Criteria of self-similarity and self-affinityof dynamical chaos

Abstract

This work is devoted to study out the following question: does any qualitative criteria of realization of such universal phenomena as self-organization exist in open systems? Self-organization is also called the appearance of order from chaos under the conditions of non-linearity, non-equilibrium and non closure. Information entropy and fractal dimension of a set of physical values are usually used as quantitative characteristics of chaos. The more detailed characteristic of dynamical chaos is the Kolmogorov-Sinay entropy. Inhomogeneity of elements of a phase space can be taken into account by use of this characteristic. Technically, precise calculation of Kolmogorov – Sinay entropy can’t be realized. Uncertain questions are: What is the minimum of increasing of entropy, how much it decreases at self-organization? Also it was not ascertained the connection between entropy criterion of self-similarity and self-affine with fractal dimensions characterized corresponding chaotic processes.

In the paper the values of information at fixed points of probability function of density of information and entropy  have been defined.Physical meaning of these values as criteria of self-affinity and self-similarity in chaotic processes have been explained. The Kolmogorov-Sinay entropy and fractal dimensions corresponding to scale-invariant sets have been described also.