Bernoulli Mapping with Hole and a Saddle-Node Scenario of the Birth of Hyperbolic Smale–Williams Attractor
DOI:
https://doi.org/10.5890/DNC.2020.03.002Abstract
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.References
[1] Smale, S. (1967), Differentiable dynamical systems, Bull. Amer. Math. Soc. , 73, 747-817.
[2] Williams, R.F. (1974), Expanding attractors, Publ. Math. de l’IHES , 43, 169-203.
[3] Isaeva, O.B., Kuznetsov, S.P., and Sataev, I.R. (2012), A “saddle-node” 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), 267-294.
[5] Buljan, H. and Paar, V. (2001), Many-hole 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), 23-35.
[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), 1208-1228.
[10] Sidorov, N. (2014), Supercritical holes for the doubling map, Acta Mathematica Hungarica, 143(2), 298-312.
[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), 347-365.
[12] Tuval, I., Schneider, J., Piro, O., and Tel, T. (2004), Opening up fractal structures of three-dimensional 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 time-dependent hole in the doubling map, Physical review E, 89, 052913.
[17] Procaccia, I., Thomae, S., and Tresser, C. (1987), First-return 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 non-autonomous van der Pol oscillators: Numerical test for expanding and contracting cones, Physics Letters A, 365(1-2), 97-104.
[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, 3-29.
[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), 1263-1283.
[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.
Article Metrics
Usage tracking begins September 1, 2026.