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In this paper, analytical solutions for periodic motions in a parametric hardening Duffing oscillator are presented using the finite Fourier series expression, and the corresponding stability and bifurcation analysis for such periodic motions are carried out. The frequency-amplitude characteristics of asymmetric period-1 and symmetric period-2 motions are discussed. The hardening Mathieu–Duffing oscillator is also numerically simulated to verify the approximate analytical solutions of periodic motions. Period-1 asymmetric and period-2 symmetric motions are illustrated for a better understanding of periodic motions in the hardening Mathieu–Duffing oscillator.
Luo et al. (Wed,) studied this question.