Poincaré-Dulac theorem

Consider a (formal) differential equation in -variables,

(a1)

A collection of eigen values is said to be resonant if there is a relation of the form

for some , with , . The Poincaré theorem on canonical forms for formal differential equations says that if the eigen values of the matrix in (a1) are non-resonant, then there is a formal substitution of variables of the form (higher degree) which makes (a1) take the form

(a2)

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

(a3)

where is a power series of which all monomials are resonant. Here a monomial , where is the -th element of the standard basis, is called resonant if , where the are the eigen values of .

A point (a collection of eigen values) belongs to the Poincaré domain if 0 is not in the convex hull of the ; the complementary set of all such that is in the convex hull of the 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 of is in the Poincaré domain, then there is a holomorphic change of variables (higher degree) taking (a1) to a canonical form (a3), with a polynomial in consisting of resonant monomials.

A point is said to be of type , where is a constant, if for all ,

The Siegel theorem says that if the eigen values of constitute a vector of type 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 (-) case there are related results, [a3]. Consider a vector field (or the corresponding autonomous system of differential equations ). A critical point of , i.e. a point such that , , is called an elementary critical point

if the real part of each eigen value of the matrix is non-zero. Let be a vector field with 0 as an elementary critical point. Then in a neighbourhood of zero, decomposes as a sum of vector fields and satisfying , and with respect to a suitable coordinate system , is of the form with the matrix similar to a diagonal matrix, and the linear part of can be represented by a nilpotent matrix (Chen's decomposition theorem). This is a non-linear analogue of the decomposition of a matrix into commuting semi-simple and nilpotent parts, cf. Jordan decomposition. Now let be a second vector field with 0 as an elementary critical point and let and be the Taylor series of and around 0. Then there exists a transformation around 0 which carries to if and only if there exists a formal transformation which carries the formal vector field to the formal vector field .

A -linearization result due to S. Sternberg says the following [a4], [a5]. If the matrix of linear terms of the equations , , is semi-simple and the set of eigen values of this matrix is non-resonant, then there is a change of coordinates which linearizes the equations. For results in the and 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 . In this form it generalizes to arbitrary nilpotent Lie algebras. Let be a finite-dimensional nilpotent Lie algebra over (cf. Lie algebra, nilpotent). Let be the Lie algebra of formal vector fields , . A formal non-linear representation of is a homomorphism of in (where is the homogeneous part of degree in ). Such a representation is holomorphic if for each the series converges in some neighbourhood of . Then is a linear representation of , called the linear part of . A formal vector field is called resonant with respect to a linear representation of if for all . The representation is normal if each is resonant with respect to the semi-simple part (cf. Jordan decomposition) of the linear representation . The Poincaré–Dulac theorem for nilpotent Lie algebras, [a8], now says that is a holomorphic non-linear representation of a nilpotent Lie algebra over , and if satisfies the Poincaré condition, then 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 does not belong to the convex hull of the weights (cf. Weight of a representation of a Lie algebra) of the linear part of .

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

In control theory one studies equations with a control parameter ; for instance, . This naturally leads to linearization problems for families of vector fields. In this setting more general notions of equivalence, involving, in particular, feedback laws , 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
[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