Spectrum invariance dilemma for nonuniformly kinematically similar systems

We unveil instances where nonautonomous linear systems manifest distinct nonuniform \(\mu \)-dichotomy spectra despite admitting nonuniform \((\mu , \varepsilon )\)-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 \(\mu \)-dichotomy such as in the notion of nonuniform \((\mu , \varepsilon )\)-kinematic similarity. To effectively comprehend these dynamics, we introduce the stable and unstable optimal ratio maps, along with the \(\varepsilon \)-neighborhood of the nonuniform \(\mu \)-dichotomy spectrum. These new concepts provide a framework for understanding scenarios governed by the noninvariance of the nonuniform \(\mu \)-dichotomy spectrum.