C. R. Acad. Sci. Paris, t. 332, Série I, p. 551–555, 2001Systèmes dynamiques/Dynamical Systems

Unique normal forms for the Takens–Bogdanovsingularity in a special caseXiaofeng WANG a, Guoting CHEN b, Duo WANG c

a Department of Mathematical Sciences, Tsinghua University, Beijing 100084, Chinab UMR AGAT CNRS, UFR de mathématique, université de Lille-1, 59655 Villeneuve d’Ascq, Francec School of Mathematical Sciences, Peking University, Beijing 100871, China

E-mail: xfwang@math.tsinghua.edu.cn; xwang@agat.univ-lille1.fr; guoting.chen@univ-lille1.fr;dwang@math.pku.edu.cn

(Reçu le 1er décembre 2000, accepté le 29 janvier 2001)

Abstract. We consider further reduction of normal forms for nilpotent planar vector fields. We givea unique normal form for a special case of an open problem for the Takens–Bogdanovsingularity. 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS

Unicité des formes normales pour la singularité de Takens–Bogdanovdans un cas particulier

Résumé. On étudie l’unicité des formes normales des champs de vecteurs nilpotents. On donneune réponse dans un cas particulier d’un problème ouvert pour la singularité de Takens–Bogdanov. 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS

Version française abrégée

Nous considérons des champs de vecteurs en dimension 2 avec une singularité à l’origine et une partielinéaire nilpotente non nulle. Nous écrirons X(x, y) = y∂x + · · ·, où ∂x = ∂

∂x , ∂y = ∂∂y et les pointillés

représentent des termes de degrés supérieurs ou égaux à 2. On sait (voir [3,4,6]) qu’une forme normaleclassique de X peut être choisie sous la forme :

F (x, y) = y∂x +

∞∑j=1

(αjx

j+1 + βjxjy)∂y.

Soient µ = inf{j : αj �= 0} et ν = inf{j : βj �= 0}. Les réductions supplémentaires des formes normalesou leur unicité sont très différentes dans les trois cas suivants (voir [1]) : µ < 2ν, µ > 2ν ou µ = 2ν.Dans [1] l’unicité des formes normales est étudiée pour les champs de vecteurs de la seconde catégorieet, pour certains cas, de la première catégorie. Le problème d’unicité des formes normales pour le cas oùν ≡ 0 (mod µ+ 2) dans la première catégorie, est toujour ouvert.

Note présentée par Bernard MALGRANGE.

S0764-4442(01)01869-9/FLA 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS. Tous droits réservés 551

X. Wang et al.

On considère seulement un cas particulier du problème, c’est-à-dire que µ = 1, i.e. α1 �= 0. Alors par unchangement linéaire de coordonnées x �→ x/α1, y �→ y/α1, on réduit α1 à 1. Nous supposerons que α1 = 1pour simplifier les calculs. Donc nous supposons que notre forme normale classique est :

F (x, y) = y∂x +(x2 + β1xy

)∂y +

∞∑j=2

(αjx

j+1 + βjxjy)∂y. (1)

Le but de cette Note est de donner une réduction supplémentaire des formes normales classiques ci-dessus.Nous donnerons une unique forme normale dans le cas ν = 1 et ν = 3 ; ce dernier cas donne une réponsed’un cas particulier du problème ouvert cité ci-dessus.

Le résultat principal de cette Note est le théorème suivant.

THÉORÈME. – SoitF un champ de vecteurs de la forme(1). Alors on peut le réduire à un nouveau champde la forme:

V (1) = y∂x +

[x2 +

∞∑p=0

(α

(1)3p+2x

3p+3 + β(1)3p+1x

3p+1y + β(1)3p+3x

3p+3y)]

∂y,

où en particulierβ(1)1 = β1 etα(1)

2 = α2.

On considère le cas oùβ1 = 0. Supposons que110β(1)4 − 183α

(1)2 β

(1)3 �= 0. Une forme normale deV (1)

est la suivante:

V (8) = y∂x +

[x2 + α

(1)2 x3 + β

(1)3 x3y + α

(1)5 x6 +

∞∑p=1

(β

(8)3p+1x

3p+1y + β(8)3p+3x

3p+3y)]

∂y.

Elle est unique via des changements de coordonnées tangents à l’identité.

Pour des détails et des preuves nous renvoyons à [2].

1. Introduction

We consider unique formal normal forms for nilpotent vector fields in dimension 2. The work in [1]clarifies them into three categories where a result is given for the second category and for some cases of thefirst category. We shall give a unique normal form for a special case of an open problem concerning vectorfields of the first category. We use the method of linear grading function introduced in [5] in which theyhave given unique normal forms for nilpotent vector fields of the third category under a non degeneracycondition.

