Few-Freedom Turbulence

Abstract

The results of numerical experiments on the structure of chaotic attractors in the Khalatnikov - Kroyter model of two freedoms are presented. This model was developed for a qualitative description of the wave turbulence of the second sound in helium. The attractor dimension, size, and the maximal Lyapunov exponent in dependence on the single dimensionless parameter $F$ of the model are found and discussed. The principal parameter $F$ is similar to the Reynolds number in hydrodynamic turbulence. We were able to discern four different attractors characterized by a specific critical value of the parameter ($F=F_{cr}$), such that the attractor exists for $F>F_{cr}$ only. A simple empirical relation for this dependence on the argument ($F-F_{cr}$) is presented which turns out to be universal for different attractors with respect to the dimension and dimensionless Lyapunov exponents. Yet, it differs as to the size of attractor. In the main region of our studies the dependence of all dimensionless characteristics of the chaotic attractor on parameter $F$ is very slow (logarithmic) which is qualitatively different as compared to that of a multi-freedom attractor, e.g., in hydrodynamic turbulence (a power law). However, at very large $F\sim 10^7$ the transition to a power-law dependence has been finally found, similar to the multi-freedom attractor. Some unsolved problems and open questions are also discussed.