4·1. Linear curves

Theorem 4·1. If the Bautin ideal is regular, then there exists a linear mic ζ. Inparticular,

Cycl(Xλ , (Γ, λ0)) � Mult(Xλ , (Γ, λ0)) � Index(

Xζ (ε), (Γ, 0))

= orderM1(s0),

where M1 is the first Melnikov function of (Xζ (ε))ε∼0, with

ζ (ε) = λ0 + aε, a ∈ Rp , ‖a‖ = 1.

Proof. We can take a minimal set of generators for I, say {ϕ1, . . . , ϕl}, adaptedto s0, such that the differentials {Dϕ1(λ

0), . . . , Dϕl(λ0)} are linearly independent (cf.

[9]). Therefore, as observed in [9],

Cycl(Xλ , (Γ, λ0)) � Mult(Xλ , (Γ, λ0)) � Index(Xλ , (Γ, λ0)) = orderHl(s0).





Now we consider a linear curve ζ(ε) =λ0 + aε, a∈Rp . A displacement map for the

induced 1-parameter family (Xζ (ε))ε∼0 is given by δ(s, ζ(ε)) = εM1(s) +O(ε2), ε→ 0,

whereM1(s) =∑l

i=1 D(ϕi)λ0(a)Hi(s). Since I is regular, there exists a constant a∈Rp

with ‖a‖=1 such that M1(s) = cHl(s), for some c > 0. Hence, the linear curve

ζ(ε) =λ0 + aε is an mic.

From example (2·2) with (2·9), we have:

Theorem 4·2. There exist algebraic families having a regular Bautin ideal, withouta linear mmc.

From example (2·2) with (2·5), we get the following:

Theorem 4·3. There exist linear families having a regular Bautin ideal, without alinear mcc.


From example (2·2) with (2·10) and g(λ2) = sin λ2, we immediately get:

Theorem 4·4. There exist analytic families having a regular Bautin ideal, withoutan algebraic mcc, or an algebraic mmc.

4·3. Regularity of the Bautin ideal

From the proof of Theorem 4·1, we already know that the Bautin ideal can not beregular if all first order Melnikov functions, induced by a linear curve, are identical

zero. This result can be improved:

Theorem 4·5. The Bautin ideal is regular and its dimension equals the dimension ofthe parameter space if and only if for every linear curve in parameter space through 0,the first Melnikov function is not identically zero.

Proof. Take a minimal set of generators ϕ1, . . . , ϕl adapted to s0. By Taylor’stheorem, we can write, ∀j =1, . . . , l:

ϕj (λ) =

p∑

i=1

∂

∂λi

ϕj (0)λi +O(‖λ‖2), λ −→ 0.

Write the displacement function as:

δ(s, λ) =

l∑

j=1

ϕj (λ)hj (s, λ) =

l∑

j=1

p∑

i=1

∂

∂λi

ϕj (0)λiHj (s) +O(‖λ‖2), λ −→ 0,

where Hj ≡hj (·, 0), ∀j =1, . . . , l. Take a linear curve through 0, λ= ελ, with λ�0,then we have δ(s, ελ) = ε

∑lj=1

∑pi=1(∂/∂λi)ϕj (0)λiHj (s) +O(ε2), ε→ 0. Therefore, the

first order Melnikov function is defined as:M1(s, λ) =∑l

j=1

∑pi=1(∂/∂λi)ϕj (0)λiHj (s).

If-part. Since the factor functions Hj have a strictly increasing order at s= s0,the first order Melnikov function M1(·, λ) is not identically zero if there exists aninteger j ∈{1, . . . , l} such that

∑pi=1(∂/∂λi)ϕj (0)λi�0. Therefore, the fact that for

every λ�0, the first order Melnikov function M1(·, λ) is not identical to zero, isequivalent to the fact that the map ϕ= (ϕ1, . . . , ϕl) is an immersion at 0. Since the

set of generators is minimal, this fact implies that l= p; as a consequence, the Bautinideal is regular.




Only if-part. We take l= p: suppose to the contrary that M1(·, λ)≡ 0, for someλ� 0. Since the factor functions are independent over R, it follows that

∀j = 1, . . . , l :

p∑

i=1

∂

∂λi

ϕj (0)λi = 0.

This is impossible since the matrix of the linear system is non-singular.

5. Principal Bautin ideal

5·1. Linear curves

Theorem 5·1. If the Bautin ideal I is principal, then there exists a linear mic.

Proof. Let ϕ be a generator for I. We can write, for a certain k � 1,

ϕ(λ) = Pk (λ − λ0) +Rk (λ),

where Pk is a homogeneous polynomial in (λ−λ0) of degree k, and

Rk (λ) = O(‖λ − λ0‖k+1), λ −→ λ0.

Hence, the displacement function can locally be written as

δ(s, λ) = ϕ(λ)h(s, λ) = (Pk (λ − λ0) +Rk (λ))h(s, λ).

For a linear curve γ(ε) =λ0 + aε, the displacement function of (Xγ (ε))ε gets:

δ(s, γ(ε)) = εkPk (a)H(s) +O(εk+1), ε −→ 0.

We can choose a ∈ Rp with ‖a‖ = 1 and Pk (a)� 0; then Mk (s) = Pk (a)H(s), and therequired result follows.

From example (2·2) with (2·10) and g(λ2) =λ22, we have:

Theorem 5·2. There exist algebraic families with a principal Bautin ideal, without alinear mmc or a linear mcc.


