A Melnikov analysis on a class of second order discontinuous differential equations

This paper focuses in providing a Melnikov-like function that controls the existence of periodic solutions bifurcating from period annuli present in some families of the second-order discontinuous differential equation given by $\ddot{x}+\alpha\; \text{sgn}(x)=\eta x+\varepsilon \;f(t,x,\dot{x})$. This family has garnered extensive attention from various researchers, especially when considering specific instances of $f(t,x,\dot{x})$. The interest in studying this type of differential equation is due to its relevance in modeling systems with abrupt state changes in both natural and engineering contexts.