Borysov S. Statistical analysis of behaviour of self-organized complex systems

Statistical properties of the self-organized complex systems whose phase space contains limit cycles or is hierarchically arranged are investigated. For a deterministic non-linear system, the criteria of limit cycle stability, which does not require to make an additional transformation of dynamical variables, is found. A mechanism of oscillations suppressing under effect of L?vy noises is studied. For the systems which possess a fractal phase space the field formalism based on different calculi is developed. Equations for of the systems which possess an internal symmetry and constraints are obtained. The dependency of generating functionals, one-site statistical sum and moments of free fields on deformation is defined. It is shown that the consistent theoretical-probabilistic description of hierarchical structures requires using of the Tsallis algebra. Distributions over hierarchical levels are studied analytically and numerically.