Example (2·2) with (2·10) and g(λ2) = sin λ2 induces the following result:

Theorem 5·3. There exist analytic families with a principal Bautin ideal, without analgebraic mmc or an algebraic mcc.

6. Open subanalytic sets and algebraic curves

The aim of this section is to prove the existence of an algebraic mcc ζc (respectively

mmc ζm ) for analytic families of planar vector fields ‘having a stratum of maximal

cyclicity (respectively stratum of maximal multiplicity) with non-empty interior at

λ0’. In Section 6·2, one can find a precise definition of this condition.First, we derive an interesting specification of the curve selection lemma for open

subanalytic sets (Section 6·1), implying the existence of an algebraic curve enteringa given subanalytic set. The results and proofs of this section are purely situated in

the theory of analytic geometry.

Next, we apply these results to analytic families of vector fields ‘having a stratum

of maximal cyclicity (respectively multiplicity) with non-empty interior at λ0’. Since




the stratum of maximal cyclicity (respectively multiplicity) is not necessarily a sub-

analytic set, the required result cannot be proven in a straightforward way.

In Section 6·3 (respectively Section 6·4), we first construct an appropriate opensubanalytic set W ; then we apply the specified curve selection lemma to this open

subanalytic set W, proving the required results.

6·1. Algebraic curves and determining jets

Essentially we will rely on the following result:

Theorem 6·1. For any open subanalytic set V ⊂Rp , that accumulates at λ0 � V , therealways exists an algebraic curve γ,

γ(t) = (P1(t), . . . , Pp (t)), γ(0) = λ0,

where P1, . . . , Pp are polynomials in t, such that the curve γ(t) lies in V for all t > 0small enough.

This theorem is clearly a consequence of the curve selection lemma (Lemma 6·4)and:

Theorem 6·2. Let V be an open subanalytic set in Rp , that accumulates on λ0, and letγ be an analytic curve that starts at λ0 (i.e. γ(0) =λ0). Suppose that γ(t) ∈ V \ {λ0}, forall t > 0 small enough. Then there exists a positive integer n such that for every analyticcurve γ with jn (γ − γ)(0) = 0, we also have

γ(t) ∈ V \ {λ0} for t > 0 sufficiently small.

Remark 6·3. Let I be a regular ideal in the local ring of analytic function germs,(Rp , λ0) → R. Let ϕ1, . . . , ϕl be a minimal set of generators for I. Let h1, . . . , hl be

analytic functions such that for Hi ≡ hi(·, λ0):

orderHi(s0) = i − 1, i = 1, . . . , l. (6·1)

Let f be the analytic function defined by

f (s, λ) =

l∑

i=1

ϕi (λ)hi (s, λ) ,

then there exists a neighbourhood W of λ0, and a neighbourhood V of s0 in R such

that

Zf = {λ ∈ W : f has exactly l − 1 zeroes in V }

is a non-empty open subanalytic set, that accumulates at λ0. (From (6·1) it followsthat the functions H1, . . . , Hl form a Chebychev system, and therefore the set Zf

is non-empty and accumulates at λ0. By Rolle’s theorem there exist W1, V1 suchthat the zeroes are simple, and then by the implicit function theorem there exist

W ⊂W1, V ⊂V1 such that Zf is open.)

Probably Theorem 6·2 is known by specialists in analytic geometry, but we havenot found it in the literature. In this paper we will give a quick sketch of how to prove

it. A detailed elaboration can be found in [3]. Let us first recall the curve selection

lemma for subanalytic sets (for a proof see [1, 4 and 6]), and state a proposition

which is the key step in the proof of Theorem 6·2.




Lemma 6·4 (Curve selection lemma for subanalytic sets). Suppose that V is a sub-

analytic set in Rp and λ0 is an accumulation point of V , then there exists an analyticcurve γ: [0, 1]→Rp such that γ(]0, 1])⊂V and γ(0) =λ0.

Let B(0, 1) = {µ ∈ Rp−1 : ‖µ‖� 1} and Cn = {(λ, µ)∈ ]0, 1]× B1(0) : ‖µ‖� λn},n∈N.

Proposition 6·5. Let V be an open subanalytic set, V ⊂ Rp such that

(]0, 1]× {0}) � ({1} × B(0, 1)) ⊂ V.

Then, there exists an integer n∈N1 with Cn ⊂V.

Proof. By a compactness-argument we clearly see that for any 0< r1 < 1, theremust exist n∈N1 with the property that

Cn � {(λ, µ) ∈ Rp : r1 � λ � 1} ⊂ V.

Hence, the problem is concentrated near the origin. As such, if we suppose that no

Cn ⊂V, we do not only find a sequence (λn , µn )∈Cn\V, but we also may assume thatsome subsequence (λnk

) tends to 0 for k→∞. Since V c and [0, 1]×{0} are closedsubanalytic sets, they are regularly situated (cf. [6]); it means that there exist a

neighbourhood W of (0, 0) in Rp , and positive constants C, r such that ∀(λ, µ)∈W :

d((λ, µ), V c) + d((λ, µ), [0, 1]× {0}) � Cd((λ, µ), (0, 0))r .

Since (λnk, µnk

)→ (0, 0), there is a positive integer N ∈N such that ∀k � N :(λnk

, µnk)∈W �V c and λnk −r

nk< C. Therefore, ∀k � N :

‖µnk‖ = d((λnk

, µnk), [0, 1]× {0}) � C‖(λnk

, µnk)‖r

� Cλrnk

> λnknk

.

