Generalization of natural system parameters: examples, theory and rules

А method is proposed to determine quantitative relationships between observed parameters of natural systems basing on their generalized (normalized) values. The normalized values are presented by two relations: 1) the difference between the current and minimum values to the difference between the maximum and the minimum, and 2) the difference between the maximum and current values to the same difference between the maximum and the minimum. With their help, the whole set of variables is clearly representable within the range of 0...1 or 1...0. This reduces the amount of factual data required to establish quantitative relationships between them, and makes it possible to correctly compare the diverse values, which makes the solution universal for a large class of tasks. The sum of these relations is one, and each component can increase (or decrease) only due to a decrease (or increase) in the other. This is a typical dichotomy, embodying the law of unity and struggle of opposites, which can serve as a model of any two-component system. It is shown that in most cases establishing the laws of any system development is sufficient to consider the interaction between only its two major and most influential groups, for example, woody and herbaceous vegetation (but not individual plants from their composition), forests and steppes, land and sea, heat and cold, etc. Geometric interpretation of a two-component system can serve as a single segment consisting of two parts: a larger one - dominant (ф), and a smaller one - subdominant (1 - ф). Balance and maximum sustainability of this model are achieved through the division of a line segment in the golden section, the proportion between the whole and its two parts maintained, with ф = 0,618... м 0,62. But a single segment can be divided into parts according to the multitude (q) of other sections as well. The expression ф is found in the general case when 0 < q < да. It is ф + ф =1, where n = q + 1, n - number of particles in the system, q - the number of sections. The roots of this equation for different integers n are a sequence of the generalized golden sections (GGS). They are the most common ratios of components in many systems of the world, including biological ones ensuring the consistency of their interaction (harmony), sustainability and long existence [1, 2]. The article shows that the patterns of GGS are relevant not only with whole but also with fractional n. The integers correspond to the correct geometrical forms (plane, cube, etc.). Fractional - to fractals with their complex branching forms of a tree crown type, river systems, blood vessels, etc. Together with the increase in n the number of dividing lines borders, which stress concentrators, is also growing. These places (ecotones, coastal and snow line, off-season - spring and autumn (morning and evening), borders and ethnic groups, frontier and the like) are most sensitive to changes in the external environment and most exposed to deformation. These system sections are the ones with the least resistance, and more likely to fracture. A correlation was found between GGS and relative durability of solids, frozen soils and ice in particular, and through it with global factors. The relationships between parameters of different natural systems - biological, climatic, cryogenic are established and quantified. Particularly, the formula of dependence of the diameter of a tree trunk and its productivity on age, weight wood greenery - on trunk diameter. Primarily polynomial or exponential nature of these relationships is shown with numerical coefficients of calculation formulas close to the proportions of the golden section. The examples of using the proposed method, indicating a good convergence of estimates and actual data, are given. The proposed method is a powerful multi-disciplinary (synthetic) means of generalization and identification of the common in different natural systems.

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