The entry-exit function and nilpotent corrections in multidimensional slow-fast systems

We study multidimensional entry-exit phenomena in slow-fast systems whose fast linearization contains a nilpotent component. We show that the classical spectral balance still determines the exit point as $\epsilon \to 0$, however the nilpotent component induces an algebraic amplification invisible to the spectrum, producing a $\epsilon \mathrm{\mid }\log \epsilon \mathrm{\mid }$-shift of the exit point for $0<\epsilon \ll 1$. We further extend the analysis to systems with several spectral branches and show that mode dominance follows a hierarchical, lexicographic selection principle: accumulated spectral growth determines the leading order, while the nilpotent component resolves degeneracies between spectrally equivalent modes.