We describe and experimentally explore a nonlinear stiffness model of the trapping of a solid particle in a single-axis acoustic levitator. In contrast to the commonly employed linear stiffness assumption, our nonlinear model accurately predicts the response of the system. Our nonlinear model approximates the acoustic field in the vicinity of the trap as a one-dimensional sinusoid and solves the resulting dynamics using numerical continuation. In particular, we predict a softening of stiffness with amplitude as well as period-doubling bifurcations, even for small excitation amplitudes of ≈2% of the wavelength. These nonlinear dynamic features are observed experimentally in a single-axis levitator operating at 40 kHz and trapping millimetre-scale expanded polystyrene spheres. Excellent agreement between the observed and predicted behaviour is obtained suggesting that this relatively simple model captures the relevant physical phenomena. This new model enables the dynamic instabilities of trapped particles to be accurately predicted, thereby benefiting contactless transportation and manipulation applications.