View source for Poincaré-Dulac theorem

Consider a (formal) differential equation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073030/p0730301.png" />-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/p0730302.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

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

<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>

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

<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>

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

<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>

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" />.

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.

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" />,

<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>

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).

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

<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>

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" />.

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" />.

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]]].

====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>