  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.
