Performance measures after perturbations in the presence of inertia

Abstract Synchronization phenomena occur among populations of interacting elements and is of great importance for the functionality of several types of complex systems. Much effort has been devoted to understanding its emergence especially on the Kuramoto model, and the research now on coupled oscillators takes advantage of the recent theory of the interplay between intrinsic dynamics and topological structures. However, the underlying mechanism of the interplay-induced synchronization of the model with inertia remains elusive. Here we investigate the dynamical-structural interplay in the Kuramoto model with inertia from two perspectives; namely, one for stationary states, and the other for transition processes of the system. For stationary states, we decompose the concept of alignment function as a combination of the Laplacian matrix and the natural frequency, and use the Gershgorin disk theorem to quantify the ensemble average of alignment functions induced by a variety of structural modifications. For transition processes, we derive the solution of the performance metric, especially with respect to the inertia term. Additionally, we show that, when the natural frequency is tangent to the dominant eigenvector of the Laplacian matrix, both the ensemble average of the alignment functions and the performance metric approach optimization (minimization). Finally, we perform numerical simulations to support our theoretical analysis.