非线性方程组求解的混沌最小二乘法及平面四杆函数机构综合

机构学问题的数学模型常可化为多元非线性方程组，一般求解多元非线性方程组需要初始值，而初始值的选择是相当困难的；同伦方法不需初始值就能求出全部解，为求解这一问题提供了可行的方法，但需要编写专用的程序，且计算量比较大；本文结合MATLAB6．5．1高级程序设计语言采用简单的最小二乘法迭代，并将非线性方程视为非线性的动力学系统，利用使得系统产生混沌的Julia集的点求解方程的全实数解，而Julia集的点集用二周期逆像函数求得，再在其邻域内求解即可。运用该算法编写了MATLAB程序，对平面四杆机构综合问题进行了研究，从而找到了实现最大精确点时该问题的全部的解，为实际机构的设计提供了多种选择方案，为机构学设计提供了全新的方法。

Chaos Least Square Algorithm for Finding Solution of Non-linear Equations and its Application to Mechanism Synthesis

The mathematical model of mechanism can be translated into a multi-variables non-linear equations, and, it is difficult to give the initial value for the non-linear equations. The homology continuation algorithm has overcomed this difficult without given initial value and can find all solutions, but a special programming must be compiled and the calculating load is very larger at the same time. For the first time, a numerical method is be found, taking the iteration of least square algorithm as a nonlinear dynamitic system in which Julia sets leads to Chaos. The point of Julia set is found by constructing repulsion two-cycle point function and making use of inverseimage iterative methods. Then in the neighborhood of Julia point all solutions of non-linear equations. The problem of function generation for planar four-link mechanism is solved by this method, and thus all solutions for this problem with maximum precision positions are obtained. This provide a simple realization method for mechanics design.