During middle ear reconstruction, surgeons typically have to adjust the size and position of a medical prosthesis to fit the needs of each patient. In this paper, a prosthesis made of shape memory alloy is proposed to reduce operation time while accounting for the individual variability in middle ear lengths. This shape memory prosthesis (SMP) is activated by heat, so it is important to measure the regular and irregular behaviour prior to recommendation for medical practice. We present a two-degrees-of-freedom model of a reconstructed middle ear with an SMP that has two nonlinear springs, representing the properties of the annular ligament and the prosthesis. The properties of the SMP are described via polynomial dependence. Experiments on a temporal bone are used to determine the parameters of an intact ossicular chain and test a conventional titanium prosthesis. The dynamics of the reconstructed middle ear are investigated under different external excitation parameters. Bifurcation analysis was conducted and the stability of periodic solutions was examined to monitor the behaviour of the stapes. The results of this study provide interesting chaotic vibrations and regular oscillations with fluctuation. We determine that only the harmonic response of the stapes is advisable from a practical point of view, and this can only be achieved for small excitation amplitudes. Therefore, we show the importance of selecting appropriate prosthetic parameters to avoid irregular motions and deterministic chaos.