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Consider a (formal) differential equation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p0730301.png" />-variables,
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p0730302.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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A collection of eigen values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p0730303.png" /> is said to be resonant if there is a relation of the form
+
Consider a (formal) differential equation in  $  n $ variables,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p0730304.png" /></td> </tr></table>
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$$ \tag{a1 }
 +
\dot{x}  = A x + ( \textrm{ higher  degree  } ) .
 +
$$
  
for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p0730305.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p0730306.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p0730307.png" />. The Poincaré theorem on canonical forms for formal differential equations says that if the eigen values of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p0730308.png" /> in (a1) are non-resonant, then there is a formal substitution of variables of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p0730309.png" /> (higher degree) which makes (a1) take the form
+
A collection of eigen values ( \lambda _ {1}, \dots, \lambda _ {n} ) $
 +
is said to be resonant if there is a relation of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303010.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
$$
 +
\lambda _ {r}  = m _ {1} \lambda _ {1} + \dots + m _ {n} \lambda _ {n}  $$
  
Part of the Poincaré–Dulac theorem says that there is for any equation of the form (a1) a formal change of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303011.png" /> (higher degree) which transforms (a1) into an equation of the form
+
for some  $  r \in \{ 1, \dots, n \} $,
 +
with  $  m _ {i} \in \mathbf N \cup \{ 0 \} $,
 +
$  \sum _ {i=1}  ^ {n} m _ {i} \geq  2 $.
 +
The Poincaré theorem on canonical forms for formal differential equations says that if the eigen values of the matrix  $  A $
 +
in (a1) are non-resonant, then there is a formal substitution of variables of the form  $  y = x+ $(higher degree) which makes (a1) take the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303012.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
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$$ \tag{a2 }
 +
\dot{y}  = A y.
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303013.png" /> is a power series of which all monomials are resonant. Here a monomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303015.png" /> is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303016.png" />-th element of the standard basis, is called resonant if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303017.png" />, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303018.png" /> are the eigen values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303019.png" />.
+
Part of the Poincaré–Dulac theorem says that there is for any equation of the form (a1) a formal change of variables  $  y = x + $(higher degree) which transforms (a1) into an equation of the form
  
A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303020.png" /> (a collection of eigen values) belongs to the Poincaré domain if 0 is not in the convex hull of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303021.png" />; the complementary set of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303022.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303023.png" /> is in the convex hull of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303024.png" /> is called the Siegel domain. The second part of the Poincaré–Dulac theorem now says that if the right-hand side of (a1) is holomorphic and the eigen value set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303025.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303026.png" /> is in the Poincaré domain, then there is a holomorphic change of variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303027.png" /> (higher degree) taking (a1) to a canonical form (a3), with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303028.png" /> a polynomial in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303029.png" /> consisting of resonant monomials.
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$$ \tag{a3 }
 +
\dot{y}  = Ay + w( y) ,
 +
$$
  
A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303030.png" /> is said to be of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303031.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303032.png" /> is a constant, if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303033.png" />,
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where  $  w( y) $
 +
is a power series of which all monomials are resonant. Here a monomial  $  y  ^ {m} e _ {r} $,  
 +
where $  e _ {r} $
 +
is the  $  r $-th element of the standard basis, is called resonant if $  \lambda _ {r} = m _ {1} \lambda _ {1} + \dots + m _ {n} \lambda _ {n} $,
 +
where the  $  \lambda _ {i} $
 +
are the eigen values of  $  A $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303034.png" /></td> </tr></table>
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A point  $  \lambda = ( \lambda _ {1}, \dots, \lambda _ {n} ) \in \mathbf C  ^ {n} $ (a collection of eigen values) belongs to the Poincaré domain if 0 is not in the convex hull of the  $  \lambda _ {1} \dots \lambda _ {n} $;  
 +
the complementary set of all  $  \lambda $
 +
such that  $  0 $
 +
is in the convex hull of the  $  \lambda _ {1}, \dots, \lambda _ {n} $
 +
is called the Siegel domain. The second part of the Poincaré–Dulac theorem now says that if the right-hand side of (a1) is holomorphic and the eigen value set  $  ( \lambda _ {1}, \dots, \lambda _ {n} ) $
 +
of  $  A $
 +
is in the Poincaré domain, then there is a holomorphic change of variables  $  y = x + $(higher degree) taking (a1) to a canonical form (a3), with  $  w( y) $
 +
a polynomial in  $  y $
 +
consisting of resonant monomials.
  
