Orbital Linearization of Smooth Completely Integrable Vector Fields
The main purpose of this paper is to prove the smooth local orbital linearization theorem for smooth vector fields which admit a complete set of first integrals near a nondegenerate singular point. The main tools used in the proof of this theorem are the formal orbital linearization theorem for form...
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irk-123456789-1492712019-02-20T01:24:46Z Orbital Linearization of Smooth Completely Integrable Vector Fields Zung, N.T. The main purpose of this paper is to prove the smooth local orbital linearization theorem for smooth vector fields which admit a complete set of first integrals near a nondegenerate singular point. The main tools used in the proof of this theorem are the formal orbital linearization theorem for formal integrable vector fields, the blowing-up method, and the Sternberg-Chen isomorphism theorem for formally-equivalent smooth hyperbolic vector fields. 2017 Article Orbital Linearization of Smooth Completely Integrable Vector Fields / N.T. Zung // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 15 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 37G05; 58K50; 37J35 DOI:10.3842/SIGMA.2017.093 http://dspace.nbuv.gov.ua/handle/123456789/149271 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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The main purpose of this paper is to prove the smooth local orbital linearization theorem for smooth vector fields which admit a complete set of first integrals near a nondegenerate singular point. The main tools used in the proof of this theorem are the formal orbital linearization theorem for formal integrable vector fields, the blowing-up method, and the Sternberg-Chen isomorphism theorem for formally-equivalent smooth hyperbolic vector fields. |
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Zung, N.T. Orbital Linearization of Smooth Completely Integrable Vector Fields Symmetry, Integrability and Geometry: Methods and Applications |
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Zung, N.T. |
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Zung, N.T. |
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Orbital Linearization of Smooth Completely Integrable Vector Fields |
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Orbital Linearization of Smooth Completely Integrable Vector Fields |
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Orbital Linearization of Smooth Completely Integrable Vector Fields |
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Orbital Linearization of Smooth Completely Integrable Vector Fields |
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Orbital Linearization of Smooth Completely Integrable Vector Fields |
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orbital linearization of smooth completely integrable vector fields |
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Інститут математики НАН України |
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2017 |
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Orbital Linearization of Smooth Completely Integrable Vector Fields / N.T. Zung // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 15 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT zungnt orbitallinearizationofsmoothcompletelyintegrablevectorfields |
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2025-07-12T21:12:38Z |
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1837477152137674752 |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 093, 11 pages
Orbital Linearization of Smooth Completely
Integrable Vector Fields
Nguyen Tien ZUNG †‡
† School of Mathematics, Shanghai Jiao Tong University,
800 Dongchuan Road, Minhang District, Shanghai 200240, P.R. China
‡ Institut de Mathématiques de Toulouse, UMR5219 CNRS, Université Paul Sabatier,
118 route de Narbonne, 31062 Toulouse, France
E-mail: tienzung.nguyen@math.univ-toulouse.fr
Received July 04, 2017, in final form November 30, 2017; Published online December 12, 2017
https://doi.org/10.3842/SIGMA.2017.093
Abstract. The main purpose of this paper is to prove the smooth local orbital linearization
theorem for smooth vector fields which admit a complete set of first integrals near a non-
degenerate singular point. The main tools used in the proof of this theorem are the formal
orbital linearization theorem for formal integrable vector fields, the blowing-up method, and
the Sternberg–Chen isomorphism theorem for formally-equivalent smooth hyperbolic vector
fields.
Key words: integrable system; normal form; linearization; nondegenerate singularity
2010 Mathematics Subject Classification: 37G05; 58K50; 37J35
1 Introduction
The main purpose of this paper is to show the following orbital linearization theorem for
smooth (C∞) vector fields which admit a complete set of first integrals near a nondegener-
ate singular point:
Theorem 1.1. Let X be a smooth vector field in a neighborhood of O = (0, . . . , 0) in Rn, which
vanishes at O and satisfies the following conditions:
i) (complete integrability): X admits n − 1 functionally independent smooth first integrals
F1, . . . , Fn−1, i.e., X(F1) = · · · = X(Fn−1) = 0 and dF1 ∧ · · · ∧ dFn−1 6= 0 almost every-
where;
ii) (nondegeneracy 1): the semisimple part of the linear part of X at O is non-zero, and the∞-
jets of F1, . . . , Fn−1 at O are functionally independent (i.e., the ∞-jet of dF1∧ · · · ∧dFn−1
at O is non-zero);
iii) (nondegeneracy 2): If moreover 0 is an eigenvalue of X at O with multiplicity k ≥ 1, then
the differentials of the functions F1, . . . , Fk are linearly independent at O: dF1(O) ∧ · · · ∧
dFk(O) 6= 0. Then there exists a local smooth coordinate system (x1, . . . , xn) in which X
can be written as
X = FX(1),
where X(1) is a semisimple linear vector field in (x1, . . . , xn), and F is a smooth first
integral of X(1), i.e., X(1)(F ) = 0, with F (O) = 1.
