<p>This study examines a Mathieu oscillator subjected to parametric forcing, and it is stabilized by applying negative derivative feedback control under the most severe resonance conditions. The multiple-scales-approach is used to obtain an approximate solution that includes parametric forcing in the control system. Validation of the derived approximate solution is performed by comparing it with the numerical solution computed via the fourth-order Runge-Kutta method. For resonance cases, solvability conditions and characteristic exponents are derived, whereas the Routh-Hurwitz criteria are applied to determine the stability of fixed points linked to steady-state solutions. Frequency response curves, bifurcation diagrams, and Poincaré maps are presented to capture the system’s dynamic behavior and responses under varying conditions. Lyapunov exponent spectra, Kaplan–Yorke dimension, and basin stability were plotted to quantify system sensitivity, attractor complexity, and multistability across varying initial conditions and parameter ranges. These analyses provide deeper insights into the complex behavior of the oscillator. The equation of Mathieu oscillator is widely applied across disciplines due to its ability to model systems with periodic coefficients. Its applications include analyzing mechanical vibrations, parametric resonance, and structural stability, such as in bridges and aircraft wings. Additionally, it is employed in plasma physics, quantum mechanics, signal processing, biological rhythms, and optical and electronic systems, owing to its versatility in representing oscillatory and stability phenomena driven by periodic variations.</p>