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 simulations 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 experiments 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.

中文翻译：

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颞下颌关节盘非线性数学模型的辨识与分析

本文研究了与识别描述颞下颌关节 (TMJ) 盘动力学的非线性数学模型相关的问题。基于真实盘的测试，建立并验证了非线性模型，然后进行数值模拟，目的是分析模型在各种激励条件下的行为。其中包括绘制最大李雅普诺夫指数的多色分布图，根据该图识别周期性运动区和混沌运动区的出现区域。生成李亚普诺夫图样本部分的稳态分岔图以及周期解和混沌解的相流。对于相同的部分，进行数值模拟以确定共存的解决方案。这些研究是使用显示共存解的数量及其周期性的图表进行的。论文中提出的研究表明计算机模拟结果与实验室实验记录的数据非常吻合。由于系统中存在非常强的阻尼，混沌吸引子的几何形状类似于准周期解。强阻尼也会显着影响多个解，这在分析模型中相对罕见。大多数混沌响应和多重解发生在影响关节盘组织的动态载荷的低振幅值范围内。数值实验的结果清楚地表明，在作用于系统的外部载荷的低频值范围内，周期为1T的单周期解占主导地位。随着载荷幅值的增加，此类解出现的面积也随之增加。