This is in contradiction with (λnk, µnk

) ∈ Cnk.

Proof of Theorem 6·2 Without loss of generality we can assume that λ0 = 0 andthat V lies entirely in the first quadrant I:

I = {(λ1, . . . , λp ) ∈ Rp : λi > 0, ∀i = 1, . . . , p}.

It is clear that every non-constant analytic curve γ, like in Lemma 6·4, enteringthe first quadrant, has a so-called “Puiseux expansion” [1] of the form

τ �−→ (τ k , ϕ2(τ ), . . . , ϕp (τ )), (6·2)

for a certain k∈N, where τ =h(t) is an analytic regular reparametrisation withh(0) = 0, and ϕj is analytic in τ, for τ close to 0 (∀j =2, . . . , p).

We now treat three different cases depending on specific properties of γ. At firstwe suppose that γ is a regular parametrisation of a straight line. This case is a directconsequence of Proposition 6·5. If the analytic curve is regular, but not straight,then we can straighten it by an analytic coordinate change, clearly leading to the

required result. To treat a non-regular analytic curve, with a Puiseux expansion

(6·2), we perform a blow-up map Φ defined by

Φ(µ1, µ2, . . . , µp ) =(

µn11 , µ2, . . . , µp

)

.

This blow-up reduces the general case to the second one.




6·2. Definition and examples

Suppose that Cycl(Xλ , (Γ, λ0)) =n and letM > 0 and letW be a neighbourhood of

λ0 such that δλ has at most n zeroes in [s0−M, s0 +M ]. Define the set ZWM as the

subanalytic set

ZWM = {λ ∈ W : ∃ξ1, . . . , ξn ∈]s0 − M, s0 +M [: ξ1 < · · · < ξn

and δλ (ξi) = 0, ∀i = 1, . . . , n}.

Denote the interior of ZWM by ZW

M . Notice that the set ZWM can decrease when M

shrinks to 0. The problem is that for λ→λ0, we need that the corresponding zeroesof δλ tend to s0. Therefore, unlike in many examples that we have used till now, itis not always possible to define the stratum of maximal cyclicity Smc by a subset in

parameter space, independent ofM as is demonstrated by example (2·2) with (2·11).We now give a precise definition of what we mean by ‘a family (Xλ ) having a stratum

of maximal cyclicity with non-empty interior at λ0’.

Definition 6·6. Under the same notations as above, we say that the family (Xλ )λhas a stratum of maximal cyclicity with non-empty interior at λ0 or an mc-stratumwith non-empty interior at λ0, if there exist an M > 0 and a neighbourhood W of

λ0 such that there exist a sequence (λj )j∈N in ZWM with λj →λ0, and such that the

corresponding zeroes ξ1(λj ) < · · · < ξn (λj ) tend to s0 (if j →∞).

Remark that we cannot really define the notion of ‘stratum of maximal cyclicity’

in a sense of germs (M → 0), but only linked to a fixed M > 0.

The same holds concerning the set of parameter values of maximal multiplicity

Smm . Suppose that Mult(Xλ , (Γ, λ0)) =m and let M > 0 and let W be a neighbour-

hood of λ0 such that if δλ has n zeroes in [s0−M, s0 +M ] with respectivemultiplicities

m1, . . . , mn , then∑n

i=1 mi � m. Define the set ϑWM as the subanalytic set

ϑWM =

{

λ ∈ W |∃n ∈ N : ∀1 � i � n : ∃mi ∈ N� : ∃ξ1, . . . , ξn ∈]s0 − M, s0 +M [:

ξ1 < · · · < ξn and ∀i = 1, . . . , n : 0 � j < mi : δ(j )λ (ξi) = 0,

and ∀i = 1, . . . , n : δ(m i )λ (ξi)� 0}.

Denote the interior of ϑWM by ϑW

M .Notice also here that the set ϑWM can decrease when

M shrinks to 0. A precise definition of ‘family having a stratum of maximal multi-

plicity with non-empty interior at λ0’ can now be given similar to Definition 6·6.

In case the family does not have a mc-stratum with non-empty interior at λ0, weare not yet sure whether an algebraic mcc exists. In example (2·2) with (2·10), themc-stratum can be defined at λ0, but it has an empty interior. If g2(λ2) = sin λ2, thenthere does not exist an algebraic mcc; but if g2(λ2) =λ22, then there obviously existsan algebraic mcc (of degree at least 2).

It is not clear at all if this phenomenon is possible in linear families. In most

examples encountered in the literature, nearby vector fields with maximal cyclicity

are structurally stable and hence occur in open subanalytic sets of the parameter

space. In the rest of this section, we will now limit to this case, proving the existence

of algebraic mcc’s.

Of course, the analoguous remarks are valid when the notions cyclicity, mc-stratum

and mcc are replaced by multiplicity, mm-stratum and mmc respectively.




Remark thatW in the definition of ZWM does not play a very important role: if we

take a smaller neighbourhoodW1, then the set ZW 1

M =ZWM �W1 is just the intersection

of ZWM with W1. For fixed M > 0, the germs of {ZW

M : W a neighbourhood of λ0}remain the same at λ0. When M shrinks to 0, then the germs of {ZW

M :M >0} canchange completely, as recalled above by example (2·2) with (2·11). Therefore, ingeneral, M > 0 in the definition of the set ZW

M plays a crucial role. For this reason,

we will omit the dependence onW in our notation ZWM and we will write simply ZM .

