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

Heli Elorreaga, Gonzalo Robledo, David Urrutia-Vergara

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. Peña and S. Rivera--Villagrán.

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

Abstract

In this article, we complete the study of the equivalences between the properties of --dichotomy, --noncriticality and --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 --dichotomy. It should be noted that all the results have been obtained by using a characterization of the --dichotomy by a group theory approach recently developed by J.F. Peña and S. Rivera--Villagrán.

Paper Structure

This paper contains 13 sections, 17 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. The uniform $h$--dichotomy: a short review
  3. Uniform $h$--dichotomies from a group theory perspective
  4. Uniform bounded growth
  5. Uniform $h$--noncriticality and $h$--expansiveness
  6. $h$--dichotomies on $J$
  7. Preliminary results
  8. Generalization of Proposition \ref{['P2']} to $\mathcal{I}=J$.
  9. Uniform $h$--dichotomy on $J$
  10. Generalized Floquet theory and uniform $h$--dichotomy
  11. Technical results
  12. Uniform $h$--dichotomy for generalized Floquet systems
  13. An example

Key Result

Proposition 1

The system lin has a uniform $h$--dichotomy on $\mathcal{I}$ if and only if, given a fundamental matrix $\Phi(t)$, there exists a constant projector $P$ and constants $K\geq 1$ and $\alpha>0$ such that:

Theorems & Definitions (41)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 3
  • Proposition 2
  • Definition 4
  • ...and 31 more