  Solitary states emerge in oscillator networks when one oscillator separates
from the fully synchronized cluster and becomes incoherent with the rest of the
network. Such chimera-type patterns with an incoherent state formed by a single
oscillator were observed in various oscillator networks; however, there is
still a lack of understanding of how such states can stably appear. Here, we
study the stability of solitary states in Kuramoto networks of identical
two-dimensional phase oscillators with inertia and a phase-lagged coupling. The
presence of inertia can induce rotatory dynamics of the phase difference
between the solitary oscillator and the coherent cluster. We derive asymptotic
stability conditions for such a solitary state as a function of inertia,
network size, and phase lag that may yield either attractive or repulsive
coupling. Counterintuitively, our analysis demonstrates that (i) increasing the
size of the coherent cluster can promote the stability of the solitary state in
the attractive coupling case and (ii) the solitary state can be stable in
small-size networks with all repulsive coupling. We also discuss the
implications of our stability analysis for the emergence of rotatory chimeras.