For similar reasons we will write simply ϑM instead of ϑWM .

6·3. Algebraic mcc

Suppose that the family (Xλ )λ has an mc-stratum with non-empty interior at λ0.We would like to apply Theorem 6·1 to guarantee the existence of an algebraic mcc.The existence of an algebraic mcc does however not follow in a straightforward way

from Theorem 6·1. It is important to construct an open subanalytic set Z�

M ⊂ ZM

with the following property: if an analytic curve ζ enters Z�

M at λ0, then ζ is an mcc.Indeed, although there exists an algebraic curve ζ inside ZM (by Theorem 6·1), ζ

is not necessarily an mcc, since the zeroes do not necessarily need to tend to s0 fort→ 0, where t = 0 corresponds to λ=λ0. In fact, this convergence only needs to beverified for a discrete sequence (tn )n∈N with tn ↓ 0, which reveals to be an interestingobservation for the proof of Theorem 6·7.What we need is that, ∀0 < M ′ � M, there exists an ε(M ′) > 0 such that

∀0 < t < ε(M ′) : ζ(t) ∈ ZM ′ .

In other words, we have to find an analytic curve ζ, such that its germ at λ0 lies inthe germs of the sets ZM ′ � {λ0}, ∀M ′ ↓ 0. Hence, we are left with the limiting setZ0 =

⋂

M ′↓0 ZM ′ . If this set contains an open and subanalytic subset, having λ0 in itsclosure, then there exists an algebraic mcc, by Theorem 6·1. A priori, it is not clearat all whether this property holds; it is even not clear whether Z0 is non-empty, butthis is not essential.

Of course, we have ZM ′ ⊂ZM , ∀M ′ � M. It is possible that the sequence of thegerms of ZM at λ0 stabilizes, i.e. there exists an M > 0 such that for every

0<M ′ < M, there exists an neighbourhood VM ′ of λ0 with ZM ′ � VM ′ =ZM � VM ′ .Under this condition, it is clear how Theorem 6·1 can be applied in a straightforwardway to ZM to show the existence of an algebraic mcc.

However, the sequence of the germs of ZM at λ0 does not always stabilize, as isillustrated by example (2·2) with (2·11). In this example, the limiting set Z0 containsan algebraicmcc. This is not always the case; a non-empty Z0 does not need to containan algebraic mcc. Consider for instance the family (X(µ1,µ2)) obtained after application

of the analytic coordinate transformation (λ1, λ2) �→h(µ1, µ2) = (λ1 + sin λ2, λ2) inparameter space in example (2·2) with (2·11); more precisely, X(µ1,µ2) =X(µ1−sin µ2,µ2).

Then, the limiting set Z0 is given by the graph of sin : Z0 = {(µ1, µ2) : µ1 = sin µ2}.It is hence clear that in a non-stabilizing situation we can not prove the existence of

an (algebraic) mcc by considering Z0. In proving the existence of an mcc, we insteadwork with the subanalytic set

WM = {(λ, ξ1, . . . , ξn ) ∈ ZM × [s0 − M, s0 +M ]n :

ξ1 < · · · < ξn , δλ (ξi) = 0, ∀1 � i � n}.




If ω = (ζ, ξ1, . . . , ξn ) is an analytic curve inWM , that starts at (λ0, s0, . . . , s0), then ζ isclearly an mcc. The existence of such a curve ω, and hence the existence of an mcc isensured by the curve selection lemma applied to the subanalytic set WM . Since WM

is not open, we cannot apply Theorem 6·1 to WM , to guarantee the existence of an

algebraic mcc. However, we can construct another open subanalytic set Z�

M inducing

the existence of an algebraic mcc, in case that the family has an mc-stratum with

non-empty interior at λ0.

Theorem 6·7. If the family (Xλ )λ has a stratum of maximal cyclicity with non-empty

interior at λ0, then there exists an algebraic mcc ζ. Moreover, there exists a positive integerk, such that every analytic curve ζ with jk (ζ − ζ)(0) = 0, is an mcc.

Proof. Without loss of generality, we can suppose that λ0 = 0∈Rp , s0 =0∈R.There exist M > 0, a neighbourhood W of 0 in Rp and an analytic curve (ζ, ξ) =(ζ, ξ1, . . . , ξn ) : I = [0, 1]→ Rp × [−M, M ]n such that

ζ(0) = 0 and ξi (0) = 0, ∀i = 1, . . . , nδζ (ε)(ξi(ε)) = 0, ∀ε > 0, ∀i = 1, . . . , n−M < ξ1(ε) < · · · < ξn (ε) < M, ∀ε > 0

ζ(ε) ∈ ZM , ∀ε > 0.

By Theorem 6·2, it suffices to reduce ZM to an open subanalytic set Z�

M in such a

way that

ζ(ε) ∈ Z�

M , ε ↓ 0, ε� 0 (6·3)

and such that any analytic curve ζ with

ζ(0) = 0 and ζ(ε) ∈ Z�

M , ε ↓ 0, ε� 0, (6·4)

is necessarily an mcc. To obtain the second property, the set Z�

M will be constructed

such that parameter values λ in ZM are omitted, as soon as some of their corres-

ponding zeroes are situated at relatively large positions.

