Poincaré-Dulac theorem

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
[a11]	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
[a12]	B. Jakubczyk, W. Respondek, "On the linearization of control systems" Bull. Acad. Polon. Sci. Ser. Math. Astron. Phys. , 28 (1980) pp. 517–522
[a13]	E.S. Livingston, D.L. Elliott, "Linearization of families of vectorfields" J. Diff. Equations , 55 (1984) pp. 289–299