Conditions for uniform $h$--dichotomy in terms of uniform non criticality, expansiveness and via generalized Floquet theory

TL;DR

This paper explores the conditions under which linear nonautonomous ODE systems exhibit uniform $h$--dichotomy, linking it to noncriticality and expansiveness, and extends generalized Floquet theory with new necessary and sufficient conditions.

Contribution

It completes the equivalence study of $h$--dichotomy, noncriticality, and expansiveness, and extends Floquet theory using a group theory approach.

Findings

Established equivalences between $h$--dichotomy, noncriticality, and expansiveness.

Provided a necessary and sufficient condition for $h$--dichotomy.

Extended generalized Floquet theory with new criteria.

Abstract

In this article, we complete the study of the equivalences between the properties of $h$--dichotomy, $h$--noncriticality and $h$--expansiveness of a linear nonautonomous ODE system which had been initiated in a previous work. Moreover, we extend a result of the generalized Floquet theory developed by T.A. Burton and J.S. Muldowney by providing a necessary and sufficient condition for $h$--dichotomy. It should be noted that all the results have been obtained by using a characterization of the $h$--dichotomy by a group theory approach recently developed by J.F. Pe\~na and S. Rivera--Villagr\'an.