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