mailto:tienzung.nguyen@math.univ-toulouse.fr
https://doi.org/10.3842/SIGMA.2017.093
2 N.T. Zung
The above theorem is in fact more than mere orbital linearization: not only that X is orbitally
equivalent to its linear part X(1), but also the factor F in the expression X = FX(1) in a nor-
malized coordinate system is a first integral of X and X(1). In [13], this kind of linearization is
called geometric linearization.
The formal and analytic case of the above theorem also holds and was shown in [13] in a more
general context of integrable non-Hamiltonian systems of type (p, q), i.e., with p commuting
vector fields and q common first integrals, where p + q = n is the dimension of the manifold.
The vector fields that we study in this paper are integrable of type (1, n−1), i.e., just one vector
field and n− 1 first integrals.
The nondegeneracy condition in Theorem 1.1 is a bit stronger than the nondegeneracy con-
dition in [13]: in [13] the (formal or analytic) vector field X is called integrable nondegenerate
if it satisfies the above conditions i) and ii), without the need of condition iii). (A priori, the
condition that dF1 ∧ · · · ∧ dFn−1(O) 6= 0 is quite stronger than the condition that the ∞-jet of
dF1 ∧ · · · ∧ dFn−1 at O is not zero.) However, in fact, in the formal and analytic case, condi-
tion iii) is a simple consequence of the first two conditions and the theorem about the existence
of (formal or analytic) Poincaré–Dulac normalization [12, 13]. On the other hand, in the smooth
case, we don’t have a proof of the fact that condition iii) follows from conditions i) and ii) in
general, though we do have a proof of this fact for dimension 2.
The rest of this paper is organized as follows. Section 2 is devoted to some preliminary
results, including the classification of nondegenerate singularities of completely integrable vector
fields into (strong/weak) elliptic and hyperbolic cases (Lemma 2.2), and the normalization up
to a flat term (Proposition 2.3). These preliminary results are used in the proof of Theo-
rem 1.1 which is presented in Section 3. Finally, in Section 4, we show that, at least in the case
n = 2, condition iii) of Theorem 1.1 is a consequence of the first two conditions, and can be
dropped from the formulation of the theorem (Theorem 4.1). We conjecture that condition iii)
is redundant in the higher-dimensional case as well.
This paper is part of our program of systematic study of the geometry and topology of
integrable non-Hamiltonian systems. In particular, Theorem 4.1, which is a refinement of Theo-
rem 1.1 in the case of dimension 2, is the starting point of our joint work with Nguyen Van Minh
on the local and global smooth invariants of integrable dynamical systems on 2-dimensional
surfaces [15]. In connection with our results, we would like to mention the theorem of Chape-
ron on smooth equivalence of formally equivalent weakly hyperbolic systems [2], and its recent
application to smooth geometric linearization of some classes of integrable non-Hamiltonian
systems by Jiang [6]. We believe that Chaperon’s techniques will be a key element in our
smooth linearization problem, see also [14].
2 Preliminary results
2.1 Adapted first integrals
We have the following simple lemma, which is similar to the well-known Ziglin’s lemma [11].
Lemma 2.1. Let G1, . . . , Gm be m formal series in n variables which are functionally indepen-
dent. Then there exist m polynomial functions of m variables P1, . . . , Pm such that the homo-
geneous (i.e., lowest degree) parts of the formal series of P1(G1, . . . , Gm), . . . , Pm(G1, . . . , Gm)
are functionally independent.
The proof of the above lemma follows exactly the same lines as the proof of Ziglin of his lemma
in [11], and our situation is simpler than the situation of meromorphic functions considered by
Ziglin.
Orbital Linearization of Smooth Completely Integrable Vector Fields 3
Let X be a smooth completely integrable vector field with a singularity at O. We will say
that the smooth first integrals F1, . . . , Fn−1 of X are adapted first integrals if
dH1 ∧ · · · ∧ dHn−1 6= 0 a.e.,
where Hi = F
(hi)
i denotes the homogeneous part (consisting of non-constant terms of lowest
degree in the Taylor expansion) of Fi at O. Using the above lemma to replace the first integrals
F1, . . . , Fn−1 of X by appropriate polynomial functions of them if necessary, from now on we
can assume that F1, . . . , Fn−1 are adapted.
