# Stability of Lyapunov Exponents, Weak Integral Separation and Nonuniform Dichotomy Spectrum

## TL;DR

This paper establishes a necessary and sufficient condition for the stability of Lyapunov exponents in linear differential systems using weak integral separation, and explores the existence of nonuniform exponential dichotomy spectrum under this condition.

## Contribution

It introduces a weaker form of integral separation as a criterion for Lyapunov exponent stability and demonstrates the existence of the nonuniform exponential dichotomy spectrum.

## Key findings

- Lyapunov exponents stability characterized by weak integral separation
- Existence of full nonuniform exponential dichotomy spectrum proven
- Weak integral separateness suffices for spectral analysis in linear systems

## Abstract

In this paper, a necessary and sufficient condition for the stability of Lyapunov exponents of linear differential system are proved in the sense that the equations satisfy the weaker form of integral separation instead of its classical one. Furthermore, the existence of full nonuniform exponential dichotomy spectrum under the condition of weak integral separateness is also presented.