The Siegel theorem says that if the eigen values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303035.png" /> constitute a vector of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303036.png" /> and (a1) is holomorphic, then in a neighbourhood of zero (a1) is holomorphically equivalent to (a2), i.e. there is a holomorphic change of coordinates taking (a1) to (a2).
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A point  $  \lambda \in \mathbf C  ^ {n} $
 +
is said to be of type ( C, \nu ) $,  
 +
where  $  C $
 +
is a constant, if for all  $  r = 1, \dots, n $,
  
In the differentiable (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303037.png" />-) case there are related results, [[#References|[a3]]]. Consider a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303038.png" /> vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303039.png" /> (or the corresponding autonomous system of differential equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303040.png" />). A critical point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303041.png" />, i.e. a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303042.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303043.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303044.png" />, is called an elementary critical point
+
$$
 +
\left | \lambda _ {r} - \sum _ { i= 1} ^ { n }  m _ {i} \lambda _ {i} \right | \geq  \
 +
C \left ( \sum _ { i= 1} ^ { n }  m _ {i} \right ) ^ {- \nu } .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303045.png" /></td> </tr></table>
+
The Siegel theorem says that if the eigen values of  $  A $
 +
constitute a vector of type  $  ( C, \nu ) $
 +
and (a1) is holomorphic, then in a neighbourhood of zero (a1) is holomorphically equivalent to (a2), i.e. there is a holomorphic change of coordinates taking (a1) to (a2).
  
if the real part of each eigen value of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303046.png" /> is non-zero. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303047.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303048.png" /> vector field with 0 as an elementary critical point. Then in a neighbourhood of zero, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303049.png" /> decomposes as a sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303050.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303051.png" /> vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303053.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303054.png" />, and with respect to a suitable coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303056.png" /> is of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303057.png" /> with the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303058.png" /> similar to a diagonal matrix, and the linear part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303059.png" /> can be represented by a nilpotent matrix (Chen's decomposition theorem). This is a non-linear <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303060.png" /> analogue of the decomposition of a matrix into commuting semi-simple and nilpotent parts, cf. [[Jordan decomposition|Jordan decomposition]]. Now let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303061.png" /> be a second vector field with 0 as an elementary critical point and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303062.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303063.png" /> be the [[Taylor series|Taylor series]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303064.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303065.png" /> around 0. Then there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303066.png" /> transformation around 0 which carries <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303067.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303068.png" /> if and only if there exists a formal transformation which carries the formal vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303069.png" /> to the formal vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303070.png" />.
+
In the differentiable ( $  C  ^  \infty  $-) case there are related results, [[#References|[a3]]]. Consider a $  C  ^  \infty  $
 +
vector field $  X= \sum a ^ {i} ( x) ( \partial  / {\partial  x  ^ {i} } ) $ (or the corresponding autonomous system of differential equations  $  {\dot{x} } {}  ^ {i} = a ^ {i} ( x) $).  
 +
A critical point of  $  X $,  
 +
i.e. a point  $  p $
 +
such that  $  a ^ {i} ( p) = 0 $,
 +
$  i = 1, \dots, n $,  
 +
is called an elementary critical point
  
A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303071.png" />-linearization result due to S. Sternberg says the following [[#References|[a4]]], [[#References|[a5]]]. If the matrix of linear terms of the equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303072.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303073.png" />, is semi-simple and the set of eigen values of this matrix is non-resonant, then there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303074.png" /> change of coordinates which linearizes the equations. For results in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303075.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303076.png" /> case cf. [[#References|[a6]]], [[#References|[a7]]].
+
$$
 +
$$
  
