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Spectrum invariance dilemma for nonuniformly kinematically similar systems

Claudio A. Gallegos, Néstor Jara

Abstract

We unveil instances where nonautonomous linear systems manifest distinct nonuniform $μ$-dichotomy spectra despite admitting nonuniform $(μ, \varepsilon)$-kinematic similarity. Exploring the theoretical foundations of this lack of invariance, we discern the pivotal influence of the parameters involved in the property of nonuniform $μ$-dichotomy such as in the notion of nonuniform $(μ, \varepsilon)$-kinematic similarity. To effectively comprehend these dynamics, we introduce the stable and unstable optimal ratio maps, along with the $\varepsilon$-neighborhood of the nonuniform $μ$-dichotomy spectrum. These concepts provide a framework for understanding scenarios governed by the noninvariance of the nonuniform $μ$-dichotomy spectrum.

Spectrum invariance dilemma for nonuniformly kinematically similar systems

Abstract

We unveil instances where nonautonomous linear systems manifest distinct nonuniform -dichotomy spectra despite admitting nonuniform -kinematic similarity. Exploring the theoretical foundations of this lack of invariance, we discern the pivotal influence of the parameters involved in the property of nonuniform -dichotomy such as in the notion of nonuniform -kinematic similarity. To effectively comprehend these dynamics, we introduce the stable and unstable optimal ratio maps, along with the -neighborhood of the nonuniform -dichotomy spectrum. These concepts provide a framework for understanding scenarios governed by the noninvariance of the nonuniform -dichotomy spectrum.
Paper Structure (7 sections, 13 theorems, 81 equations, 1 figure)

This paper contains 7 sections, 13 theorems, 81 equations, 1 figure.

Table of Contents

  1. Introduction
  2. kinematic similarity and dichotomy spectrum
  3. Main results and contributions
  4. Consequences about the noninvariance of the nonuniform spectra
  5. Optimal dichotomy constants
  6. Kinematic similarity and spectra noninvariance
  7. Endgame remarks

Key Result

Lemma 2.10

Assume the system 601 has $({\mathrm{N}\mu},\epsilon)$-growth with constants $a\geq 0$, $\widehat{K}\geq 1$, and also admits ${\mathrm{N}\mu\mathrm{D}}$ with parameters $(\mathrm{P};\alpha,\beta,\theta,\nu)$. For $\mathrm{P}\neq 0$, we have $-(a+\epsilon)\leq\alpha$. In particular Analogously, for $\mathrm{P}\neq \mathrm{Id}$, we have $\beta\leq a+\epsilon$. In particular

Figures (1)

  • Figure 1: Contentions from Remark \ref{['619']}

Theorems & Definitions (48)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7
  • Definition 2.8
  • Remark 2.9
  • Lemma 2.10
  • ...and 38 more