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Discontinuity, Nonlinearity, and Complexity 9(1) (2020) 13--26 | DOI:10.5890/DNC.2020.03.002

Olga B. Isaeva$^{1}$,$^{2}$, Igor R. Sataev$^{1}$

$^{1}$ Kotel’nikov’s Institute of Radio-Engineering and Electronics of RAS, Saratov Branch, Zelenaya 38, Saratov, 410019, Russian Federation

$^{2}$ Saratov State University, Astrakhanskaya 83, Saratov, 410026, Russian Federation

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Abstract

One-dimensional Bernoulli mapping with hole is suggested to describe the regularities of the appearance of a chaotic set under the saddle-node scenario of the birth of the Smale–Williams hyperbolic attractor. In such a mapping, a non-trivial chaotic set (with non-zero Hausdorff dimension) arises in the general case as a result of a cascade of period-adding bifurcations characterized by geometric scaling both in the phase space and in the parameter space. Numerical analysis of the behavior of models demonstrating the saddle-node scenario of birth of a hyperbolic chaotic Smale–Williams attractor shows that these regularities are preserved in the case of multidimensional systems. Limits of applicability of the approximate 1D model are discussed.

Acknowledgments

The work was supported by the grant of the Russian Scientific Foundation No 17-12-01008. Authors acknowledge Prof. S.P. Kuznetsov and Prof. A. Pikovsky for useful discussion.

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