Scale invariance criteria of dynamical chaos
Z. Zh. Zhanabaev, Y.T. Kozhagulov, Serik A. Khokhlov · Studies in Applied Mathematics · 2013
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 ofnon-linearity,non-equilibrium and nonclosure. 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 – Sinayentropy can’t be realized. Uncertain questions are: What is the minimum of increasingof entropy, how muchitdecreases atself-organization? Also it was not ascertained the connection between entropy criterion of selfsimilarity 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. It is shownthat self-organizationoccurs whennormalizedinformation entropyStakes valuesin the interval , where = 0.567 , = 0.806 . The precision of thesefindings is proved by calculation the valueS. Applications of these resultsin modernscientific and engineering areas are possible.