We suppose that at least one of the ξi , 1� i� n is non-identically zero. The func-tions f (ε) = ‖ξ(ε)‖2 =

∑ni=1(ξi(ε))

2 and g(ε) = ‖ζ(ε)‖2 are analytic and there exist somer, q ∈N� and a, b > 0 such that f (ε) = aεr + o(εr ), g(ε) = bεq + o(εq ), ε ↓ 0. Hence thereexists some d∈Q+\{0} such that for all ε sufficiently small, ‖ξ(ε)‖< ‖ζ(ε)‖d . We now

define Z�

M = ZM \π(K), where

K ={

(λ, s1, . . . , sn ) : −M � s1 < · · · < sn � M,

δλ (si) = 0, ∀1 � i � n and ‖(s1, . . . sn )‖ � ‖λ‖d}

and π: Rp ×R→Rp: (λ, s) �→λ is the natural linear projection. By construction, Z�

M

is an open subanalytic set and is non-empty since ζ(ε) ∈ Z�

M , for all ε sufficientlysmall. Moreover, let ζ be an analytic curve satisfying (6·4), then ζ is necessarilyan mcc. Indeed, denote the corresponding zeroes by ξ1(ε)< · · ·< ξn (ε); by the com-pactness of [−M, M ], there exists a sequence (εm )m∈N with εm ↓ 0 and ξj (εm )→sj ∈ [−M, M ], ∀1� j � n. In particular, 0� ‖(s1, . . . , sn )‖= limm→∞ ‖ξ(εm )‖�

limm→∞ ‖ζ(εm )‖=0. As a consequence, sj =0, ∀1� j � n.

The case that ξi ≡ 0, ∀1� i� n, is even easier to treat (e.g., take d=1). (Notice thatthen n=1, if n denotes the cyclicity; however, this is of no importance.)




Proposition 6·8. Suppose that ζ: [0, 1]→ Rp is an mcc. Denote the zeroes of δζ (t) by

ξ1(t)< · · · < ξn (t). Then, there exist 0<E0 < 1 and r∈N∗ such that ∀t∈ [0, E0] : ξi(t) =ξi(t

1/r ) and ξi : [0, E0]→Rp is analytic, ∀1 � i � n.

Proof. Consider the subanalytic set Wζ = WM � π−1(ζ([0, 1])), where

WM = {(λ, s1, . . . , sn ) : λ ∈ ZM , s1 < · · · < sn ,

s0 − M < si < s0 +M, δλ (si) = 0, ∀1 � i � n, }

and π: Rp ×Rn → Rp: (λ, s1, . . . , sn )→ λ. By the curve selection lemma, there existsan analytic curve ω: [0, 1]→ Rp × Rn: τ → (λ(τ ), s1(τ ), . . . , sn (τ )) with

∀0 < τ � 1 : ω(τ ) ∈ Wζ and ω(0) = (λ0, s0, . . . , s0). (6·5)

From (6·5) it follows that

λ(τ ) ∈ ζ([0, 1]), ∀0 � τ � 1. (6·6)

Hence, there exists 0 < E < 1 such that ∀0 � t � E: ∃0 � τt � 1 with λ(τt) = ζ(t).Then, by (6·5), the zeroes of δζ (t) are given by si(τt), ∀0 < t < E.By (6·6), the analyticset A = {(τ, t) ∈ [0, 1]2 : ζ(t) = λ(τ )} accumulates at (0, 0), since ζ(0) = λ(0) = λ0.Then, by the curve selection lemma, there exists an analytic curve

h: [0, 1] −→ [0, 1]2: χ −→ (h1(χ), h2(χ))

with h1(0) = h2(0) = 0 and h(t) ∈ A, ∀t ∈ [0, 1]. There exist a positive integer r and aconstant a > 0 such that h2(χ) = aχr +o(χr ), χ ↓ 0, inducing that κ(χ) = h2(χ)

1/r is a

local analytic diffeomorphism at χ = 0. Hence, ∃0 < E0 < E such that ∀0 � t � E0:

τt = h1 ◦ κ−1(t1/r ). Hence, ∀1 � i � n : ξi(t) = si(h1 ◦ κ−1(t1/r )) ≡ ξi(t1/r ).

Combining Theorem 6·7 and Proposition 6·8, we get the following corollary.

Corollary 6·9. If (Xλ )λ has a stratum of maximal cyclicity with non-empty interior

at λ0, then there exist an analytic curve ζ : [0, 1] → Rp , a constant M > 0 and a

positiveinteger k such that if ζ: [0, 1] → Rp is an analytic curve with jk (ζ − ζ)(0) = 0,then:

(i) ζ is an mcc;(ii) furthermore, for every 0 < M1 < M, there exists an ε(M1) > 0 such that

∀0 < t < ε(M1) : ζ(t) ∈ ZM 1.

6·4. Algebraic mmc

In case the family has an mm-stratum with non-empty interior at λ0, we canprove analogous results as in previous section. Since the proofs of these results also

are analogous, we refer to [3] for the details.

7. Final remarks and open problems

In this section, we want to introduce some interesting notions, related to the res-

ults in this paper, and we state a few open problems concerning these notions. For

example, consider example (2·2) with (2·4). We have seen that there is no linear mcc(or mmc), however it is clear that there exists a quadratic mcc (and mmc). We could

now ask whether in general for analytic families there does exist a uniform bound

n, inducing an algebraic mcc ζ (resp. mmc) of degree n? The answer also is negative;




counterexamples are provided by a simple adaptation of the previous example (re-

place λ1 in (2·4) by λk1 with 2k > n). However, under rather generic conditions, there

always exists an algebraic mcc ζ (respectively mmc) without (uniform) limitation onthe degree, as we have seen in Section 6.

