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.