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arXiv · 2205.07391

Nonuniform $\mu$-dichotomy spectrum and kinematic similarity

César M. Silva
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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.

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César M. Silva. 2022-05-15. Nonuniform $\mu$-dichotomy spectrum and kinematic similarity. https://arxiv.org/abs/2205.07391

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