Melnikov functions of first and second order for 2-dimensional piecewise non-Hamiltonian systems with $n$ regions

In this paper, we examine piecewise non-Hamiltonian systems where the switching manifold consists of $n$ half straight lines $r_i$ originating from the origin, and each domain $D_i$ is bounded by two consecutive half straight lines $r_i$ and $r_{i+1}$. Our main goal is to derive the expressions of the first- and second-order Melnikov functions. We address the question of the bifurcation of limit cycles in several interesting models by performing a first-order Melnikov analysis on (1) a buck power converter, (2) a Lotka--Volterra model, and (3) a piecewise smooth system with cubic vector fields. Additionally, we conduct a second-order Melnikov analysis on (4) a piecewise linear vector field with heteroclinic connections.