We consider planar vector fields in equilibrium at the origin with a nonzero nilpotent linear part. We shallwrite X(x, y) = y∂x + · · ·, where ∂x = ∂

∂x , ∂y = ∂∂y and the dots represent terms of higher degrees. It is

known (see[3,4,6]) that a classical normal form can be chosen to be:

F (x, y) = y∂x +

∞∑j=1

(αjx

j+1 + βjxjy)∂y.

Let µ = inf{j : αj �= 0} and ν = inf{j : βj �= 0}. The three categories correspond respectively to µ < 2ν,µ> 2ν or µ= 2ν. For the first category, the problem of unique normal forms in the cases ν ≡ 0 (mod µ+2)is still open.

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Unique normal forms

We only consider a special case of the first category, that is the case where µ= 1, i.e., α1 �= 0. Then by alinear change of variables x �→ x/α1, y �→ y/α1, one reduces α1 to 1. We suppose that it is done to simplifycalculations. So we assume that the classical normal form is:

F (x, y) = y∂x + (x2 + β1xy)∂y +

∞∑j=2

(αjx

j+1 + βjxjy)∂y. (1)

The purpose of this Note is to give a further reduction of the above classical normal form and to give aunique normal form in the cases ν = 1 and ν = 3, the latter solves a special case of the open problemmentioned above.

Let H be the vector space of all 2 dimensional real or complex formal vector fields. The Lie bracket isdefined by [·, ·] : H ×H → H by [U,V ] = DV · U −DU · V for any U,V ∈H . Then {H, [·, ·]} formsa graded Lie algebra with respect to the classical degree. The idea of grading function is to define a newgrading so that H remains a graded Lie algebra with respect to the new grading. For details we refer to [5,2].For our purpose we need the following linear grading function: for any m,n∈N,

δ(xmyn∂x

)= 2m+ 3n− 2, δ

(xmyn∂y

)= 2m+ 3n− 3.

Let Hk (k � 0) be the vector space spanned by all monomial vector fields of degree k with respect to δ.One has [Hi,Hj] ⊂Hi+j . Hence H =

∑kHk is a graded Lie algebra. We have δ(y∂x) = δ(x2∂y) = 1.

We shall denote V1 = y∂x + x2∂y , which is the leading term in the new grading.We can now rewrite the classical normal form F (x, y) in (1) according to our new grading function,

V =∑∞

k=1 Vk, where:

Vk =

y∂x + x2∂y, for k = 1;

βjxjy∂y, for k = 2j (j � 1);

αjxj+1∂y, for k = 2j − 1 (j > 1).

For any k ∈N, we define a linear operator:

L(1)k :Hk −→Hk+1, Yk → [V1, Yk].

Let C(1)k+1 be a complementary subspace of imL

(1)k in Hk+1, i.e., Hk+1 = imL

(1)k ⊕ C

(1)k+1. It is easy to see

that there exists a sequence of near identity transformations such that the vector field V be transformed toa new vector field:

V (1) = V1 + V(1)2 + · · ·+ V

(1)k+1 + · · · ,

where V(1)k+1 ∈ C

(1)k+1 for all k � 1. We shall call it a first ordernormal form.

In order to make further reduction of a first order normal form, we define a sequence of linearoperators L(m)

k , m,k = 1,2, . . . , as follows. Let V =∑∞

j=0 Vj+1 be a formal vector field, where Vk ∈Hk

for all k � 1. We define L(1)k as above for any k. If L(m)

k is already defined for an m � 1 and k � 1, thenwe define:

L(m+1)k : kerL

(m)k ×Hm+k →Hm+k+1

with L(m+1)k (Yk, . . . , Yk+m) =

∑mi=0[Vi+1, Yk+m−i].

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X. Wang et al.

It is obvious that (Yk, . . . , Yk+m−1) ∈ kerL(m)k if and only if:

j∑i=0

[Vi+1, Yk+j−i] = 0 for j = 0, . . . ,m− 1.

DEFINITION 1. – A formal vector field V = V1 + V2 + · · ·, where Vj ∈ Hj for each j � 1, is called

an N -th order normal form, if Vi+1 ∈ C(i)i+1(1 � i � N − 1) and Vj+1 ∈ C

(N)j+1(j � N), where C

(m)k+1 is a

complementary subspace to the image of L(m)k−m+1 in Hk+1 for each m� 1 and k � 1.

