Exact analytical solutions for forced undamped Duffing oscillator

• Forcing functions composed of Jacobi elliptic functions.
• Exploration of conceivable choices of the elliptic function parameter.
• Exact analytical solution for forced steady-state oscillation response.
• Exact analytical expressions for frequency–amplitude relation.

The steady-state response of an undamped Duffing oscillator to periodic external forces is studied. The forcing functions are chosen such that the time course of the displacement can be described by exact analytical expressions. The displacement and forcing functions are governed by Jacobi elliptic functions and thus are periodic but generally non-harmonic. The parameter of the elliptic functions is deliberately chosen. For certain parameter choices, exact analytical expressions are found for the frequency–amplitude relation.