Nonautonomous Linear Systems: Exponential Dichotomy and its Applications
Álvaro Castañeda, Gonzalo Robledo
TL;DR
This work develops a comprehensive framework for nonautonomous linear systems by formalizing exponential dichotomy, its spectrum, and invariant projections as a nonautonomous analogue of hyperbolicity. It establishes fundamental tools—fundamental/transition matrices, adjoint systems, Liouville's formula, and bounded growth—and builds a robust theory of kinematical similarity, Floquet theory, and Perron-style triangularization. The text then extends these ideas to both the full line and half-lines, deriving splitting of solutions, admissibility via Green functions, and various characterizations of dichotomy, along with concrete examples and applications to linearization and perturbation analysis. Collectively, the results provide a cohesive pipeline for stability analysis, spectral-like decomposition, and nonlinear perturbation theory in nonautonomous settings with implications for both theory and applied stability assessment.
Abstract
The first purpose of this work is to provide a friendly introduction to the theory of nonautonomous linear systems of ordinary differential equations, the property of exponential dichotomy and its corresponding spectral theory. The second purpose of this work is disseminate the linearization results carried out by the authors in a nonautonomous framework. The actual structure of this work is a consequence of several elective courses (2014, 2016, 2019, 2021 and 2023) carried out by the authors for undergraduate and graduated students at the Department of Mathematics of the Universidad de Chile. The monography assumes a good knowledge of multivariate calculus, linear algebra and ordinary differential equations.