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

This paper aims to provide a Melnikov-like function that governs the existence of periodic solutions bifurcating from period annuli in certain families of second-order discontinuous differential equations of the form \(\ddot{x}+\alpha \textrm{sign}(x)=\eta x+\varepsilon \;f(t,x,\dot{x})\) $\ddot{x}+\alpha \text{sign}(x)=\eta x+\epsilon f(t,x,\dot{x})$ . This family has attracted considerable attention from researchers, particularly in the analysis of specific instances of \(f(t,x,\dot{x})\) $f(t,x,\dot{x})$ . The study of this type of differential equation is motivated by its significance in modeling systems with abrupt state changes, both in natural and engineering contexts.