A simple nonlinear undamped Duffing oscillator has been studied for its snap-through behavior at large-amplitude vibrations. We present an algorithm that uses the harmonic balance (HB) method to find amplitude and frequency relationships in two- and three-term approximations for solutions that lie outside the separatrix in the phase space. Trends of the approximate solution properties are examined with reference to an analysis of the limit as the trajectory approaches the separatrix.