Master stability function for networks of coupled piecewise-smooth oscillators

The Master Stability Function (MSF) is a well-established framework for analyzing the synchronization stability of networks composed of identical oscillators. It simplifies the analysis by reformulating the stability problem in terms of a low-dimensional variational equation defined in the vicinity of the synchronization manifold. However, for oscillators governed by piecewise-smooth vector fields, such a variational equation cannot be directly derived due to the presence of non-differentiable points in the system dynamics. This paper introduces a novel extension of the MSF framework specifically designed to address synchronization stability in networks of identical piecewise-smooth oscillators. The proposed method builds upon a previously developed Jacobian matrix estimation technique, enabling its application to piecewise-smooth systems. The formulation is presented in detail, along with a discussion of its advantages and disadvantages compared to alternative methods. Additionally, the MSF is computed for both linear and nonlinear impact oscillator systems using the proposed method, and a detailed description of its application is provided. The effectiveness of the method is demonstrated through validation against an alternative MSF estimation approach.