2.2 The eigenvalues of X
The fact that X admits n−1 first integrals implies that X is very resonant at O. More precisely,
we have:
Lemma 2.2. Let (X,F1, . . . , Fn−1) be smooth nondegenerate at O, i.e., they satisfy the con-
ditions of Theorem 1.1. Then the linear part of X at O is semisimple, and there is a positive
number λ > 0 such that either all the eigenvalues of X at O belong to λZ, or all of them belong
to
√
−1λZ.
Proof. We can assume that H1, . . . ,Hn−1 are functionally independent, where Hi denotes the
homogeneous part of Fi. The equality X(Fi) = 0 implies that
Xss(Hi) = X(1)(Hi) = 0 ∀ i = 1, . . . , n− 1,
where X(1) is the linear part of X, and Xss is the semisimple part of X(1) in the Jordan
decomposition (see, e.g., [12, 13]). We can write
Xss =
n∑
i=1
λizi
∂
∂zi
in a complex coordinate system. Recall that the ring of polynomial first integrals of
n∑
i=1
λizi
∂
∂zi
is generated by the monomial functions
n∏
i=1
zaii , which satisfy the resonance relation
n∑
i=1
aiλi = 0. (2.1)
The fact that H1, . . . ,Hn−1 are independent implies that equation (2.1) has n− 1 linearly inde-
pendent solutions which belong to Zn+, which in turn implies that there is a complex number λ
such that λ1, . . . , λn ∈ λZ. Remark that if the spectrum of Xss contains a complex eigenvalue
λ1 ∈ C \ (R ∪
√
−1R), then its complex conjugate λ1 is also in the spectrum because X is real,
and λ1 and λ1 cannot belong to λZ at the same time for any λ. Thus any eigenvalue of Xss
is either real or pure imaginary. If there is one real non-zero eigenvalue, then we can choose
λ ∈ R+, otherwise we can choose λ ∈
√
−1R+. Notice that λ 6= 0 because at least one eigenvalue
of Xss is non-zero by our assumptions.
The common level sets of H1, . . . ,Hn−1 are 1-dimensional almost everywhere, and since
both X(1) and Xss are tangent to these common level sets, we have that X(1) ∧ Xss = 0,
which implies that X(1) is semisimple, i.e., X(1) = Xss. �
With the above lemma, we can divide the problem into 4 cases (here R∗ = R \ {0}):
4 N.T. Zung
I. Strongly hyperbolic (or hyperbolic without eigenvalue 0): λi ∈ λR∗ ∀ i.
II. Weakly hyperbolic (or hyperbolic with eigenvalue 0): λi ∈ λR∗ ∀ i > k ≥ 1, λ1 = · · · =
λk = 0.
III. Strongly elliptic (or elliptic without eigenvalue 0): λi ∈
√
−1λR∗ ∀ i.
IV. Weakly elliptic (or elliptic with eigenvalue 0): λi ∈
√
−1λR∗ ∀ i > k ≥ 1, λ1 = · · · = λk
= 0.
2.3 Linearization up to a flat term
Using the geometric linearization theorem of [13] in the formal case, we get the following propo-
sition:
Proposition 2.3 (linearization up to a flat term). Assume that X satisfies the hypotheses of
Theorem 1.1. Then there is a local smooth coordinate system (x1, . . . , xn) in which X can be
written as
X = FX(1) + flat,
where X(1) is the linear part of X in the coordinate system (x1, . . . , xn), F is a smooth first
integral of X(1), and flat means a smooth term which is flat at O.
Proof. Denote by X̂ (resp. F̂i) the∞-jet of X (resp. Fi) at O: X̂ is a formal vector field (resp.
