ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

9. Ordinary Differential Equations and Dynamical Systems

1991 MS Classification: 34C20 Siegmund, Stefan, Universität Augsburg, Deutschland Normal Forms for Nonautonomous Differential Equations of Carathéodory Type Smooth transformations of equations ẋ = Ax + f(x) into normal form are a well-established tool in dynamical systems theory. Crucial for such transformations is the non-resonance condition λ_{j} − ∑_{i=1}^{n} k_{i}λ_{i} ≠ 0 for j = 1, …, n and k ∈ ℕ_{0}^{n} with 2 ≤ ∑_{i=1}^{k} k_{i} and λ_{i} the distinct eigenvalues of A ∈ ℝ^{d×d}. With a new technique from the theory of integral manifolds we generalize normal form results to nonautonomous differential equations ẋ = A(t)x + f(t, x) with A : ℝ → ℝ^{d×d} locally integrable and f : ℝ^{1+d} → ℝ^{d} a Carathéodory function (i. e. f(t, ·) is continous f. a. a. t, f(·, x) is measurable f. a. x). Under the assumption that A has bounded Sacker-Sell spectrum λ_{1} = [a_{1}, b_{1}], …, λ_{n} = [a_{n}, b_{n}] and is reduced to diagonal form A = diag(A_{1}, …, A_{n}) we construct a smooth formal linearization provided the following non-resonance condition

λ_{j} ∩ ∑_{i=1}^{n} k_{i}λ_{i} = ∅

holds. Here λ_{i} + λ_{j} is the intervall [a_{i} + a_{j}, b_{i} + b_{j}], analogous for k_{i}λ_{i}.