Mathematical Theory of Uniformity and its Applications in Ecology and Chaos

The distribution of a finite number of points in a polygon is referred to as a pattern in this book. Aggregation and uniformity are two typical opposite features of a pattern. Based on the uniform degree defined in this context, a pattern with uniform degree zero, like a periodic orbit, is the most aggregate pattern and the uniform degree of any chaotic orbit must be greater than 0. For any segment of a chaotic orbit, it may behave in either an aggregate or a uniform way. However, its average uniform degree is invariant, which is the reason for measuring chaotic behaviour by the uniform degree. Based on this, the word “chaometry” has been invented, meaning the measurement of chaos. The uniform degree of the pattern generated by a distribution function F in a polygon is abbreviated as the uniform degree of a distribution function, denoted by \(L_F\).

The uniform degree L defined in the previous chapter is in accordance with the intuition on uniformity. The k-step average uniform degree is an estimation of the expected average uniform degree, and k-step chaometry defined below can be used to evaluate the degree of chaos. As known to all, the uniform distribution is an ultimate case of chaos. In Chap. 1, the uniformity theorem is proved and the conjecture verified by a wide range of numerical simulations.

We use uniform degree to describe the periodic and uniform distributions, the two extremes of chaos. Their uniform degrees are 0 and \(\frac{1}{V_n(1)}\), respectively. The role of uniform degree is similar to entropy, while its accuracy is much better than entropy. Therefore, uniform degree may also be employed to investigate the problems arising from thermodynamics. In this chapter, more numerical simulations will be carried out, even though some numerical results have been shown previously.

Forestry and ecology are inseparable subjects. The investigation of patterns arising from these two areas gives rise to theory of uniformity, and the theory of uniformity is applicable to these two areas in reverse.