Analytical approximation and dynamic response of the driven cubic–quintic Duffing oscillator

This study investigates the dynamic behavior of a periodically driven cubic–quintic Duffing oscillator, an extended form of the classical Duffing system that accounts for stronger nonlinear stiffness effects. The motivation lies in understanding nonlinear oscillations at large amplitudes, where classical analytical approaches often fail. To address this, two semi-analytical techniques—HPM and Ms-DTM—are employed to derive approximate solutions expressed as infinite power series. The accuracy and convergence of these methods are validated through comparison with numerical integration results obtained using MATLAB’s ode45 solver. Excellent agreement among the three approaches confirms the reliability of both HPM and Ms-DTM in capturing the system’s nonlinear response. The comparative analysis reveals consistent displacement, velocity, and phase behavior, including frequency shifts due to nonlinear stiffness and amplitude variations under harmonic excitation. These findings demonstrate that HPM and Ms-DTM provide efficient and accurate tools for modeling complex nonlinear oscillatory systems.