The mathematical description of uniformity and its relationship with chaos

Based on the definition of monopolized sphere, instantaneous chaometry and k step chaometry are defined, which are the stable characteristic of chaotic orbit. Uniform index is defined by monopolized sphere,and its description of uniformity is quite consistent with people's understanding. The contained uniform index, which is a transitionary concept, is similar to uniform index and has good mathematical property. For random orbit, the contained uniform index converges to 1/V-n (1) (V-n (1) is the volume of the sphere with radius 1), when the point number of an orbit is great enough, uniform index is approximately equal to the contained uniform index. Only by properly selecting the polyhedron containing the basin of attraction of a discrete dynamic system, the ratio of instantaneous chaometry and uniform index is constant. The application of uniform index on Logistic map shows that as the parameter of Logistic map r increases, the orbits are more and more uniform but the expectation uniform index will be less than 0.5 which is that of random orbit, so asymptotical periodic orbit and random pattern are the two extreme status of chaotic orbits.