The inertial Kuramoto model (IKM) is a straightforward generalization of the Kuramoto model (KM), a well-known synchronization model. Although considering the inertia is natural due to Newton's law and the KM has been studied widely, the IKM has not been studied as much as the KM. In this thesis, we study the properties of the IKM and various IKM-type models.

First, we study the cardinality of collisions between Kuramoto oscillators in the (asymptotic) phase-locking process under inertia. In a small inertia regime, the finiteness of collisions is also equivalent to the phase-locking, like the KM. In contrast, in a large inertia regime, a homogeneous Kuramoto ensemble with the same natural frequency can exhibit the phase-locking while there are a countable number of collisions between Kuramoto oscillators. This is in contrast to the KM.

Second, we study the relaxation dynamics toward a phase-locking state for the IKM from generic initial configurations. For this, we provide a sufficient framework in terms of the generic initial data and the system and free parameters.

Third, we study the emergent dynamics of Kuramoto oscillators under the interplay between inertia and adaptive couplings. We use the inertial Kuramoto model with time-dependent mutual coupling strengths for the phase dynamics. We provide several sufficient frameworks for the phase and frequency synchronization regarding system parameters and initial data for Hebbian and anti-Hebbian coupling functions.