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人们普遍将描述骨再造过程的微分方程看作有多个自由度的非线性动力系统 ,它相对优化目标不断变化 ,可以导致很多组可能解。因此关于理论解稳定性条件的分析是非常必要的。本文的目的是阐述非线性动力系统分析的数学方法应用于一种骨自优化控制方程 ,研究该骨再造方程的全局稳定性。分析了两个理论模型 :两单元模型和 2× 2单元模型 ,该模型特点是其中各个单元的应力、应变都是相关的 ,这样与实际有限元模型更加接近。以朱兴华等[1] 提出的高阶非线性骨再造速率方程作为控制方程 ,重点考察其中再造率系数B(t)取为指数形式和引入非线性再造方程的阶数时 ,骨自优化方程获得稳定解的条件。并以两个经典的二维平面问题作为算例 ,与上述两个模型的理论分析结果进行对比 ,使分析得到的结果得以确认。这种模型的数学分析方法对于研究骨再造控制过程所采用的这一复杂的非线性动力系统中的有关生理参数的作用有着十分重要的理论意义
The differential equation describing the process of bone remodeling is generally regarded as a nonlinear dynamical system with multiple degrees of freedom. Its relative optimization goal is constantly changing and can result in many possible solutions. Therefore, it is very necessary to analyze the condition of theoretical solution stability. The purpose of this paper is to illustrate the application of a mathematical method for nonlinear dynamic system analysis to a self-optimizing control equation for bone to study the global stability of the bone remodeling equation. Two theoretical models are analyzed: two-element model and 2 × 2 element model. The model is characterized in that the stress and strain of each element are related, which is closer to the actual finite element model. Taking the high-order nonlinear bone remodeling rate equation proposed by Zhu et al. [1] as the governing equation, the Bone Self-Optimization Equation was obtained when the recycle factor B (t) was taken as the exponential form and the order of the nonlinear reconstruction equation was introduced Conditions for stable solution. Two classical two-dimensional plane problems are taken as examples to compare with the theoretical analysis results of the above two models so as to confirm the analysis results. The mathematical analysis of this model is of great theoretical significance for studying the role of physiological parameters in this complex nonlinear dynamical system used in the process of bone remodeling control