Definition 7·1. The ‘detectibility degree of maximal cyclicity’ (resp. multiplicity) istheminimal degree of an algebraicmcc (resp. mmc), shortly denoted by “ddmc” (resp.

“ddmm”). In case no algebraic mcc (resp. mmc) exists, we say that ddmc(Xλ ) =∞(resp. ddmm(Xλ ) =∞).

Problem 7·2. Does there exist a uniform finite upperbound for the ddmc (resp.

ddmm) depending on the degree N of the algebraic families given in (1·3)?

In Section 3·1 we have already observed that ddmm (Xλ ) > 1 and ddmc (Xλ ) > 1

in case Xλ linearly depends on λ.

Definition 7·3. The ‘conic degree of maximal cyclicity’ (resp. multiplicity) is theminimal value of n ∈ N1 such that there exists an analytic curve γ entering thestratum Vmax of maximal cyclicity (resp. multiplicity) with the property that ananalytic curve γ also enters Vmax if jn (γ − γ)(0) =0. It is shortly denoted by cdmc(resp. cdmm).

Problem 7·4. Does there exist a uniform finite upperbound on the cdmc (respect-

ively cdmm), depending on the degree N of the family (1·3)?

We have two general theorems that provide a starting point in answering these

questions. First we will give an auxiliary lemma.

Lemma 7·5. Let h: V ⊂Rp →Rp be a Cω diffeomorphism with h(0) = 0, where V is an

open subanalytic set with 0∈ V , and let be W =h(V ). Suppose that there is an algebraiccurve γW of degree at most n, with γW (0) = 0, entering W, with the property that ananalytic curve γW also entersW if jk (γW − γW )(0) = 0, where k is a fixed integer, n � k.Then there exists an algebraic curve γV of degree at most k that enters V, with γV (0) = 0,with the property that an analytic curve γV also enters V if jk (γV − γV )(0) = 0.

Proof. Denote ρ=h−1 ◦ γW , and define the algebraic curve of degree at most k:

γV = jk (ρ)(0). (7·1)

Suppose that γV is an analytic curve with

jk (γV − γV )(0) = 0. (7·2)

Then we only have to prove that γV enters V. From (7·1) and (7·2), we also have that

jk (ρ − γV )(0) = 0.

It then follows that jk (h ◦ ρ− h ◦ γV )(0) = 0. As a consequence: h ◦ γV entersW at 0,or γV enters V at 0.

Proposition 7·6. Suppose that for a given analytic family of planar vector fields(Xα )α the displacement function takes the following form: put S = s− s0, for s2 =x2 + y2, s0 > 0

Sl + α1Sl−1 + · · · + αl−1S + αl , where α = (α1, . . . , αl) (7·3)




respectively

α0Sl + α1S

l−1 + · · · + αl−1S + αl where α = (α0, α1, . . . , αl). (7·4)

Then, in studying the cyclicity and multiplicity of the family along Γ= {s= s0} and atα0 =0∈Rl (respectively Rl+1), there exists an algebraic mcc and mmc γ of degree smallerthan [l/2], respectively [l/2] + 1, with the property that an analytic curve γ also entersVmax if jk (γ − γ)(0) = 0, where k= [(l+1)/2]. As a consequence,

(i) cdmc,cdmm � [(l + 1)/2], respectively cdmc,cdmm � [(l + 1)/2] + 1.(ii) ddmc,ddmm � [l/2], respectively ddmc,ddmm � [l/2] + 1.

Proof. We only need to prove the facts concerning the cyclicity, since for such a

family (Xα )α , we have that

Cycl(Xα , (Γ, α0)) = Mult(Xα , (Γ, α0)).

Hence, anmcc also is anmmc.Moreover, we only have to prove this proposition in case

that the displacement map takes the form (7·3). The other case then follows. Indeed,if γ = (γ1, . . . , γl) is an mcc for (7·3), then the analytic curve t �→ (t, tγ1(t), . . . , tγl(t))is an mcc for the family (7·4). For the second property, we can use the following fact:if

jn (γ − γ)(t) = 0, γ0(t) = t +O(tn+1); t → 0,

then jn (γ − γ ◦ γ−10 )(0) = 0, γ0 ◦ γ−1

0 (t) = t.

By construction of an algebraic mcc with the required conical contact, we now

prove the proposition for the family (7·3) subsequently in the folllowing 3 cases: l=2,l is even and l is odd. Consider first the quadratic case, i.e. l=2, with displacementmaps

δ(S, a, b) = S2 + aS + b,

then the curve [0, 1]→R2 : t �→ t(a, b) is a linear mcc, ∀a∈ [1/4, 1], ∀b < 0, and thelinear cone C = {t(a, b) : 1/4� a� 1, b < 0, 0<t � 1} is contained in Vmax . Remarkthat zeroes of δ(·, a, b) stay outside the interval [−1, 1/8]if 1

4< a < 1, b � − 3/4.