function) at O. If (X,F1, . . . , Fn−1) is smooth nondegenerate at O, then (X̂, F̂1, . . . , F̂n−1) is
a nondegenerate formal integrable system of type (1, n − 1) at p. According to the geometric
linearization theorem of [13], this formal integrable system can be linearized geometrically, i.e.,
there is a formal coordinate system (x̂1, . . . , x̂n) in which we have
X̂ = F̂ X̂(1), (2.2)
where X̂(1) is the linear part of X̂ in the formal coordinate system (x̂1, . . . , x̂n), and F̂ is
a formal first integral of X̂(1). By the classical Hilbert–Weyl theorem (see, e.g., Theorem 4.2 of
Chapter XII of [5]) applied to the torus action associated to X(1) (see [12, 13] for this associated
torus action), we can write
F̂ = f̂
(
Q1(x̂1, . . . , x̂n), . . . , Qm(x̂1, . . . , x̂n)
)
, (2.3)
where f̂ is a formal series and Q1(x̂1, . . . , x̂n), . . . , Qm(x̂1, . . . , x̂n) are homogeneous polynomials
generating the ring of polynomial first integrals of X̂(1). Using Borel theorem, we get a smooth
coordinate system (x1, . . . , xn) whose ∞-jet is (x̂1, . . . , x̂n), and a smooth function f of m vari-
ables whose ∞-jet is f̂ . Put
F (x1, . . . , xn) = f
(
Q1(x1, . . . , xn), . . . , Qm(x1, . . . , xn)
)
. (2.4)
Then equations (2.2), (2.3) and (2.4) imply that X = FX(1) + flat in the smooth coordinate
system (x1, . . . , xn). �
2.4 Reduction to the case without eigenvalue 0
Assume that X has zero eigenvalue at O with multiplicity k, and dF1 ∧ · · · ∧ dFk = 0, i.e.,
we can use F1, . . . , Fk as the first k coordinates in our local coordinate systems. Since the
vector field X preserves x1, . . . , xk, we can view it as a k-dimensional family of vector fields on
(n− k)-dimensional spaces
Uc1,...,ck = {F1 = c1, . . . , Fk = ck}
Orbital Linearization of Smooth Completely Integrable Vector Fields 5
(for c1, . . . , ck small enough). It follows from the usual implicit function theorem and the nonde-
generacy condition that on each Uc1,...,ck there is a unique point Oc1,...,ck such that X(Oc1,...,ck)
= 0, and moreover the point Oc1,...,ck depends smoothly on c1, . . . , ck, the eigenvalues of X
at c1, . . . , ck, are non-zero. It also follows from the formal independence of F1, . . . , Fn at O,
that the functions Fk+1, . . . , Fn are formally independent at every point Oc1,...,ck provided that
c1, . . . , ck are sufficiently small. In other words, we have a k-dimensional family of nondegener-
ate singularities of smooth completely integrable (n − k)-dimensional vector fields Xc1,...,ck . In
order to normalize X, it suffices to normalize Xc1,...,ck in a way which depends smoothly on the
parameter.
3 Proof of Theorem 1.1
We will always assume that the vector field X satisfies the hypotheses of Theorem 1.1. The
fact that the linear part of X is semisimple is established by Lemma 2.2. In view of Sec-
tion 2.4, it suffices to prove Theorem 1.1 for the cases without zero eigenvalue, by a proof whose
parametrized version also works the same.
3.1 The hyperbolic case
Assume that X is hyperbolic without eigenvalue 0. According to Proposition 2.3, we can write
X = Y + flat, where Y = FX(1) is a smooth hyperbolic integrable vector field in normal form.
Since X and Y are hyperbolic and coincide up to a flat term, Sternberg–Chen theorem [3, 10]
says that X is locally smoothly isomorphic to Y , i.e., there is a smooth coordinate system in
which X can be written as X = FX(1), where F is a smooth first integral of X(1). Theorem 1.1
is proved in the hyperbolic case without eigenvalue 0. This is also a special case of a result of
Kai Jiang [6] on smooth linearization of weakly hyperbolic integrable vector fields.
3.2 The elliptic case
In this subsection, we will assume that all the eigenvalues of X at O are non-zero pure imaginary.
Using Proposition 2.3, we can assume that X = FX(1)+flat in a local smooth coordinate system
(x1, . . . , xn), where F is a smooth function such that F (O) = 1. Put Y = X/F . Then Y has
the same first integrals as X, and
Y = X(1) + flat.
The fact that X is of strong elliptic type implies immediately that the dimension n is even, the
eigenvalues of X at O are ±
√
−1a1, . . . ,±
√
−1an/2 where a1, . . . , an/2 are positive real numbers,
and we can choose the coordinates (x1, . . . , xn) such that
X(1) =
n/2∑
i=1
ai
(
x2i−1
∂
∂x2i
− x2i
∂
∂x2i−1
)
. (3.1)
According to Lemma 2.2, we can choose λ > 0 such that a1/λ, . . . , an/2/λ are natural numbers
whose greatest common divisor is 1.
Lemma 3.1. Locally near O all the orbits of Y = X/F (except the fixed point O) are periodic,
with periods which are uniformly bounded above and below.
