Analytical expressions for stability regions in the Ince–Strutt diagram of Mathieu equation

Simple analytical expressions are suggested for transition curves that separate, in the Ince–Strutt diagram, different types of solutions to the famous Mathieu equation. The derivations of these expressions in this paper rely on physically meaningful periodic solutions describing various regular motions of a familiar nonlinear mechanical system—a rigid planar pendulum with a vertically oscillating pivot. The paper is accompanied by a relevant simulation program.