The Poincaré–Dulac theorem can be seen as a result on canonical forms of non-linear representations of the one-dimensional nilpotent Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303077.png" />. In this form it generalizes to arbitrary nilpotent Lie algebras. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303078.png" /> be a finite-dimensional nilpotent Lie algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303079.png" /> (cf. [[Lie algebra, nilpotent|Lie algebra, nilpotent]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303080.png" /> be the Lie algebra of formal vector fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303081.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303082.png" />. A formal non-linear representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303083.png" /> is a homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303084.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303085.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303086.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303087.png" /> is the homogeneous part of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303088.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303089.png" />). Such a representation is holomorphic if for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303090.png" /> the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303091.png" /> converges in some neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303092.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303093.png" /> is a [[Linear representation|linear representation]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303094.png" />, called the linear part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303095.png" />. A formal vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303096.png" /> is called resonant with respect to a linear representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303097.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303098.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p07303099.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p073030100.png" />. The representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p073030101.png" /> is normal if each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p073030102.png" /> is resonant with respect to the semi-simple part (cf. [[Jordan decomposition|Jordan decomposition]]) of the linear representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p073030103.png" />. The Poincaré–Dulac theorem for nilpotent Lie algebras, [[#References|[a8]]], now says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p073030104.png" /> is a holomorphic non-linear representation of a nilpotent Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p073030105.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p073030106.png" />, and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p073030107.png" /> satisfies the Poincaré condition, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p073030108.png" /> is holomorphically equivalent to a polynomial normal representation. In this setting the Poincaré condition (i.e., belonging to the Poincaré domain) takes the form that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p073030109.png" /> does not belong to the convex hull of the weights (cf. [[Weight of a representation of a Lie algebra|Weight of a representation of a Lie algebra]]) of the linear part <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p073030110.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p073030111.png" />.
+
if the real part of each eigen value of the matrix  $  ( {\partial  a  ^ {i} } / {\partial  x  ^ {j} } ) ( p) $
 +
is non-zero. Let  $  X $
 +
be a  $  C  ^  \infty  $
 +
vector field with 0 as an elementary critical point. Then in a neighbourhood of zero,  $  X $
 +
decomposes as a sum  $  X = S+ N $
 +
of  $  C  ^  \infty  $
 +
vector fields  $  S $
 +
and  $  N $
 +
satisfying  $  [ S, N] = 0 $,
 +
and with respect to a suitable coordinate system  $  y $,
 +
$  S $
 +
is of the form  $  S = \sum _ {i,j} c _ {j}  ^ {i} y  ^ {j} ( \partial  / {\partial  y _ {j} } ) $
 +
with the matrix  $  ( c _ {j}  ^ {i} ) $
 +
similar to a diagonal matrix, and the linear part of  $  N $
 +
can be represented by a nilpotent matrix (Chen's decomposition theorem). This is a non-linear  $  C  ^  \infty  $
 +
analogue of the decomposition of a matrix into commuting semi-simple and nilpotent parts, cf. [[Jordan decomposition|Jordan decomposition]]. Now let  $  Y = \sum b  ^ {i} ( x) ( \partial  / {\partial  x  ^ {i} } ) $
 +
be a second vector field with 0 as an elementary critical point and let  $  {\widehat{a}  } {}  ^ {i} ( x) $
 +
and  $  {\widehat{b}  } {}  ^ {i} ( x) $
 +
be the [[Taylor series|Taylor series]] of  $  a  ^ {i} ( x) $
 +
and  $  b  ^ {i} ( x) $
 +
around 0. Then there exists a  $  C  ^  \infty  $
 +
transformation around 0 which carries  $  X $
 +
to  $  Y $
 +
if and only if there exists a formal transformation which carries the formal vector field  $  \sum {\widehat{a}  } {}  ^ {i} ( x) ( \partial  / {\partial  x  ^ {i} } ) $
 +
to the formal vector field  $  \sum {\widehat{b}  } {}  ^ {i} ( x) ( \partial  / {\partial  x  ^ {i} } ) $.
 +
 