Proof. The vector field (dF1 ∧ · · · ∧ dFn−1)y
(
∂
∂x1
∧ · · · ∧ ∂
∂xn
)
is tangent to Y , and therefore it
is divisible by Y (by de Rham division theorem, because of the nondegeneracy condition, see,
e.g., Appendix A.2 of [4]; in that appendix the de Rham theorem is given for differential forms,
6 N.T. Zung
but it works the same for vector fields, via an isomorphism between the tangent bundle and the
contangent bundle over a manifold), i.e., we can write
(dF1 ∧ · · · ∧ dFn−1)y
(
∂
∂x1
∧ · · · ∧ ∂
∂xn
)
= GY, (3.2)
where G is a smoth non-flat function at O. Notice that the singular locus of the map (F1, . . . ,
Fn−1) : U → Rn−1, where U 3 O is a small neighborhood of O in Rn, coincides with the zero
locus of G.
It is clear that, by continuity, the set of all points x ∈ U such that the orbit of Y through x
is periodic of period ≤ 3π/λ is a closed subset of U . We want to show that this set is actually
equal to U (provided that U is small enough). Consider the singular locus
S = {x ∈ U |G(x) = 0} = {x ∈ U |dF1 ∧ · · · ∧ dFn−1(x) = 0}.
Since G is non-flat at O, we can choose a coordinate system (z1, . . . , zn) which is a linear
transformation of the coordinate system (x1, . . . , xn), such that the homogeneous part G(h) of G
has the form
G(h) = zh1 + · · · (3.3)
(where h denotes the degree of G(h)), which implies that ∂hG
∂zh1
6= 0 in U .
Because ∂hG
∂zh1
does not vanish in U , by the classical Rolle’s theorem on each line {z2 =
const, . . . , zn = const} in U there are at most h zeros of the function G, the intersection of the
singular locus S with each line {z2 = const, . . . , zn = const} in U consists of at most n − 1
points, and the function G is not flat at any point of S.
Due to the ellipticity of the vector field X(1), there must be at least one index j 6= 1 such
that the coefficient of the monomial term z1
∂
∂zj
in X(1) in the coordinate system (z1, . . . , zn) is
not zero. Without loss of generality, we may assume that the coefficient of the monomial term
z1
∂
∂zn
in X(1) is not zero.
Consider the local hyperplane
P = {zn = 0} ⊂ U
with the coordinate system (z1, . . . , zn−1). For each ε > 0 sufficiently small, denote by
qε = (z1 = ε, z2 = 0, . . . , zn = 0) ∈ P
the point in U whose coordinate z1 is equal to ε and the other coordinates zi vanish for every
i ≥ 2. Then the vector fields X(1) and Y are transversal to P at every point qε such that ε > 0
is sufficiently small.
Recall from Section 2.1 that we can, and will, assume that F1, . . . , Fn to be adapted first
integrals, i.e., the homogeneous part of dF1 ∧ · · · ∧ dFn is equal to dH1 ∧ · · · ∧ dHn, where
Hi = F
(hi)
i is the homogeneous part of Fi for each i = 1, . . . , n − 1 and degHi = hi. In
particular, we have h+ 1 =
n−1∑
i=1
(hi − 1), where h is the degree of the homogeneous part G(h) of
the function G in formulas (3.2) and (3.3), and h+ 2 ≥ hi for any i = 1, . . . , n− 1.
Denote by Bn−1(qε, ε
h+2) the (n − 1)-dimensional ball of center qε and radius εh+2 on the
local hyperplane P = {zn = 0}. We observe that, for every ε sufficiently small, the restriction
of the map (F1, . . . , Fn−1) to Bn−1(qε, ε
h+2) is an injective map from Bn−1(qε, ε
h+2) to Rn−1.
