Insights into the behavior of the damped parametric oscillator: Analytical and numerical perspectives

In this study, we analyze the behavior of a damped parametric oscillator, incorporating a damping factor to examine its influence on system dynamics. We derive the equation of motion using the Lagrangian formulation and the Euler–Lagrange equation, which leads to a nonlinear differential equation governing the system’s motion. The equation is solved numerically using the 4th-order Runge-Kutta method and analytically using the Multistep Differential Transformation Method (MS-DTM), which provides an efficient and accurate approximation. The results obtained from MS-DTM are in good agreement with the numerical solutions, demonstrating its capability to capture the system’s dynamics with reduced computational effort. By analyzing two distinct scenarios—one with a small damping coefficient (β = 0.01) and the other with a larger damping coefficient (β = 0.5)—we observe that the damping parameter significantly influences system behavior. Specifically, in the first scenario, the system exhibits stable damped harmonic motion with constant energy, while in the second scenario, the system experiences energy dissipation and becomes unstable. This study highlights the critical role of damping in determining the system’s stability and energy dissipation, showcasing the effectiveness of both MS-DTM and numerical methods in analyzing nonlinear oscillatory systems.