  In networks of coupled oscillators, it is of interest to understand how
interaction topology affects synchronization. Many studies have gained key
insights into this question by studying the classic Kuramoto oscillator model
on static networks. However, new questions arise when network structure is
time-varying or when the oscillator system is multistable, the latter of which
can occur when an inertial term is added to the Kuramoto model. While the
consequences of evolving topology and multistability on collective behavior
have been examined separately, real-world systems such as gene regulatory
networks and the brain can exhibit these properties simultaneously. How does
the rewiring of network connectivity affect synchronization in systems with
multistability, where different paths of network evolution may differentially
impact system dynamics? To address this question, we study the effects of
time-evolving network topology on coupled Kuramoto oscillators with inertia. We
show that hysteretic synchronization behavior occurs when the network density
of coupled inertial oscillators is slowly varied as the dynamics evolve.
Moreover, we find that certain fixed-density rewiring schemes induce
significant changes to the level of global synchrony, and that these changes
remain after the network returns to its initial configuration and are robust to
a wide range of network perturbations. Our findings suggest that the specific
progression of network topology, in addition to its initial or final static
structure, can play a considerable role in modulating the collective behavior
of systems evolving on complex networks.
