2017/10/21 by С. П. Кузнецов, Kuznetsov, Sergey P.
Mathematics · Physics and Astronomy · #37D45 #Chaos control and synchronization #Chaotic Dynamics (nlin.CD) #FOS: Physical sciences #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.1710.07828
openalex publication_date 2017/10/21 · openalex created_date 2017/11/10 · openalex updated_date 2026/07/28
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.