Indeed, consider any two distinct points p,q ∈ Bn−1(qε, ε
h+2), p 6= q. Consider the constant
unit vector field Zp,q = q−p
‖q−p‖ on U with respect to the coordinate system (z1, . . . , zn) which
Orbital Linearization of Smooth Completely Integrable Vector Fields 7
maps U onto a neighborhood of the origin in the Euclidean space Rn. The difference q − p
and the norm ‖q− p‖ (i.e., the distance from p to q) are taken with respect to this Euclidean
structure. Due to the fact that our first integrals are adapted, at least one of the functions
Zp,q(F1), . . . , Zp,q(Fn−1), say Zp,q(Fk), has a homogeneous part of the type
cp,qz
hk−1
1 + · · · ,
with a coefficient cp,q 6= 0, and k can be chosen as a function of p and q such that |c(p,q)|
is uniformly bounded by positive constants. Using standard division techniques (see, e.g., the
Malgrange’s preparation theorem [8], but the situation here is much simpler than this general
theorem), one can decompose Zp,q(Fk) as
Zp,q(Fk) = f1,p,qz
hk−1
1 +
n∑
i=2
zifi,p,q,
where fi,p,q are smooth functions on U depending uniformly continuously on the unit vec-
tor Zp,q (as long as the index k remains the same), and with f1,p,q(0) = cp,q. For the points on
Bn−1(qε, ε
h+2) we have that ε− εh+2 ≤ z1 ≤ ε+ εh+2 while |zi| ≤ εh+2 ≤ εhk for all i ≥ 2, hence
in the above expression of Zp,q(Fk) the terms zifi,p,q with i ≥ 2 are very small compared to the
term f1,p,qz
hk−1
1 , and so in particular the sign of Zp,q(Fk) does not change on Bn−1(qε, ε
h+2)
for ε sufficiently small, which implies that Fk(q)−Fk(p) 6= 0 by the mean value theorem. Thus,
we have shown the injectivity of (F1, . . . , Fn−1) on Bn−1(qε, ε
h+2) for every ε sufficiently small.
Consider now the Poincaré map (i.e., the first return map), denoted by φ, defined on the ball
Bn−1(qε, ε
h+2) ⊂ P of the flow of Y in U (a priori the image of this map may lie a bit outside of
Bn−1(qε, ε
h+2) but on the same local hyperplane P ). A priori this map does not necessarily fix
the point qε. But due to the fact the flow of X(1) is periodic (in particular, the Poincaré map
for X(1) is the identity map) and the fact that Y = X(1) + flat (which implies that the Poincaré
map φ for Y deviates from the Poincaré map for X by a flat term), we have that the distance
from qε to φ(qε) is a flat function in ε. In particular, for every ε sufficiently small we have
d(qε, φ(qε)) < εh+2,
where d denotes the Euclidean distance in the coordinate system (z1, . . . , zn), and hence the
point φ(qε) lies in the (n− 1)-dimensional ball Bn−1(qε, ε
h+2).
Due to the invariance of the functions Fi with respect to the vector field Y , and hence
with respect to the Poincaré map φ, we also have that the points qε and φ(qε) have the same
image under the map (F1, . . . , Fn−1). But this map is injective on the ball Bn−1(qε, ε
h+2) which
contains these two points, so in fact these two points must coincide, i.e., we have qε = φ(qε), and
the orbit of the flow of Y through the point qε is a periodic orbit, and the period of this orbit is
equal to 2π/λ plus a small error term which tends to 0 faster than any power of ε when ε tends
to 0.
Denote by V the path-connected component of U \S which contains the points qε. (V is not
equal U \ S in general). Then the orbit of Y through any point q ∈ V is also periodic and its
period is close to 2π/λ (the difference between the period and 2π/λ tends to 0 uniformly when
the radius of U tends to 0). This fact can be proved easily by showing that the set of points
of V which satisfies the mentioned property is closed and open in V at the same time: closed
due to the continuity, and open because (F1, . . . , Fn−1) is regular in V and is preserved by the
flow of Y .
Let q ∈ S be a point in the locus S which also lies on the boundary of V . Then by continuity,
there is also a number T near 2π/λ such that the time-T flow of Y fixes the point q. In other
words, the orbit of Y through q is also periodic, and the period is equal to T or a fraction T/m
8 N.T. Zung
of T for some natural number m. As before, consider a (n−1)-dimensional ball Bn−1(q, δ) which
is centered at q and orthogonal to Y (q), for some δ > 0 small enough. Consider the Poincaré
map φ of Y on Bn−1(q, δ) corresponding to the time T (i.e., if the period of the orbit through q
is T/m then consider the m-time iteration of the usual Poincaré map). Since the intersection
of Bn−1(q, δ) with V contains an open subset of Bn−1(q, δ) whose closure contains q, and the
Poincaré map is identity on that open subset by the above considerations, the Poincaré map on
Bn−1(q, δ) is equal to the identity map plus a flat term at q. On the other hand, this Poincaré
map must preserve the map (F1, . . . , Fn−1)|Bn−1(q,δ), and the determinant of the differential
of this map is not flat at q. It implies that the Poincaré map must be identity in a small
neighborhood of q in Bn−1(q, δ). Thus, we can “engulf” the set of points shown to have periodic
orbits from V to a larger open subset of U which contains the boundary of V . Continuing
this engulfing process, we get that the set of points in U having periodic orbits is actually the
whole U . �
We will linearize Y = X/F orbitally, and then deduce the normalization of X from this
linearization. In order to do that, let us consider the blow-up of Rn atO, which will be denoted by
p : E → U,
where U 3 O is a neighborhood of O in Rn and p−1(O) ∼= RPn−1 is the exceptional divisor of
the blow-up in E. We will need the following simple lemma, whose proof is straightforward:
Lemma 3.2. With the above notations, a function G or a vector field Z is flat at O in U if and
only if its pull-back to E via the projection map p is flat along p−1(O) in E.
