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Smoothness of linearization by mixing parameters of dichotomy, bounded growth and perturbation

Álvaro Castañeda, Ignacio Huerta, Gonzalo Robledo

TL;DR

The paper addresses the smoothness of the global, time-parametrized topological conjugacy between a linear nonautonomous system with a nonuniform exponential dichotomy and a quasilinear perturbation. It employs a Green function framework and a parameterized Banach-space setup to construct explicit conjugacy maps $H$ and $G$, proving the existence of a topological conjugacy under a set of conditions that couple dichotomy, bounded growth, and perturbation data. The authors then establish differentiability of the conjugacy and provide a diffeomorphism result on a finite interval by leveraging additional regularity assumptions and Plastock's criterion. The work extends prior results by allowing nonuniform growth and unbounded perturbations, offering explicit parameter relations that widen the smoothness interval and providing a detailed comparison with Jara’s results. This contributes to a more versatile understanding of nonlinear linearization for nonautonomous systems with nontrivial projectors and nonuniform dynamics, with potential implications for global linearization techniques in nonuniform settings.

Abstract

We study the smoothness properties of a global and nonautonomous topological conjugacy between a linear system and a quasilinear perturbation. The linear system exhibits a nonuniform exponential dichotomy with a nontrivial projector and nonuniform bounded growth property. Additionally, the quasilinear perturbation is dominated by an increasing exponential function. Emphasis is placed on employing a set of parameters to describe the conditions of dichotomy, bounded growth and quasilinear perturbations. Finally, we prove that modifying these conditions enables us to achieve a broader smoothness interval.

Smoothness of linearization by mixing parameters of dichotomy, bounded growth and perturbation

TL;DR

The paper addresses the smoothness of the global, time-parametrized topological conjugacy between a linear nonautonomous system with a nonuniform exponential dichotomy and a quasilinear perturbation. It employs a Green function framework and a parameterized Banach-space setup to construct explicit conjugacy maps and , proving the existence of a topological conjugacy under a set of conditions that couple dichotomy, bounded growth, and perturbation data. The authors then establish differentiability of the conjugacy and provide a diffeomorphism result on a finite interval by leveraging additional regularity assumptions and Plastock's criterion. The work extends prior results by allowing nonuniform growth and unbounded perturbations, offering explicit parameter relations that widen the smoothness interval and providing a detailed comparison with Jara’s results. This contributes to a more versatile understanding of nonlinear linearization for nonautonomous systems with nontrivial projectors and nonuniform dynamics, with potential implications for global linearization techniques in nonuniform settings.

Abstract

We study the smoothness properties of a global and nonautonomous topological conjugacy between a linear system and a quasilinear perturbation. The linear system exhibits a nonuniform exponential dichotomy with a nontrivial projector and nonuniform bounded growth property. Additionally, the quasilinear perturbation is dominated by an increasing exponential function. Emphasis is placed on employing a set of parameters to describe the conditions of dichotomy, bounded growth and quasilinear perturbations. Finally, we prove that modifying these conditions enables us to achieve a broader smoothness interval.
Paper Structure (23 sections, 6 theorems, 104 equations)

This paper contains 23 sections, 6 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. The Green function approach
  3. State of art of continuity results
  4. State of art of smoothness results
  5. Novelty of this work
  6. Structure
  7. Preliminaries.
  8. Nonuniform exponential dichotomy and nonuniform bounded growth
  9. Topological conjugacy
  10. Proof of Theorem \ref{['Teo1']}
  11. About the map $(t,\tau,\eta)\mapsto w^{*}(t;(\tau,\eta))$.
  12. About the map $\Gamma_{(\tau, \xi)}: \mathcal{A}\rightarrow \mathcal{A}$.
  13. The property (TC2) is verified
  14. The property (TC3) is verified.
  15. The property (TC1) is verified: Bijectivity.
  16. ...and 8 more sections

Key Result

Lemma 1

If the linear system lin1 has the properties of nonuniform exponential dichotomy and nonuniform bounded growth on $\mathbb{R}_{0}^{+}$ with constants $(K,\alpha,\mu)$ and $(K_{0},a,\varepsilon)$ respectively, then it follows that

Theorems & Definitions (22)

  • Definition 1
  • Definition 2
  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 3
  • Remark 4
  • Lemma 1
  • proof
  • Remark 5
  • ...and 12 more