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Dimension estimates for nonlinear nonautonomous systems in arbitrary normed spaces

John Ioannis Stavroulakis

Abstract

We calculate explicit estimates for the dimension of trajectories satisfying a certain growth bound. We generalize the classic results of Kurzweil by considering nonlinear nonautonomous and uniformly compact dynamical systems on normed spaces over arbitrary fields. We furthermore refine the results in the case of delay equations, greatly simplifying the relevant growth bounds.

Dimension estimates for nonlinear nonautonomous systems in arbitrary normed spaces

Abstract

We calculate explicit estimates for the dimension of trajectories satisfying a certain growth bound. We generalize the classic results of Kurzweil by considering nonlinear nonautonomous and uniformly compact dynamical systems on normed spaces over arbitrary fields. We furthermore refine the results in the case of delay equations, greatly simplifying the relevant growth bounds.

Paper Structure

This paper contains 10 sections, 16 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Dimension Estimates for Nonautonomous Systems
  3. Preliminaries and standing assumptions
  4. Lemmata and proof of main Theorem
  5. Remarks on nonlinear systems
  6. General comments and possible generalizations
  7. Growth/Decay Estimates for Delay Equations
  8. Framework and bounds for linear equations
  9. Variational equation
  10. Acknowledgements

Key Result

Lemma 4

The determinant defined in the above definition enjoys the following properties:

Theorems & Definitions (24)

  • Definition 2
  • Definition 3
  • Lemma 4
  • Lemma 5: kurz1kurz73
  • Theorem 6
  • Lemma 7: kurz72kurz1, cf. pietsch
  • Lemma 8: kurz72kurz1, cf. pietsch
  • Remark 9
  • Definition 10
  • Lemma 11: kurz1
  • ...and 14 more