The Bohl spectrum for nonautonomous differential equations

TL;DR

This paper introduces the Bohl spectrum for nonautonomous linear differential equations, showing its properties, differences from the Sacker--Sell spectrum, and implications for stability analysis.

Contribution

It develops the Bohl spectrum concept, proves its structure as finitely many intervals, and compares it with existing spectra, highlighting new stability insights.

Findings

Bohl spectrum is union of finitely many intervals.

Bohl spectrum can differ from the Sacker--Sell spectrum.

Higher-order nonlinear perturbations can be exponentially stable despite spectrum differences.

Abstract

We develop the Bohl spectrum for nonautonomous linear differential equation on a half line, which is a spectral concept that lies between the Lyapunov and the Sacker--Sell spectrum. We prove that the Bohl spectrum is given by the union of finitely many intervals, and we show by means of an explicit example that the Bohl spectrum does not coincide with the Sacker--Sell spectrum in general. We demonstrate for this example that any higher-order nonlinear perturbation is exponentially stable, although this not evident from the Sacker--Sell spectrum. We also analyze in detail situations in which the Bohl spectrum is identical to the Sacker-Sell spectrum.