 +
A  $  C  ^  \infty  $-linearization result due to S. Sternberg says the following [[#References|[a4]]], [[#References|[a5]]]. If the matrix of linear terms of the equations  $  {\dot{x} } {}  ^ {i} = \sum a  ^ {i} ( x) $,
 +
$  a  ^ {i} ( 0) = 0 $,
 +
is semi-simple and the set of eigen values of this matrix is non-resonant, then there is a  $  C  ^  \infty  $
 +
change of coordinates which linearizes the equations. For results in the  $  C  ^ {1} $
 +
and  $  C  ^ {0} $
 +
case cf. [[#References|[a6]]], [[#References|[a7]]].
 +
 
 +
The Poincaré–Dulac theorem can be seen as a result on canonical forms of non-linear representations of the one-dimensional nilpotent Lie algebra $  \mathfrak g = \mathbf C $.  
 +
In this form it generalizes to arbitrary nilpotent Lie algebras. Let $  \mathfrak g $
 +
be a finite-dimensional nilpotent Lie algebra over $  \mathbf C $ (cf. [[Lie algebra, nilpotent|Lie algebra, nilpotent]]). Let $  V _ {n} $
 +
be the Lie algebra of formal vector fields $  \sum a  ^ {i} ( x) ( \partial  / {\partial  x  ^ {i} } ) $,  
 +
$  a  ^ {i} ( 0) = 0 $.  
 +
A formal non-linear representation of $  \mathfrak g $
 +
is a homomorphism $  \rho = \sum _ {n \geq  1 }  \rho  ^ {n} $
 +
of $  \mathfrak g $
 +
in $  V _ {n} $ (where $  \rho  ^ {n} $
 +
is the homogeneous part of degree $  n $
 +
in $  \rho $).  
 +
Such a representation is holomorphic if for each $  X $
 +
the series $  \sum \rho  ^ {n} ( X) $
 +
converges in some neighbourhood of 0 $.  
 +
Then $  \rho  ^ {1} $
 +
is a [[Linear representation|linear representation]] of $  \mathfrak g $,  
 +
called the linear part of $  \rho $.  
 +
A formal vector field $  \xi \in V _ {n} $
 +
is called resonant with respect to a linear representation $  \sigma  ^ {1} $
 +
of $  \mathfrak g $
 +
if $  [ \sigma  ^ {1} ( X), \xi ] = 0 $
 +
for all $  X \in \mathfrak g $.  
 +
The representation $  \rho $
 +
is normal if each $  \rho ( X) $
 +
is resonant with respect to the semi-simple part (cf. [[Jordan decomposition|Jordan decomposition]]) of the linear representation $  \rho  ^ {1} $.  
 +
The Poincaré–Dulac theorem for nilpotent Lie algebras, [[#References|[a8]]], now says that $  \rho $
 +
is a holomorphic non-linear representation of a nilpotent Lie algebra $  \mathfrak g $
 +
over $  \mathbf C $,  
 +
and if $  \rho $
 +
satisfies the Poincaré condition, then $  \rho $
 +
is holomorphically equivalent to a polynomial normal representation. In this setting the Poincaré condition (i.e., belonging to the Poincaré domain) takes the form that 0 $
 +
does not belong to the convex hull of the weights (cf. [[Weight of a representation of a Lie algebra|Weight of a representation of a Lie algebra]]) of the linear part $  \rho  ^ {1} $
 +
of $  \rho $.
  
 
For rather complete accounts of the Poincaré–Dulac and Siegel theorems cf. [[#References|[a9]]], [[#References|[a10]]].
 
For rather complete accounts of the Poincaré–Dulac and Siegel theorems cf. [[#References|[a9]]], [[#References|[a10]]].
  
