Nonuniform $\mu$-dichotomy spectrum and kinematic similarity

TL;DR

This paper introduces a new family of spectra called nonuniform μ-dichotomy spectrum for linear nonautonomous differential equations, generalizing existing spectra and linking them to Lyapunov exponents to analyze reducibility and polynomial behaviors.

Contribution

It defines the nonuniform μ-dichotomy spectrum, describes its possible forms, relates it to Lyapunov exponents, and applies it to reducibility and normal form results.

Findings

The nonuniform μ-dichotomy spectrum generalizes existing spectra.

Connected components of the spectrum relate to Lyapunov exponents.

Examples include polynomial behavior and spectrum computation.

Abstract

For linear nonautonomous differential equations we introduce a new family of spectrums defined with general nonuniform dichotomies: for a given growth rate $\mu$ in a large family of growth rates, we consider a notion of spectrum, named nonuniform $\mu$-dichotomy spectrum. This family of spectrums contain the nonuniform dichotomy spectrum as the very particular case of exponential growth rates. For each growth rate $\mu$, we describe all possible forms of the nonuniform $\mu$-dichotomy spectrum, relate its connected components with adapted notions of Lyapunov exponents, and use it to obtain a reducibility result for nonautonomous linear differential equations. We also give an illustrative examples where the spectrum is obtained, including a situation where a normal form is obtained for polynomial behavior.