(In the above lemma, G and Z are arbitrary, and the pull back from U to E means the con-
tinuous extention from E \ p−1(O) to E of the pull-back by the diffeomorphism p : E \ p−1(O)
→ U \ {O}, if such a continuous extension exists.)
Denote by G̃ (resp. Z̃) the pull-back of a function G (resp. vector field Z) via the projection
map π : E → U of the blow-up. Then we have
Ỹ = X̃(1) + Z̃
in E, where Z̃ is vector field which is flat along p−1(O), and X̃(1) is a smooth periodic vector
field in E of period 2π/λ. By Lemma 3.1, the orbits of Ỹ are closed, with periods close to the
period of X̃(1). Due to the flatness of Z along p−1(O), the period of Ỹ at the points in E is
equal to 2π/λ plus a smooth function on E which is flat along p−1(O). Projecting Ỹ back to U
and using Lemma 3.2, we get a smooth period function P = 2π/λ + flat (which is invariant
on the orbits) such that PY is periodic of period 1. In other words, PY generates a smooth
T1-action. Using the classical Cartan–Bochner smooth linearization theorem for compact group
actions, we find a smooth coordinate system, which we will denote again by (x1, . . . , xn), in
which PX/F = PY is a linear vector field, i.e., in which we have
X = GX(1), (3.4)
where G is a smooth function and X(1) is a linear vector field which satisfies formula (3.1).
A priori, the function F given by Proposition 2.3 is not a first integral of X (though it is a first
integral of the linear part of X in some coordinate system), and so the function G = 2πF/Pλ
in formula (3.4) is not a first integral of X either. But we can normalize further in order to
change G into a first integral. Indeed, by the arguments presented above, we can assume that G
is a smooth first integral of X plus a flat term, or we can write G = G1(1 + flat), where G1 is
a first integral of X. Normalizing the new vector field Y = X/G1 instead of the old Y = X/F ,
Orbital Linearization of Smooth Completely Integrable Vector Fields 9
we get a new smooth coordinate system in which PY = (2π/λ)X(1), where P is the period
function of the new vector field Y , and it is a smooth first integral of the type const + flat. In
this new coordinate system we have that X is equal to its linear part times a first integral, and
Theorem 1.1 is proved in the elliptic case, i.e., without eigenvalue 0.
Since our proof for the strong hyperbolic case and the strong elliptic case also works for
smooth families of integrable vector fields, Theorem 1.1 is proved.
Remark 3.3. According to a theorem of Schwarz [9], the smooth first integral F in the normal
form in the elliptic case can also be written as
F = f
(
Q1(x1, . . . , xn), . . . , Qm(x1, . . . , xn)
)
,
where Q1(x1, . . . , xn), . . . , Qm(x1, . . . , xn) are homogeneous polynomials which generate the ring
of polynomial first integrals of the linear vector field X(1).
4 The case of dimension 2
The aim of this section is to show that condition iii) in Theorem 1.1 is redundant at least in the
case of dimension 2. More precisely, we have:
Theorem 4.1. Let X be a smooth vector field in a neighborhood of O = (0, 0) in R2, which
vanishes at O and satisfies the following conditions:
i) (complete integrability): X admits a smooth first integral F1;
ii) (nondegeneracy): the semisimple part of the linear part of X at O is non-zero, and the
∞-jet of F1 at O is non-constant. Then there exists a local smooth coordinate system (x, y)
in which X can be written as
X = FX(1),
where X(1) is a semisimple linear vector field in (x, y), and F is a smooth first integral
of X(1).
Proof. Remark that, in the case of dimension 2, there are only 3 possibilities: elliptic without
zero eigenvalue, hyperbolic without zero eigenvalue, and hyperbolic with zero eigenvalue. The
first two possibilities are covered by Theorem 1.1. It remains to prove Theorem 4.1 for the case
when X has one eigenvalue equal to 0. By Proposition 2.3, we can assume that
X = F (y)x
∂
∂x
+ flat1
∂
∂x
+ flat2
∂
∂y
in a smooth coordinate system (x, y), where flat1 and flat2 are two flat functions, and F (0) 6= 0.
