Abstract
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.
Highlights
Several scenarios of a turbulence transition have been proposed since 1883 when the turbulence concept was firstly introduced by an English engineer Osborne Reynolds
It was obtained that a chaotic system with a non stationary diffusion satisfies a nonlinear dispersion law
This law leads to instabilities in a phase space and to the appearance of the clustering properties for the initially uniform system
Summary
Several scenarios of a turbulence transition have been proposed since 1883 when the turbulence concept was firstly introduced by an English engineer Osborne Reynolds. Consent has been obtained that an unpredictability of chaos is consequence of two conditions: a finite resolution of generalized phase space, instability of phase trajectories and mixing of phase trajectories [5]. These conditions can be defined in the following way – (1), (2) and (3): xi x i min (1). We may define complex portrait of chaotic system – it is a complex dynamic system that corresponds to given obligatory properties: a finite resolution of phase space, instability of phase trajectories, mixing of phase trajectories and a positive entropy production. All properties that have been shown above are assumed to be satisfied
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