Belykh attractor in Zaslavsky map and its transformation under smoothing

TL;DR

This paper investigates the Belykh attractor in a Zaslavsky map, examining how smoothing a sawtooth function affects its chaotic dynamics, Lyapunov exponents, and the emergence of periodicity windows, with implications for electronic device applications.

Contribution

It introduces a detailed analysis of the Belykh attractor's behavior under smoothing, revealing the transition from quasi-hyperbolic chaos to periodicity and mapping the dynamic regimes.

Findings

Smoothing the sawtooth function induces periodicity windows.

Decreasing smoothing scale reduces the size of periodicity regions.

The attractor's chaotic nature diminishes with increased smoothing.

Abstract

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.