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The phenomenon of dynamic chaos and self-organization is related to stochastic instability and critical behavior of nonlinear physical systems of various nature. These processes occur, e.g., in the dynamic failure phenomenon of condensed media and fully developed turbulence. In such nonlinear systems, a cascade of dynamic dissipative structure with a fractal structure arises. Multifractal properties are characterized by the spectral function f(dfi), determined by the behavior of the number of elements l required to cover the fractal sets with the similar probabilities Pi~ . In both cases, the transit of systems from one scale-time level to the next one via a cascade of bifurcation and quantitative process characteristics at developed stages do not depend on the Hamiltonian of interatomic interaction. The fractal organization of processes at all scale-time levels testifies to the process similarity and allows us to attribute these processes to a single class of universality. The evolution of the system as a whole is determined not by the interatomic interaction Hamiltonians, but by the incipient cascades of dissipative structures. The close values of fractal dimensions on all the studied scales, which characterize the failure structure, allow us to study the formation of micro-flaws of failure and macro-failure as the scale edges of the spectrum of a single process that has common order parameters. [ABSTRACT FROM AUTHOR]