CIRCULAR NONAUTONOMOUS GENERATOR OF HYPERBOLIC CHAOS
Cite this article as:
Kruglov V. P. CIRCULAR NONAUTONOMOUS GENERATOR OF HYPERBOLIC CHAOS. Izvestiya VUZ. Applied Nonlinear Dynamics, 2010, vol. 18, iss. 5, pp. 132-147. DOI: https://doi.org/10.18500/0869-6632-2010-18-5-132-147
A scheme of circular system is introduced, which is supposed to generate hyperbolic chaos. Its operation is based on doubling of phase on each complete cycle of the signal transmission through the feedback ring. That is a criterion for the attractor of Smale–Williams type to exist. Mathematically, the model is described by the fourth order nonautonomous system of ordinary differential equations. The equations for slowly varying complex amplitudes are derived, and the Poincar ́ e return map is obtained. Numerical simulation data are presented. The attractor of Smale–Williams type is observed in the Poincar ́ e crosssection. The computations indicate that the dynamics of phases is described approximately by the Bernoulli map. Lyapunov exponents for the Poincar ́ e map are estimated, and their dependence on parameters is plotted. Smooth dependence of the largest Lyapunov exponent on parameters supports the structural stability of the observed attractor.
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BibTeX
author = {V. P. Kruglov},
title = {CIRCULAR NONAUTONOMOUS GENERATOR OF HYPERBOLIC CHAOS},
year = {2010},
journal = {Izvestiya VUZ. Applied Nonlinear Dynamics},
volume = {18},number = {5},
url = {https://old-andjournal.sgu.ru/en/articles/circular-nonautonomous-generator-of-hyperbolic-chaos},
address = {Саратов},
language = {russian},
doi = {10.18500/0869-6632-2010-18-5-132-147},pages = {132--147},issn = {0869-6632},
keywords = {hyperbolic chaos,Smale–Williams attractor,Bernoulli map,structural stability.},
abstract = {A scheme of circular system is introduced, which is supposed to generate hyperbolic chaos. Its operation is based on doubling of phase on each complete cycle of the signal transmission through the feedback ring. That is a criterion for the attractor of Smale–Williams type to exist. Mathematically, the model is described by the fourth order nonautonomous system of ordinary differential equations. The equations for slowly varying complex amplitudes are derived, and the Poincar ́ e return map is obtained. Numerical simulation data are presented. The attractor of Smale–Williams type is observed in the Poincar ́ e crosssection. The computations indicate that the dynamics of phases is described approximately by the Bernoulli map. Lyapunov exponents for the Poincar ́ e map are estimated, and their dependence on parameters is plotted. Smooth dependence of the largest Lyapunov exponent on parameters supports the structural stability of the observed attractor. }}