In control theory one studies equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p073030112.png" /> with a control parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p073030113.png" />; for instance, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p073030114.png" />. This naturally leads to linearization problems for families of vector fields. In this setting more general notions of equivalence, involving, in particular, feedback laws <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p073030115.png" />, are also natural (linearization by feedback). A selection of references is [[#References|[a11]]]–[[#References|[a13]]].
+
In control theory one studies equations $  \dot{x} = f( x, u) $
 +
with a control parameter $  u $;  
 +
for instance, $  \dot{x} = f( x) + u _ {1} g _ {1} ( x) + \dots + u _ {m} g _ {m} ( x) $.  
 +
This naturally leads to linearization problems for families of vector fields. In this setting more general notions of equivalence, involving, in particular, feedback laws $  u = h( x, v) $,  
 +
are also natural (linearization by feedback). A selection of references is [[#References|[a11]]]–[[#References|[a13]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Poincaré, , ''Oeuvres'' , '''1''' , Gauthier-Villars (1951) pp. UL-CXXXII</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Dulac,   "Recherches sur les points singuliers des equations différentielles" ''J. Ecole Polytechn. Ser. II'' , '''9''' (1904) pp. 1–25</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> K.-T. Chen,   "Equivalence and decomposition of vectorfields about an elementary critical point" ''Amer. J. Math.'' , '''85''' (1963) pp. 693–722</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> F. Bruhat,   "Travaux de Sternberg" ''Sém. Bourbaki'' , '''13''' (1960–1961) pp. Exp. 2187</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> S. Sternberg,   "On the structure of a local homeomorphism" ''Amer. J. Math.'' , '''80''' (1958) pp. 623–631</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> M. Nagumo,   K. Isé,   "On the normal forms of differential equations in the neighbourhood of an equilibrium point" ''Osaka Math. J.'' , '''9''' (1957) pp. 221–234</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> P. Hartman,   "On the local linearization of differential equations" ''Proc. Amer. Math. Soc.'' , '''14''' (1963) pp. 568–573</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> D. Arnal,   M. Ben Ammar,   G. Pinczon,   "The Poincaré–Dulac theorem for nonlinear representations of nilpotent Lie algebras" ''Lett. Math. Phys.'' , '''8''' (1984) pp. 467–476</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> V.I. Arnol'd,   "Geometrical methods in the theory of ordinary differential equations" , Springer (1977) pp. Chapt. V (Translated from Russian)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> A.D. Bryuno,   "Analytic forms of differential equations" ''Trans. Moscow Math. Soc.'' , '''25''' (1971) pp. 131–288 ''Trudy Moskov. Mat. Obshch.'' , '''25''' (1971) pp. 119–262</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> L.R. Hunt,   R. Su,   "Linear equivalents of nonlinear time-varying systems" , ''Internat. Symp. Math. Th. Networks and Systems Santa Monica, 1983'' , '''4''' , Western Periodicals (1981) pp. 119–123</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> B. Jakubczyk,   W. Respondek,   "On the linearization of control systems" ''Bull. Acad. Polon. Sci. Ser. Math. Astron. Phys.'' , '''28''' (1980) pp. 517–522</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> E.S. Livingston,   D.L. Elliott,   "Linearization of families of vectorfields" ''J. Diff. Equations'' , '''55''' (1984) pp. 289–299</TD></TR></table>
+
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Poincaré, , ''Oeuvres'' , '''1''' , Gauthier-Villars (1951) pp. UL-CXXXII {{MR|1787680}} {{MR|1401792}} {{MR|1401348}} {{MR|1401791}} {{MR|0392476}} {{MR|0392475}} {{MR|1554897}} {{ZBL|0894.01021}} {{ZBL|0894.01020}} {{ZBL|0894.01019}} {{ZBL|0072.24103}} {{ZBL|0059.00104}} {{ZBL|0059.00103}} {{ZBL|0049.44102}} {{ZBL|0041.37403}} {{ZBL|0041.37402}} {{ZBL|46.0004.01}} {{ZBL|36.0022.04}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Dulac, "Recherches sur les points singuliers des equations différentielles" ''J. Ecole Polytechn. Ser. II'' , '''9''' (1904) pp. 1–25</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> K.-T. Chen, "Equivalence and decomposition of vectorfields about an elementary critical point" ''Amer. J. Math.'' , '''85''' (1963) pp. 693–722</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> F. Bruhat, "Travaux de Sternberg" ''Sém. Bourbaki'' , '''13''' (1960–1961) pp. Exp. 2187</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> S. Sternberg, "On the structure of a local homeomorphism" ''Amer. J. Math.'' , '''80''' (1958) pp. 623–631</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> M. Nagumo, K. Isé, "On the normal forms of differential equations in the neighbourhood of an equilibrium point" ''Osaka Math. J.'' , '''9''' (1957) pp. 221–234</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> P. Hartman, "On the local linearization of differential equations" ''Proc. Amer. Math. Soc.'' , '''14''' (1963) pp. 568–573</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> D. Arnal, M. Ben Ammar, G. Pinczon, "The Poincaré–Dulac theorem for nonlinear representations of nilpotent Lie algebras" ''Lett. Math. Phys.'' , '''8''' (1984) pp. 467–476</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> V.I. Arnol'd, "Geometrical methods in the theory of ordinary differential equations" , Springer (1977) pp. Chapt. V (Translated from Russian)</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> A.D. Bryuno, "Analytic forms of differential equations" ''Trans. Moscow Math. Soc.'' , '''25''' (1971) pp. 131–288 ''Trudy Moskov. Mat. Obshch.'' , '''25''' (1971) pp. 119–262</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top"> L.R. Hunt, R. Su, "Linear equivalents of nonlinear time-varying systems" , ''Internat. Symp. Math. Th. Networks and Systems Santa Monica, 1983'' , '''4''' , Western Periodicals (1981) pp. 119–123</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top"> B. Jakubczyk, W. Respondek, "On the linearization of control systems" ''Bull. Acad. Polon. Sci. Ser. Math. Astron. Phys.'' , '''28''' (1980) pp. 517–522</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> E.S. Livingston, D.L. Elliott, "Linearization of families of vectorfields" ''J. Diff. Equations'' , '''55''' (1984) pp. 289–299</TD></TR></table>