Suppose now that l � 4 is even, i.e. l=2n (n � 2). From the quadratic case, it follows

that the displacement map

δ(S, α) =

n∏

k=1

(S2 + takS − bk t) = S2n + α1S2n−1 + · · · + α2n (7·5)

has exactly 2n disjoint zeroes,∀0<t � 1 where ∀k=1, . . . , n:

{

ak ∈ Ik =]ak − 2−k−1, ak + 2−k−1[, ak = (2

k − 1)/2k

bk ∈ In+k =]bk − 1/4, bk + 1/4[, bk = −k.(7·6)

Denote I =∏2n

i=1Ik , and a= (a1, . . . , an ), b= (b1, . . . , bn ). From equation (7·5), weget an analytic map g = (g1, . . . , g2n ) : R2n+1→R2n : (a, b, t) �→ (α1, . . . , α2n ), where




(a, b) = (a1, . . . , an , b1, . . . , bn ), and

g1(a, b) = t

n∑

i=1

ai

g2k (a, b) = tk∑

1�i1< ···<ik �n

bi1 . . . bik+O(tk+1), t → 0

g2k−1(a, b) = tk+1n∑

i=1

ai

∑

1�i 1< ···< i k �ni j�i

bi1 . . . bik+O(tk+2), t → 0.

, ∀k = 1, . . . , n

Clearly, a parameter α= g(a, b, t) belongs to Vmax if (a, b)∈ I, 0<t � 1. Moreover, the

analytic curve γ: [0, 1]→R2n : t �→ g(a, b, t) is an algebraic mcc of degree n. Now wewill prove that every analytic curve γ enters Vmax if

jn (γ − γ)(0) = 0; (7·7)

more precisely, we prove that there exists an ε > 0 such that ∀0 < t < ε : γ(t) ∈ Vmax .Let γ be an analytic curve with property (7·7). Define the analytic curves γ, γ suchthat

γj (t) = t

[

j +12

]

γj(t), γj (t) = t

[

j +12

]

γj(t), ∀j = 1, . . . , 2n.

Property (7·7) now translates into ∀j =1, . . . , 2n : γj(t) = γ

j(t) +O(tn−[ j +1

2]+1), t ↓ 0.

Therefore, ∀j =1, . . . , 2n, there exist positive constants Mj > 0, 0<εj < 1 such that

|γj(t)− γ

j(t)| � Mj |t|

n−[ j +12]+1, ∀|t| < εj . (7·8)

We also define the analytic function h= (h1, . . . , h2n+1) : R2n+1→R2n+1 by

{

hj (a, b, t) = t−[j +12 ]gj (a, b, t), ∀j = 1, . . . , 2n

h2n+1(a, b, t) = t.

It can be proven that Dh(a ,b,0) is non-singular, more precisely, one can calculate by

induction on n that det Dh(a ,b,0) = (−1)n (2!)2(3!)2 . . . ((n − 1)!)2�0. From the inverse

function theorem, it follows that there exists an open neighbourhood V of (a, b, 0) inI ×R⊂R2n+1 such that h: V →h(V ) is an analytic diffeomorphism. As a consequence,h(V ) is a neighbourhood of (γ(0), 0), and therefore there exist positive constantsA1, . . . , An , B1, . . . , Bn , ε0 > 0, such that

n∏

j=1

]

γj(0)−Aj , γj

(0)+Aj

[

×

2n∏

j=n+1

]

γj(0)−Bj , γj

(0)+Bj

[

× ]−ε0, ε0[ ⊂ h (V ) . (7·9)

Define ε > 0 by ε= min{ε0, εj , Aj/Mj , Bj/Mn+j : j =1, . . . , n}, then from (7·9) and(7·8), it follows that ∀|t| < ε : (γ(t), t) ∈ h(V ); hence, γ(t) ∈ Vmax , ∀0 < t < ε.Suppose now that l � 3 is odd, i.e. l = 2n + 1 (n � 1). From the quadratic case, it

follows that the displacement map

δ(S, α) =

n∏

k=1

(S2 + takS − bk t)(S + tc) = S2n+1 + α1S2n + · · · + α2n+1 (7·10)

has exactly 2n+1 disjoint zeroes, ∀0 < t � 1 where ak ∈ Ik , bk ∈ Ik+n (cfr (7·6)), and

c∈]

− 12, 12

[

, c = 0.




Identification of equal coefficients in (7·10) defines an analytic map

g = (g1, . . . , g2n+1) : R2n+2 −→ R2n+1 : (a, b, c, t) �−→ (α1, . . . , α2n+1),

where (a, b, c, t) = (a1, . . . , an , b1, . . . , bn , c, t). Analogously as above, we deduce that∀k=1, . . . , n:

g1(a, b, c, t) = g1(a, b, t)g2k (a, b, c, t) = g2k (a, b, t) +O(tk+1), t → 0

g2k+1(a, b, c, t) = tk+1

(

g2k+1(a, b, t) + c∑

1�i1<...<ik �n

bi1 . . . bik

)

+O(tk+2), t → 0.

Then we can define an analytic function h= (h1, . . . , h2n+2) : R2n+2→R2n+2 by

{

hj (a, b, c, t) = t−[j +12 ]gj (a, b, c, t), ∀j = 1, . . . , 2n + 1

h2n+2(a, b, c, t) = t.

It is easy to see that det Dh(a ,b,0,0) = (−1)nn! det Dh(a ,b,0) � 0. Hence, Dh(a ,b,0,0) is

non-singular; the sequel of the proof is similar to the one in the even case.

Theorem 7·7. Suppose that the Bautin ideal I satisfies the following properties:

(i) I is regular;(ii) there is a minimal set of generators {ϕ1, . . . , ϕl} for I with a local division of the

displacement map of type

δ(s, λ) =

l∑

i=1

ϕi(λ)hi(s, λ)

and there exists n ∈ N such that ∀i = 1, . . . , l

hi(s, λ) = ci(λ)(s − s0)n+i−1 + o((s − s0)

n+i−1), s → s0

with ci(0)� 0. Then:

(a) there exists a linear mic;

(b) Cycl(Xλ , (Γ, 0)) = l − 1;(c) if l � 3, then there is no linear mcc or mmc;(d) there exists an algebraic mcc and mmc of degree [(l + 2)/2]; as a consequence,

ddmc, ddmm � [(l + 2)/2];

(e) cdmc,cdmm � [(l + 2)/2].

