Clustering and Uncertainty in Perfect Chaos Systems

TL;DR

This paper derives new properties of chaotic systems using a generalized Fokker-Planck equation, revealing how uniform systems tend to unstable clustering and establishing uncertainty relations in phase space.

Contribution

It introduces a generalized Fokker-Planck framework for chaos analysis, uncovering instability mechanisms and spectral properties in chaotic systems with fixed boundaries.

Findings

Chaotic systems with uniform properties tend to unstable clustering.

Phase space diffusion determines dynamic accuracy.

Uncertainty relations are derived for conjugate variables.

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 non stationary 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 to find a principal consequence: a chaotic system with uniform dynamic properties tends to unstable 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.