Exact steady states of the periodically forced and damped Duffing oscillator

We present an analytical methodology to compute the exact steady states of the strongly nonlinear Duffing oscillator with arbitrary viscous damping and under the action of a time-periodic excitation. In contrast to current analytical approaches that consider harmonic excitations to produce approximate steady-state solutions, the excitations considered in this work are multi-harmonic and are computed in closed form. We consider the exact fundamental resonance plot of this strongly nonlinear system and present an exact depiction of the dynamic balance between the internal and external forces leading to it in terms of rotating vectors in the complex plane with modulated amplitudes and modulated frequencies of rotation. Moreover, an infinity of exact steady-state solutions seems to be possible, each corresponding to a different time-periodic excitation. Generalization of the presented methodology to systems with other type of nonlinearities or with many degrees of freedom is presumably straightforward.