  If we allow non-smooth or discontinuous functions in definition of an
evolution operator for dynamical systems, then situations of quasi-hyperbolic
chaotic dynamics often occur like, for example, on attractors in model Lozi map
and in Belykh map. The present article deals with the quasihyperbolic attractor
of Belykh in a map describing a rotator with dissipation driven by periodic
kicks, the intensity of which depends on the instantaneous angular coordinate
of the rotator as a sawtooth-like function, and also the transformation of the
attractor under smoothing of that function is considered. Reduction of the
equations to the standard form of the Belykh map is provided. Results of
computations illustrating the dynamics of the system with continuous time on
the Belykh attractor are presented. Also, results for the model with the
smoothed sawtooth function are considered depending on the parameter
characterizing the smoothing scale. On graphs of Lyapunov exponents versus a
parameter, the smoothing of the sawtooth implies appearance of periodicity
windows, which indicates violation of the quasi-hyperbolic nature of the
attractor. Charts of dynamic regimes on the parameter plane of the system are
also plotted, where regions of periodic motions ("Arnold's tongues") are
present, which decrease in size with the decrease in the characteristic scale
of the smoothing, and disappear in the limit case of the sawtooth function with
a break. Since the Belykh attractor was originally introduced in the
radiophysical context (phase-locked loops), the analysis undertaken here is of
interest from the point of view of possible exploiting the chaotic dynamics on
this attractor in electronic devices.