Latest revision as of 05:14, 24 February 2022


Consider a (formal) differential equation in $ n $ variables,

$$ \tag{a1 } \dot{x} = A x + ( \textrm{ higher degree } ) . $$

A collection of eigen values $ ( \lambda _ {1}, \dots, \lambda _ {n} ) $ is said to be resonant if there is a relation of the form

$$ \lambda _ {r} = m _ {1} \lambda _ {1} + \dots + m _ {n} \lambda _ {n} $$

for some $ r \in \{ 1, \dots, n \} $, with $ m _ {i} \in \mathbf N \cup \{ 0 \} $, $ \sum _ {i=1} ^ {n} m _ {i} \geq 2 $. The Poincaré theorem on canonical forms for formal differential equations says that if the eigen values of the matrix $ A $ in (a1) are non-resonant, then there is a formal substitution of variables of the form $ y = x+ $(higher degree) which makes (a1) take the form

$$ \tag{a2 } \dot{y} = A y. $$

Part of the Poincaré–Dulac theorem says that there is for any equation of the form (a1) a formal change of variables $ y = x + $(higher degree) which transforms (a1) into an equation of the form

$$ \tag{a3 } \dot{y} = Ay + w( y) , $$

where $ w( y) $ is a power series of which all monomials are resonant. Here a monomial $ y ^ {m} e _ {r} $, where $ e _ {r} $ is the $ r $-th element of the standard basis, is called resonant if $ \lambda _ {r} = m _ {1} \lambda _ {1} + \dots + m _ {n} \lambda _ {n} $, where the $ \lambda _ {i} $ are the eigen values of $ A $.

A point $ \lambda = ( \lambda _ {1}, \dots, \lambda _ {n} ) \in \mathbf C ^ {n} $ (a collection of eigen values) belongs to the Poincaré domain if 0 is not in the convex hull of the $ \lambda _ {1} \dots \lambda _ {n} $; the complementary set of all $ \lambda $ such that $ 0 $ is in the convex hull of the $ \lambda _ {1}, \dots, \lambda _ {n} $ is called the Siegel domain. The second part of the Poincaré–Dulac theorem now says that if the right-hand side of (a1) is holomorphic and the eigen value set $ ( \lambda _ {1}, \dots, \lambda _ {n} ) $ of $ A $ is in the Poincaré domain, then there is a holomorphic change of variables $ y = x + $(higher degree) taking (a1) to a canonical form (a3), with $ w( y) $ a polynomial in $ y $ consisting of resonant monomials.

