  The inclusion of inertia in the Kuramoto model has been long reported to
change the nature of phase transition, providing a fertile ground to model the
dynamical behaviors of interacting units. More recently, higher-order
interactions have been realized as essential for the functioning of real-world
complex systems ranging from the brain to disease spreading. Yet, analytical
insights to decipher the role of inertia with higher-order interactions remain
challenging. Here, we study the Kuramoto model with inertia on simplicial
complexes, merging two research domains. We develop an analytical framework in
a mean-field setting using self-consistent equations to describe the
steady-state behavior, which reveals a prolonged hysteresis in the
synchronization profile. Inertia and triadic interaction strength exhibit
isolated influence on system dynamics by predominantly governing, respectively,
the forward and backward transition points. This work sets a paradigm to deepen
our understanding of real-world complex systems such as power grids modeled as
the Kuramoto model with inertia.
