View source for Poincaré-Dulac theorem

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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, [[#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

$$ 
$$

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]]].

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====
<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>