Denote by
S = {q ∈ U |X(q) = 0}
the singular locus of X near O, where U denotes a small neighborhood of O in R2. The main
point is to prove that S is a smooth curve. If S is a smooth curve, then we can write S = {x = 0},
the vector field X is divisible by x, i.e., Y = X/x is still a smooth vector field, which is non-zero
at O, and therefore locally rectifiable and admits a first integral G such that dG(0) 6= 0. But G
is also a first integral of X, so condition iii) of Theorem 1.1 is also satisfied, and Theorem 4.1
is reduced to a particular case of Theorem 1.1.
10 N.T. Zung
Denote by
S1 = {(x, y) ∈ U |F (y)x+ flat1(x, y) = 0}
the set of points where the ∂
∂x -component of X vanishes. It is clear that S ⊂ S1, and S1 is
a smooth curve tangent to the line {x = 0} at O by the inverse function theorem. We will show
that S = S1.
Consider the open cone
C = {(x, y) ∈ U | |x| < |y|}.
Clearly, S1 ⊂ C ∪{O} (provided that U is small enough). The non-flat first integral F1 of X in
the coordinate system (x, y) has the type
F1 = f(y) + flat,
where f(y) = ahy
h + h.o.t. is a non-flat smooth function. It implies that the level sets of F1 in
the cone C are smooth curves (because ∂F1
∂y 6= 0 in this cone) which are nearly tangent to the
lines {y = const} (because
∣∣∂F1
∂x
∣∣ is very small compared to
∣∣∂F1
∂y
∣∣ in the cone). In particular,
each level set of F1 in C intersects with S1 at exactly 1 point. Since X is tangent to these
level sets, and the ∂
∂x -component of X vanishes at the intersection points of these level sets
with S1, it follows that X itself vanishes at these intersection points. But every point of S1 is an
intersection point of S1 with a level set of F1. Thus X vanishes on S1, and we have S = S1. �
Remark 4.2. Two-dimensional elliptic-like vector fields, i.e., those vector fields whose orbits
near a singular point are closed, are also called centers in the literature. There is a recent
interesting theorem of Maksymenko [7] about the orbital linearization of the center, without the
assumption on the existence of a first integral, but with an assumption on the periods of the
periodic orbits. Maksymenko’s theorem is similar to and a bit stronger than the elliptic case of
Theorem 4.1 because his assumptions are weaker, and the conclusions are the same. His proof
is also based on the formal normalization and the blowing-up method.
Remark 4.3. Some of the arguments of the proof of Theorem 4.1 are still valid in the n-
dimensional case where 0 is an eigenvalue with multiplicity k ≥ 1. In particular, one can still
show that, even without condition iii) of Theorem 1.1, the local singular locus of X is still
a smooth k-dimensional manifold. However, it is more difficult to show that there is still a local
regular invariant (n − k)-dimensional foliation. If one can show the existence of this regular
invariant foliation, then one can drop condition iii) from the statement of Theorem 1.1 because
it is a consequence of the first two conditions. Maybe it is possible to use the techniques of
Belitskii–Kopanskii [1] together with a kind of desingularization of the first integrals in order to
show the existence of an invariant regular foliation, but we don’t have a proof so far.
Remark 4.4. As pointed out by a referee of this paper, there is a less elementary but more
dynamical proof of Theorem 4.1 which uses a Cr-central manifold ofX, where r can be arbitrarily
large.
Acknowledgement
The first version of this manuscript was available since 2012 as an unpublished preprint (see
arXiv:1204.5701v1). It was then revised and submitted during the author’s stay at the School
of Mathematical Sciences, Shanghai Jiao Tong University, as a visiting professor in 2017. He
would like to thank Shanghai Jiao Tong University, and especially Tudor Ratiu, Jianshu Li, and
Jie Hu for the invitation, hospitality and excellent working conditions.
The authors would also like to thank the referees of this paper for many pertinent remarks
which helped improve the presentation of the paper.
Orbital Linearization of Smooth Completely Integrable Vector Fields 11
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1 Introduction
2 Preliminary results
2.1 Adapted first integrals
2.2 The eigenvalues of X
2.3 Linearization up to a flat term
2.4 Reduction to the case without eigenvalue 0
3 Proof of Theorem 1.1
3.1 The hyperbolic case
3.2 The elliptic case
4 The case of dimension 2
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