Mathieu's Equation and its Generalizations: Overview of Stability Charts and their Features

This work is concerned with Mathieu's equation - a classical differential equation, which has the form of a linear second-order ordinary differential equation with Cosine-type periodic forcing of the stiffness coefficient, and its different generalizations/extensions. These extensions include: the effects of linear viscous damping, geometric nonlinearity, damping nonlinearity, fractional derivative terms, delay terms, quasiperiodic excitation or elliptic-type excitation. The aim is to provide a systematic overview of the methods to determine the corresponding stability chart, its structure and features, and how it differs from that of the classical Mathieu's equation.