Comprehending stability and bifurcations in Mathieu cubic–quintic Duffing system using Lindstedt–Poincaré methodology

The Mathieu equation serves as a foundational model in the study of parametrically excited systems and has been extensively analyzed using both analytical and numerical techniques. Motivated by its relevance to various physical and engineering systems, we investigate its nonlinear extension – the Mathieu cubic–quintic Duffing equation – which includes both cubic and quintic stiffness terms. This nonlinear variant exhibits rich dynamical behavior including complex stability transitions and bifurcations. To explore these phenomena, we employ the Lindstedt–Poincaré method, which allows us to analytically capture both pitchfork bifurcations (subcritical and supercritical) near unstable parametric resonance tongues and subharmonic bifurcations at higher values of excitation frequencies. The analytical predictions are validated using numerical simulations based on Poincaré sections. Our results reveal the emergence of two distinct unstable tongues centered around ${\omega }_p={\omega }_n$ and ${\omega }_p=2{\omega }_n$, as well as two subharmonic bifurcations located at ${\omega }_p=4{\omega }_n$ and ${\omega }_p=6{\omega }_n$. We construct a comprehensive global bifurcation diagram in the parametric frequency–excitation strength space, showing transitions in equilibrium stability and the birth of multiple stable and unstable equilibrium points. Notably, the transition through each bifurcation boundary results in a specific number of stable and unstable equilibrium points alternatively. These findings not only extend the known behavior of the Mathieu and Mathieu–Duffing systems but also offer deeper insight into the complex dynamics induced by higher-order nonlinearities.