Proof. Properties (a) and (b) are clear. Proposition 2·2 can be generalized forany regular Bautin ideal, as a consequence of Remark 6·3 and Lemma 7·5. Thisproves part (c). Hence, we are left to prove part (d). By Lemma 7·5, we can assume(without loss of generality) that the generators of the Bautin ideal are given by

ϕj (λ) =λj , ∀j =1, . . . , l. Without loss of generality, we can also assume that n=0and s0 =0. Clearly, we can restrict ourselves to the subset in parameter spaceλl+1 = · · · =λp =0. Furthermore, by Rolle’s theorem, to have at least l zeroes, itis necessary that this curve lies in a region of parameter space with λl small but

non-zero. Therefore we consider the family

F (s, µ, λl) = µ1h1(s, (λlµ, λl)) + · · · + µl−1hl−1(s, (λlµ, λl)) + hl(s, (λlµ, λl)),




where µ= (µ1, . . . , µl−1) and ∀i = 1, . . . , l − 1 : λi =λlµi . In this way,

δ(s, λlµ1, . . . , λlµl−1, λl) = λlF (s, µ1, . . . , µl−1, λl).

For (µ, λl) = 0, we have for a certain c� 0:

F (r, 0) = Hl(r) = crl−1 + o(rl−1), r → 0.

Hence, by the preparation theorem [7], the function F can be written as:

F (r, µ, λl) = (a1 + a2R + · · · + al−1Rl−2 +Rl−1)Q(R, a),

with Q(0, 0)� 0 and R = r +O(r2). Moreover, one can deduce the following relationbetween a = (a1, . . . , al−1) and µ:

a1 = d11µ1(

1 + f 11 (λlµ, λl))

...

al−1 = dl−1l−1µl−1

(

1 + f l−1l−1 (λlµ, λl)

)

+ dl−1l−2µl−2

(

1 + f l−1l−2 (λlµ, λl)

)

+ · · · + dl−11 µ1

(

1 + f l−11 (λlµ, λl)

)

λl =λl

(7·11)

where dij ∈R, di

i�0 and f ij are analytic functions with f i

j (0) = 0, 1� j � i, 1� i� l−1.Relation (7·11) clearly defines a diffeomorphism that respects the origin. By setting

bi =λlai , ∀i=1, . . . , l − 1, we can consider the family of polynomials of degree l− 1:

P (R, b, λl) = b1 + b2R + · · · + bl−1Rl−2 + λlR

l−1, (7·12)

then we have δ(s, λlµ1, . . . , λlµl−1, λl) =λlP (R, b, λl). There is the following diffeo-morphism λ= (λ1, . . . , λl) �→ b = (b1, . . . , bl−1, λl):

b1 = d11λ1(

1 + f 11 (λ))

...

bl−1 = dl−1l−1λl−1

(

1 + f l−1l−1 (λ)

)

+ dl−1l−2λl−2

(

1 + f l−1l−2 (λ)

)

+ · · · + dl−11 λ1

(

1 + f l−11 (λ)

)

λl =λl

Lemma 7·5 and Theorem 7·6 now imply that ddmc,ddmm� [((l− 1) + 1)/2] + 1=[(l+2)/2], and cdmc,cdmm� [(l + 2)/2]. Hence, the assertion is proven.

These problems constitute a good starting point in studying the relation between

the structure of the bifurcation set of limit cycles, more precisely the structure of the

‘stratum of maximal cyclicity’ (respectively multiplicity) and the algebraic nature

of the perturbation in p-parameter perturbations from a center.

REFERENCES

[1] E. Bierstone and P. Milman. Semi-analytic and subanalytic sets. Inst. Hautes Etudes Sci.Publ. Math. 67 (1988), 5–42.

[2] M. Caubergh. Configurations of zeroes of analytic functions. C. R. Acad. Sci. Paris 333, I(2001), 307–312.

[3] M. Caubergh.Limit cycles near vector fields of center type. Phd Thesis, Limburgs UniversitairCentrum (Belgium, 2004).

[4] Z. Denkowska and J. Stasica. Ensembles sous-analytiques a la polonaise. Preprint (Krakow,1986).

[5] J. Francoise. Successive derivatives of a first returnmap, application to the study of quadraticvector fields. Ergodic Theory Dynam. Systems 16, 1 (1996), 87–96.







[6] S. �Lojasiewicz. Ensembles semi-analytiques. Preprint, Inst. Hautes Etudes Sci. (1965).[7] B. Malgrange. Ideals of Differentiable Functions (Oxford University Press, 1966).[8] J. Poggiale. Applications des varietes invariantes a la modelisation de l’heterogeneite en

dynamiques des populations. These (Universite de Bourgogne, 1994).[9] R. Roussarie. Bifurcations of planar vector fields and Hilbert’s sixteenth problem. Progr.

Math. 164 (1998).[10] R. Roussarie. Melnikov functions and Bautin ideal. Prepublication ou Rapport de Recherche

(Universite de Bourgogne, France) 219 (2000).






Documents

Algebraic curves of maximal cyclicity