Abstract

We unveil instances where nonautonomous linear systems manifest distinct nonuniform ?-dichotomy spectra despite admitting nonuniform (?,?)-kinematic similarity. Exploring the theoretical foundations of this lack of invariance, we discern the pivotal influence of the parameters involved in the property of nonuniform ?-dichotomy such as in the notion of nonuniform (?,?)-kinematic similarity. To effectively comprehend these dynamics, we introduce the stable and unstable optimal ratio maps, along with the ?-neighborhood of the nonuniform ?-dichotomy spectrum. These new concepts provide a framework for understanding scenarios governed by the noninvariance of the nonuniform ?-dichotomy spectrum. © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024.