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The new notion of Bohl dichotomy for nonautonomous difference equations and its relation to exponential dichotomy

Adam Czornik, Konrad Kitzing, Stefan Siegmund

Abstract

Bohl dichotomy is a notion of hyperbolicity for linear nonautonomous difference equations that is weaker than the classical concept of exponential dichotomy. In the class of systems with bounded invertible coefficient matrices which have bounded inverses, we study the relation between the set $\mathrm{BD}$ of systems with Bohl dichotomy and the set $\mathrm{ED}$ of systems with exponential dichotomy. It can be easily seen from the definition of Bohl dichotomy that $\mathrm{ED} \subseteq \mathrm{BD}$. Using a counterexample we show that the closure of $\mathrm{ED}$ is not contained in $\mathrm{BD}$. The main result of this paper is the characterization $\operatorname{int}\mathrm{BD} = \mathrm{ED}$. The proof uses upper triangular normal forms of systems which are dynamically equivalent and utilizes a diagonal argument to choose subsequences of perturbations each of which is constructed with the Millionshikov Rotation Method. An Appendix describes the Millionshikov Rotation Method in the context of nonautonomous difference equations as a universal tool.

The new notion of Bohl dichotomy for nonautonomous difference equations and its relation to exponential dichotomy

Abstract

Bohl dichotomy is a notion of hyperbolicity for linear nonautonomous difference equations that is weaker than the classical concept of exponential dichotomy. In the class of systems with bounded invertible coefficient matrices which have bounded inverses, we study the relation between the set of systems with Bohl dichotomy and the set of systems with exponential dichotomy. It can be easily seen from the definition of Bohl dichotomy that . Using a counterexample we show that the closure of is not contained in . The main result of this paper is the characterization . The proof uses upper triangular normal forms of systems which are dynamically equivalent and utilizes a diagonal argument to choose subsequences of perturbations each of which is constructed with the Millionshikov Rotation Method. An Appendix describes the Millionshikov Rotation Method in the context of nonautonomous difference equations as a universal tool.
Paper Structure (6 sections, 30 theorems, 206 equations)

This paper contains 6 sections, 30 theorems, 206 equations.

Table of Contents

  1. Exponential and Bohl dichotomy
  2. Bohl exponents
  3. Applying the Millionshikov rotation method
  4. Upper triangularization and subsystems
  5. Main result
  6. Appendix

Key Result

Lemma 4

The following three statements are equivalent: (i) System 1 has a Bohl dichotomy. (ii) There exists a splitting $L_1 \oplus L_2 = \mathbb{R}^d$ with (iii) There is $\alpha >0$, such that for all $x_{0}\in \mathbb{R}^{d}\setminus \{0\}$, Moreover, if system 1 has a Bohl dichotomy with splitting $L_1 \oplus L_2 = \mathbb R^d$, then statement (ii) holds with that splitting.

Theorems & Definitions (61)

  • Definition 1: Exponential dichotomy
  • Definition 2: Bohl dichotomy
  • Definition 3: Bohl exponents
  • Lemma 4: Characterization of Bohl dichotomy
  • Corollary 5: Criterion for non-existence of Bohl dichotomy
  • Lemma 6: Characterization of exponential dichotomy
  • Lemma 7: Bohl exponents for trivial exponential dichotomy
  • proof
  • Lemma 8: Exponential growth on subsequence via Bohl exponent
  • proof
  • ...and 51 more