It is called an infinite order normal form if Vm+1 ∈ C(m)m+1 for all m � 1. Notice that L(k)

1 is defined byusing V1, . . . , Vk , which are supposed to be in a k-th order normal form.

It is proved in [5] that an infinite order normal form is unique with respect to near identity transformationsand an N -th order normal form is an infinite order normal form (hence is unique) if and only ifimL

(N+m)k = imL

(N)k+m for any k � 1,m� 1.

2. Unique normal forms

One can first reduce a classical normal form V to a first order normal form that we state in the following.

THEOREM 1. – Let a vector field of the formV be given. Then its first order normal form is:

V (1) = V1 +

+∞∑p=0

(V

(1)6p+3 + V

(1)6p+2 + V

(1)6p+6

)

i.e.,

V (1) = y∂x +

[x2 +

∞∑p=0

(α

(1)3p+2x

3p+3 + β(1)3p+1x

3p+1y + β(1)3p+3x

3p+3y)]

∂y,

where in particularβ(1)1 = β1 andα(1)

2 = α2.

The above first order normal form is not unique. The following theorem gives a unique normal form inthe case where β1 �= 0, which is a special case of a result given in [1]. We have improved the method of [5]to get this result and to prove a new result in Theorem 3.

THEOREM 2. – Let V (1) be a vector field in a first order normal form as given in Theorem1. Supposethatβ1 �= 0. The second order normal form ofV (1) is the following:

V (2) = y∂x +

[x2 + α2x

3 +

∞∑p=0

(β

(2)3p+1x

3p+1y + β(2)3p+3x

3p+3y)]

∂y,

where in particularβ(2)1 = β1. Moreover, this normal form is unique with respect to near identity

transformations.

Let V (1) be a vector field in a normal form of Theorem 1, with V(1)2 = 0, i.e., β1 = 0. We now give

our main theorem, which gives a result for the special case of an open problem under a non degeneracycondition.

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Unique normal forms

THEOREM 3. – LetV (1) be a vector field in a first order normal form as given in Theorem1 with β1 = 0.Suppose that110β

(1)4 − 183α

(1)2 β

(1)3 �= 0. The eighth order normal form ofV (1) is the following:

V (8) = y∂x +

[x2 + α

(1)2 x3 + β

(1)3 x3y + α

(1)5 x6 +

∞∑p=1

(β

(8)3p+1x

3p+1y + β(8)3p+3x

3p+3y)]

∂y.

It is unique with respect to near identity transformations.

The proof of the above theorem is based on the following lemma.

LEMMA 1. – LetV (1) be a vector field in a first order normal form as described in Theorem1. Supposethatβ(1)

1 = 0 and110β(1)4 − 183α

(1)2 β

(1)3 �= 0. Then:

(a) for any integerk � 1, such thatk �= 6p+ 1 with an integerp� 1, we haveim(L(8)k ) = im(L

(1)k+7);

(b) for any integerp� 1, we haveim(L(8)6p+1) = H6p+9.

For details and proofs we refer to [2].

Acknowledgements. This paper was prepared when D. Wang was visiting the University of Lille-1 and X. Wangwas visiting Le Laboratoire AGAT, UMR 8524 du CNRS, at the same university. They appreciate the hospitality of theDepartment of Mathematics at the University of Lille-1. X. Wang is supported by NSF of China and the Basic ResearchFoundation of Tsinghua University (project 98JC071). D. Wang is supported by NSF of China and NSF of Beijing.

References

[1] Baider A., Sanders J.A., Further reductions of the Takens–Bogdanov normal form, J. Diff. Equations 99 (1992)205–244.

[2] Chen G., Wang D., Wang X., Unique normal forms of nilpotent planar vector fields, Preprint, Université de Lille-1,2000.

[3] Cushman R., Sanders J., Nilpotent normal forms and representation theory of sl2(R), in: Multiparameter BifurcationTheory, Golubitsky M., Guckenheimer J. (Eds.), Contemporary Mathematics, Vol. 56, Amer. Math. Soc.,Providence, 1986, pp. 31–35.

[4] Elphick C., Tirapegui E., Brachet M., Coulet P., Iooss G., A simple characterization for normal forms of singularvector fields, Physica D 29 (1987) 95–127.

[5] Kokubu H., Oka H., Wang D., Linear grading function and further reduction of normal forms, J. Diff. Equations 132(1996) 293–318.

[6] Takens F., Singularities of vector fields, Publ. Math. I.H.E.S. 43 (1974) 47–100.[7] Ushiki S., Normal forms for singularities of vector fields, Japan J. Appl. Math. 1 (1984) 1–34.

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