A point $ \lambda \in \mathbf C ^ {n} $ is said to be of type $ ( C, \nu ) $, where $ C $ is a constant, if for all $ r = 1, \dots, n $,

$$ \left | \lambda _ {r} - \sum _ { i= 1} ^ { n } m _ {i} \lambda _ {i} \right | \geq \ C \left ( \sum _ { i= 1} ^ { n } m _ {i} \right ) ^ {- \nu } . $$

The Siegel theorem says that if the eigen values of $ A $ constitute a vector of type $ ( C, \nu ) $ and (a1) is holomorphic, then in a neighbourhood of zero (a1) is holomorphically equivalent to (a2), i.e. there is a holomorphic change of coordinates taking (a1) to (a2).

In the differentiable ( $ C ^ \infty $-) case there are related results, [a3]. Consider a $ C ^ \infty $ vector field $ X= \sum a ^ {i} ( x) ( \partial / {\partial x ^ {i} } ) $ (or the corresponding autonomous system of differential equations $ {\dot{x} } {} ^ {i} = a ^ {i} ( x) $). A critical point of $ X $, i.e. a point $ p $ such that $ a ^ {i} ( p) = 0 $, $ i = 1, \dots, n $, is called an elementary critical point

$$ $$

if the real part of each eigen value of the matrix $ ( {\partial a ^ {i} } / {\partial x ^ {j} } ) ( p) $ is non-zero. Let $ X $ be a $ C ^ \infty $ vector field with 0 as an elementary critical point. Then in a neighbourhood of zero, $ X $ decomposes as a sum $ X = S+ N $ of $ C ^ \infty $ vector fields $ S $ and $ N $ satisfying $ [ S, N] = 0 $, and with respect to a suitable coordinate system $ y $, $ S $ is of the form $ S = \sum _ {i,j} c _ {j} ^ {i} y ^ {j} ( \partial / {\partial y _ {j} } ) $ with the matrix $ ( c _ {j} ^ {i} ) $ similar to a diagonal matrix, and the linear part of $ N $ can be represented by a nilpotent matrix (Chen's decomposition theorem). This is a non-linear $ C ^ \infty $ analogue of the decomposition of a matrix into commuting semi-simple and nilpotent parts, cf. Jordan decomposition. Now let $ Y = \sum b ^ {i} ( x) ( \partial / {\partial x ^ {i} } ) $ be a second vector field with 0 as an elementary critical point and let $ {\widehat{a} } {} ^ {i} ( x) $ and $ {\widehat{b} } {} ^ {i} ( x) $ be the Taylor series of $ a ^ {i} ( x) $ and $ b ^ {i} ( x) $ around 0. Then there exists a $ C ^ \infty $ transformation around 0 which carries $ X $ to $ Y $ if and only if there exists a formal transformation which carries the formal vector field $ \sum {\widehat{a} } {} ^ {i} ( x) ( \partial / {\partial x ^ {i} } ) $ to the formal vector field $ \sum {\widehat{b} } {} ^ {i} ( x) ( \partial / {\partial x ^ {i} } ) $.

A $ C ^ \infty $-linearization result due to S. Sternberg says the following [a4], [a5]. If the matrix of linear terms of the equations $ {\dot{x} } {} ^ {i} = \sum a ^ {i} ( x) $, $ a ^ {i} ( 0) = 0 $, is semi-simple and the set of eigen values of this matrix is non-resonant, then there is a $ C ^ \infty $ change of coordinates which linearizes the equations. For results in the $ C ^ {1} $ and $ C ^ {0} $ case cf. [a6], [a7].

