Synchronization phenomena are ubiquitous in the natural sciences and engineering, but also in social systems. Among the many models that have been proposed for a description of synchronization, the Kuramoto model is most popular. The second-order Kuramoto model has been used to investigate power grids, Josephson junctions, and other systems.

The first main part of this talk is devoted to study the networked second-order Kuramoto model in the presence of a correlation between the oscillators' natural frequencies and the network's degree.The theoretical framework in the continuum limit and for uncorrelated networks is provided for the model with an asymmetrical natural frequency distribution.

Discontinuous phase transitions indicated by the order parameter and hysteresis emerge due to different initial conditions. For very large perturbations, the system could move from a desirable state to an undesirable state. Basin stability was proposed to quantify the stability of a system to stay in the desirable state after being subjected to strong perturbations.

In the second main part of this talk, the basin stability of the synchronization of the second-order Kuramoto model is investigated via perturbing nodes separately.

To quantify the stability influence between clusters, in particular for cluster synchronization, a new concept of partial basin stability is proposed. The concept is implemented on two important real examples: neural networks and the northern European power grid.