Skip Navigation Links
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Bernoulli Mapping with Hole and a Saddle-Node Scenario of the Birth of Hyperbolic Smale–Williams Attractor

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

Download Full Text PDF



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.


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.


  1. [1]  Smale, S. (1967), Differentiable dynamical systems, Bull. Amer. Math. Soc. , 73, 747-817.
  2. [2]  Williams, R.F. (1974), Expanding attractors, Publ. Math. de l’IHES , 43, 169-203.
  3. [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. [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. [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. [6]  Paar, V. and Pavin, N. (1997),Missing preimages for chaotic logistic map with a hole, Fizika B, 6(1), 23-35.
  7. [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. [8]  Dettmann, C. (2012), Open circle maps: small hole asymptotics, Nonlinearity, 26(1), 307.
  9. [9]  Glendinning, P. and Sidorov, N. (2015), The doubling map with asymmetrical holes, Ergodic Theory and Dynamical Systems, 35(4), 1208-1228.
  10. [10]  Sidorov, N. (2014), Supercritical holes for the doubling map, Acta Mathematica Hungarica, 143(2), 298-312.
  11. [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. [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. [13]  Schneider, J., Tel, T., and Neufeld, Z. (2007), Dynamics of “leaking” Hamiltonian systems, Physical review E, 66, 066218.
  14. [14]  Altmann, E.G. and Tel, T. (2008), Poincaré recurrences from the perspective of transient chaos, Physical review letters, 100, 174101.
  15. [15]  Altmann, E.G. and Tel, T. (2009), Poincaré recurrences and transient chaos in systems with leaks, Physical review E, 79, 016204.
  16. [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. [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. [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. [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. [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. [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. [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.