Discontinuity, Nonlinearity, and Complexity
Bernoulli Mapping with Hole and a SaddleNode Scenario of the Birth of Hyperbolic Smale–Williams Attractor
Discontinuity, Nonlinearity, and Complexity 9(1) (2020) 1326  DOI:10.5890/DNC.2020.03.002
Olga B. Isaeva$^{1}$,$^{2}$, Igor R. Sataev$^{1}$
$^{1}$ Kotel’nikov’s Institute of RadioEngineering 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
Onedimensional Bernoulli mapping with hole is suggested to describe the regularities of the appearance of a chaotic set under the saddlenode scenario of the birth of the Smale–Williams hyperbolic attractor. In such a mapping, a nontrivial chaotic set (with nonzero Hausdorff dimension) arises in the general case as a result of a cascade of periodadding bifurcations characterized by geometric scaling both in the phase space and in the parameter space. Numerical analysis of the behavior of models demonstrating the saddlenode 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 171201008. Authors acknowledge Prof. S.P. Kuznetsov and Prof. A. Pikovsky for useful discussion.
References

[1]  Smale, S. (1967), Differentiable dynamical systems, Bull. Amer. Math. Soc. , 73, 747817. 

[2]  Williams, R.F. (1974), Expanding attractors, Publ. Math. de l’IHES , 43, 169203. 

[3]  Isaeva, O.B., Kuznetsov, S.P., and Sataev, I.R. (2012), A “saddlenode” bifurcation scenario for birth or destruction of a Smale–Williams solenoid, Chaos: An Interdisciplinary Journal of Nonlinear Science, 22(4), 043111. 

[4]  Isaeva, O.G.B., Kuznetsov, S.P., Sataev, I.R., and Pikovsky, A.S. (2013), On a bifurcation scenario of a birth of attractor of Smale–Williams type, Nelineinaya Dinamika [Russian Journal of Nonlinear Dynamics], 9(2), 267294. 

[5]  Buljan, H. and Paar, V. (2001), Manyhole interactions and the average lifetimes of chaotic transients that precede controlled periodic motion, Physical Review E, 63(6), 066205. 

[6]  Paar, V. and Pavin, N. (1997),Missing preimages for chaotic logistic map with a hole, Fizika B, 6(1), 2335. 

[7]  Paar, V. and Pavin, N. (1997), Bursts in average lifetime of transients for chaotic logistic map with a hole, Physical Review E, 55(4), 4112. 

[8]  Dettmann, C. (2012), Open circle maps: small hole asymptotics, Nonlinearity, 26(1), 307. 

[9]  Glendinning, P. and Sidorov, N. (2015), The doubling map with asymmetrical holes, Ergodic Theory and Dynamical Systems, 35(4), 12081228. 

[10]  Sidorov, N. (2014), Supercritical holes for the doubling map, Acta Mathematica Hungarica, 143(2), 298312. 

[11]  Hare, K.G. and Sidorov, N. (2014), On cycles for the doubling map which are disjoint from an interval, Monatshefte fur Mathematik, 175(3), 347365. 

[12]  Tuval, I., Schneider, J., Piro, O., and Tel, T. (2004), Opening up fractal structures of threedimensional flows via leaking, Europhysics letters, 65, 633. 

[13]  Schneider, J., Tel, T., and Neufeld, Z. (2007), Dynamics of “leaking” Hamiltonian systems, Physical review E, 66, 066218. 

[14]  Altmann, E.G. and Tel, T. (2008), Poincaré recurrences from the perspective of transient chaos, Physical review letters, 100, 174101. 

[15]  Altmann, E.G. and Tel, T. (2009), Poincaré recurrences and transient chaos in systems with leaks, Physical review E, 79, 016204. 

[16]  Livorati, A.L.P., Georgiou, O., Dettmann, C.P., and Leonel, E.D. (2014), Escape through a timedependent hole in the doubling map, Physical review E, 89, 052913. 

[17]  Procaccia, I., Thomae, S., and Tresser, C. (1987), Firstreturn maps as a unified renormalization scheme for dynamical systems, Physical Review A, 35(4), 1884. 

[18]  Kuznetsov, S.P. (2005), Example of a physical system with a hyperbolic attractor of the Smale–Williams type, Physical review letters, 95(14), 144101. 

[19]  Kuznetsov, S.P. and Sataev, I.R. (2007), Hyperbolic attractor in a system of coupled nonautonomous van der Pol oscillators: Numerical test for expanding and contracting cones, Physics Letters A, 365(12), 97104. 

[20]  Kuznetsov, S.P. and Sataev, I.R. (2006), Verification of hyperbolicity conditions for a chaotic attractor in a system of coupled nonautonomous van der Pol oscillators, Izvestiya VUZ. Appl. Nonlin. Dynam.(Saratov), 14, 329. 

[21]  Wilczak, D. (2010), Uniformly hyperbolic attractor of the Smale–Williams type for a Poincaré map in the Kuznetsov system, SIAM Journal on Applied Dynamical Systems, 9(4), 12631283. 

[22]  Isaeva, O.B., Kuznetsov, S.P., Sataev, I.R., Savin, D.V., and Seleznev, E.P. (2015), Hyperbolic chaos and other phenomena of complex dynamics depending on parameters in a nonautonomous system of two alternately activated oscillators, International Journal of Bifurcation and Chaos, 25(12), 1530033. 