Generalized exponential dichotomy and global linearization
The nonautonomous version of the Hartman-Grobman theorem was first proved by Palmer and establishes a topological conjugacy between a nonlinear system \[ x' = A(t)x + f(t,x) \] and its linearization \[ x' = A(t)x, \] under the assumption of \(f\) being bounded with small Lipschitz constant and the linear system admitting an exponential dichotomy. The contribution of this paper is to extend Palmer's results to systems with a linearization which only admits a generalized exponential dichotomy \[ | U(t) P U^{-1}(s)| \leq K \exp\left(- \int_s^t a(\tau) d\tau\right) , \quad t \geq s, \] for the fundamental matrix \(U\) and projection \(P\), with a positive, bounded and continuous function \(a\), and similarly for \(t \leq s\).
scientific article; zbMATH DE number 5016142