Clustering and Uncertainty in Perfect Chaos Systems

The goal of this investigation was to derive strictly new properties of chaotic systems and their mutual relations. The generalized Fokker-Planck equation with a nonstationary diffusion has been derived and used for chaos analysis. An anomalous transport turned out to be natural property of this equation. A nonlinear dispersion of the considered motion allowed us to find a principal consequence: a chaotic system with uniform dynamic properties tends to instable clustering. Small fluctuations of particles density increase by time and form attractors and stochastic islands even if the initial transport properties have uniform distribution. It was shown that an instability of phase trajectories leads to the nonlinear dispersion law and consequently to a space instability. A fixed boundary system was considered, using a standard Fokker-Planck equation. We have derived that such a type of dynamic systems has a discrete diffusive and energy spectra. It was shown that phase space diffusion is the only parameter that defines a dynamic accuracy in this case. The uncertainty relations have been obtained for conjugate phase space variables with account of transport properties. Given results can be used in the area of chaotic systems modelling and turbulence investigation. 1. Introduction Several scenarios of a turbulence transition have been proposed since 1883 when the turbulence concept was firstly introduced by an English engineer called Osborne Reynolds. The dynamic phase transition in a liquid stream was remarked by Reynolds—it was characterized by an unstable vortex appearance and two limit states of motion: laminar and turbulent types. Since an introduction of a turbulence concept, its properties have been generalized and transformed into properties of a chaos state. Several scenarios of a turbulence transition have been developed. Among them, the Landau-Hopf instability mechanism [1], the Lorenz attractor mechanism [2], the scenario of Poincare-Feigenbaum [3], and Kolmogorov-Arnold-Moser [4]. Each of the outlined mechanisms has its basic assumptions and an individual area of its application. For this reason none of them suggests some universal approaches—moreover unambiguous connections between the given mechanisms are not stated yet. Let us try to highlight common points for the formulation of chaos concept. Consent has been obtained that an unpredictability of chaos is consequence of two conditions: a finite resolution of generalized phase space and instability of phase trajectories and mixing of phase trajectories [5]. Formally these conditions can be