A Hartman–Grobman Theorem for Utkin-Selected Sliding Dynamics in Filippov Systems and Threshold Analysis in Dry Friction Oscillators

We prove a Hartman--Grobman theorem for \emph{Utkin-selected sliding dynamics} in codimension-one Filippov systems. Under uniform transversality and $C^1$ regularity, near a \emph{hyperbolic equilibrium} of the reduced sliding ODE on the switching manifold, the Utkin-selected evolution is locally \emph{orbit equivalent} to a linear product model: a transverse contracting drift $\dot\sigma=-1$ together with the linearized sliding flow $\dot z = Dz\, z$ on the manifold. The proof combines smooth Hartman--Grobman theory for the sliding ODE with a transverse hitting/impact map that yields a product chart near the manifold. As an independent application of the reduction framework, we analyze a dry friction oscillator, derive explicit sticking thresholds under constant bias and harmonic forcing, and provide reproducible numerical validation.