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The paper presents a study of issues related to the identification of a non-linear mathematical model describing
dynamics of the temporomandibular joint (TMJ) disc. Based on the tests of real disks, a non-linear model was
built and verified, and then numerical simulations were carried out, the purpose of which was to analyze the
behavior of the model for various excitation conditions. They include, among others, plotting a multi-colored
map of distribution of the largest Lyapunov exponent based on which the areas of occurrence of periodic and
chaotic motion zones are identified. Bifurcation diagrams of steady states for sample sections of the Lyapunov
map and phase flows of periodic and chaotic solutions are generated. For the same sections, numerical simu-
lations are performed to identify coexisting solutions. These studies are carried out using diagrams showing the
number of coexisting solutions and their periodicity. The research presented in the paper shows a very good
match between the results of computer simulations and the data recorded in the laboratory experiment. Due to
the very strong damping occurring in the system, the chaotic attractors resemble quasi-periodic solutions with
their geometric shape. Strong damping also significantly affects multiple solutions, which are relatively rare in
the analyzed model. Most of the chaotic responses and multiple solutions occur in the range of low amplitude
values of the dynamic load affecting the tissues of the articular disc. The obtained results of numerical experi-
ments clearly indicate that in the range of low frequency values of the external load acting on the system, single
periodic solutions with a periodicity of 1 T dominate. With the increase of the load amplitude, the area of
occurrence of such solutions increases.