The Poincaré–Dulac theorem can be seen as a result on canonical forms of non-linear representations of the one-dimensional nilpotent Lie algebra $ \mathfrak g = \mathbf C $. In this form it generalizes to arbitrary nilpotent Lie algebras. Let $ \mathfrak g $ be a finite-dimensional nilpotent Lie algebra over $ \mathbf C $ (cf. Lie algebra, nilpotent). Let $ V _ {n} $ be the Lie algebra of formal vector fields $ \sum a ^ {i} ( x) ( \partial / {\partial x ^ {i} } ) $, $ a ^ {i} ( 0) = 0 $. A formal non-linear representation of $ \mathfrak g $ is a homomorphism $ \rho = \sum _ {n \geq 1 } \rho ^ {n} $ of $ \mathfrak g $ in $ V _ {n} $ (where $ \rho ^ {n} $ is the homogeneous part of degree $ n $ in $ \rho $). Such a representation is holomorphic if for each $ X $ the series $ \sum \rho ^ {n} ( X) $ converges in some neighbourhood of $ 0 $. Then $ \rho ^ {1} $ is a linear representation of $ \mathfrak g $, called the linear part of $ \rho $. A formal vector field $ \xi \in V _ {n} $ is called resonant with respect to a linear representation $ \sigma ^ {1} $ of $ \mathfrak g $ if $ [ \sigma ^ {1} ( X), \xi ] = 0 $ for all $ X \in \mathfrak g $. The representation $ \rho $ is normal if each $ \rho ( X) $ is resonant with respect to the semi-simple part (cf. Jordan decomposition) of the linear representation $ \rho ^ {1} $. The Poincaré–Dulac theorem for nilpotent Lie algebras, [a8], now says that $ \rho $ is a holomorphic non-linear representation of a nilpotent Lie algebra $ \mathfrak g $ over $ \mathbf C $, and if $ \rho $ satisfies the Poincaré condition, then $ \rho $ is holomorphically equivalent to a polynomial normal representation. In this setting the Poincaré condition (i.e., belonging to the Poincaré domain) takes the form that $ 0 $ does not belong to the convex hull of the weights (cf. Weight of a representation of a Lie algebra) of the linear part $ \rho ^ {1} $ of $ \rho $.

For rather complete accounts of the Poincaré–Dulac and Siegel theorems cf. [a9], [a10].

In control theory one studies equations $ \dot{x} = f( x, u) $ with a control parameter $ u $; for instance, $ \dot{x} = f( x) + u _ {1} g _ {1} ( x) + \dots + u _ {m} g _ {m} ( x) $. This naturally leads to linearization problems for families of vector fields. In this setting more general notions of equivalence, involving, in particular, feedback laws $ u = h( x, v) $, are also natural (linearization by feedback). A selection of references is [a11][a13].

References

[a1] H. Poincaré, , Oeuvres , 1 , Gauthier-Villars (1951) pp. UL-CXXXII MR1787680 MR1401792 MR1401348 MR1401791 MR0392476 MR0392475 MR1554897 Zbl 0894.01021 Zbl 0894.01020 Zbl 0894.01019 Zbl 0072.24103 Zbl 0059.00104 Zbl 0059.00103 Zbl 0049.44102 Zbl 0041.37403 Zbl 0041.37402 Zbl 46.0004.01 Zbl 36.0022.04
[a2] H. Dulac, "Recherches sur les points singuliers des equations différentielles" J. Ecole Polytechn. Ser. II , 9 (1904) pp. 1–25
[a3] K.-T. Chen, "Equivalence and decomposition of vectorfields about an elementary critical point" Amer. J. Math. , 85 (1963) pp. 693–722
[a4] F. Bruhat, "Travaux de Sternberg" Sém. Bourbaki , 13 (1960–1961) pp. Exp. 2187
[a5] S. Sternberg, "On the structure of a local homeomorphism" Amer. J. Math. , 80 (1958) pp. 623–631
[a6] M. Nagumo, K. Isé, "On the normal forms of differential equations in the neighbourhood of an equilibrium point" Osaka Math. J. , 9 (1957) pp. 221–234
[a7] P. Hartman, "On the local linearization of differential equations" Proc. Amer. Math. Soc. , 14 (1963) pp. 568–573
[a8] D. Arnal, M. Ben Ammar, G. Pinczon, "The Poincaré–Dulac theorem for nonlinear representations of nilpotent Lie algebras" Lett. Math. Phys. , 8 (1984) pp. 467–476
[a9] V.I. Arnol'd, "Geometrical methods in the theory of ordinary differential equations" , Springer (1977) pp. Chapt. V (Translated from Russian)
[a10] A.D. Bryuno, "Analytic forms of differential equations" Trans. Moscow Math. Soc. , 25 (1971) pp. 131–288 Trudy Moskov. Mat. Obshch. , 25 (1971) pp. 119–262
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How to Cite This Entry:
Poincaré-Dulac theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poincar%C3%A9-Dulac_theorem&oldid=23472