# Nonuniform Dichotomy Spectrum Intervals: Theorem and Computation

## TL;DR

This paper introduces a new nonuniform exponential dichotomy spectrum for nonautonomous systems, explores its relationship with classical spectra, and discusses computational methods extending previous bounded-growth work.

## Contribution

It develops a theory for computing nonuniform dichotomy spectra, extending prior work on bounded growth to more general nonuniform conditions.

## Key findings

- Established the relationship between nonuniform exponential dichotomy spectrum and classical spectra.
- Proved stability of these spectra under small linear perturbations.
- Provided an example illustrating the theoretical results.

## Abstract

Under the condition of nonuniformly bounded growth, %nonuniform exponential dichotomy spectrum for nonautonomous linear system is proposed the relationship of the nonuniform exponential dichotomy spectrum and the other two classical spectrums (the Lyapunov spectrum and Sacker-Sell spectrum) is given, and the stability of these spectrums under small linear perturbations are summarized and presented in this paper. A main goal of this paper is to discuss the theory for the computation of these spectrums under the condition of nonuniformly bounded growth, and this extends the work of Dieci and Vleck \cite{dv-02}, which compute the Lyapunov spectrum and Sacker-Sell spectrum under the condition of bounded. Finally, an example is given to illustrate and verify the theoretical results.