NEW SERIES. SERIES: PHYSICS

Izvestiya of Saratov University.

ISSN 1817-3020 (Print)
ISSN 2542-193X (Online)


Cite this article as:

Popova E. S., Stankevich N. V., Kuznetsov A. P. Cascade of Invariant Curve Doubling Bifurcations and Quasi-Periodic Hénon Attractor in the Discrete Lorenz-84 Model. //Izvestiya of Saratov University. New series. Series: Physics. , 2020, vol. 20, iss. 3, pp. 222-232. DOI: https://doi.org/10.18500/1817-3020-2020-20-3-222-232

Published online: 
31.08.2020
Language: 
Russian
UDC: 
517.9

Cascade of Invariant Curve Doubling Bifurcations and Quasi-Periodic Hénon Attractor in the Discrete Lorenz-84 Model

Autors: 
Popova Elena Sergeevna, Saratov State University
Stankevich Nataliay Vladimirovna, Saratov Branch of Kotel’nikov Institute of Radio Engineering and Electronics of the Russian Academy of Sciences
Kuznetsov Alexander Petrovich, Saratov Branch of Kotel’nikov Institute of Radio Engineering and Electronics of the Russian Academy of Sciences
Abstract: 

Background and Objectives: Chaotic behavior is one of the fundamental properties of nonlinear dynamical systems, including maps. Chaos can be most easily and reliably diagnosed using the largest Lyapunov exponent, which will be positive for the chaotic mode. Unlike flow dynamical systems, the presence of zero Lyapunov exponent in the spectrum is not an obligatory condition for maps. The zero exponent in the spectrum of a map will indicate the possibility of embedding such a map in a flow. In the framework the present paper, using as an example a three-dimensional discrete Lorenz-84 model, it is shown that there can appear chaotic attractors whose Lyapunov exponent spectrum contains one positive, one zero, and one negative exponents. Such a specific attractor represents the production of a two-dimensional torus and the Hénon attractor and was called the Quasi-periodic Hénon attractor. A scenario of development of such kind behavior is an open problem. Materials and Methods: The discrete Lorenz-84 oscillator obtained by discretizing the three-dimensional flow Lorenz-84 model is considered as an object of the present research. The dynamics of the map is studied numerically. The main analysis is carried out using the charts of dynamical regimes based on the calculation of Lyapunov exponents. Lyapunov exponents were calculated by the Benettin method with Gramm-Schmidt orthogonalization. Results: The scenario of occurrence of the Quasi-periodic Hénon attractor via a cascade of invariant curve doubling bifurcations is described. Conclusion: We study the discrete Lorenz-84 oscillator, which is a three-dimensional map (three-dimensional diffeomorphism). In the map the possibility of implementing a steady state of equilibrium, an invariant curve, a torus-attractor, and chaos with zero Lyapunov exponent in the spectrum was shown. It is also demonstrated that the chaotic mode with zero Lyapunov exponent, the Quasi-periodic Hénon attractor, can appear as a result of a cascade of doubling bifurcations of the invariant curve. Typical structures on the parameter plane, such as CrossRoad-Area, Spring-Area, whose base mode is an invariant curve, are illustrated. These structures and chaotic oscillations can arise on the basis of various invariant curves.

DOI: 
10.18500/1817-3020-2020-20-3-222-232
References: 
  1. Mira C., Gardini L., Barugola A., Cathala J. Chaotic dynamics in two-dimensional noninvertible maps. World Scientifi c, Series on Nonlinear Science, Series A, 1996, vol. 20. 632 p. DOI: https://doi.org/10.1142/2252
  2. Schuster H. G., Just W. Deterministic chaos: an introduction. 4th ed. John Wiley & Sons, 2006. 312 p.
  3. Kuznetsov A. P., Savin A. V., Sedova Yu. V., Tyurykina L. V. Bifurkatsii otobrazheniy [Bifurcations of mappings]. Saratov, OOO Izdatel’skiy tsentr “Nauka”, 2012. 196 p. (in Russian).
  4. Gonchenko S. V., Ovsyannikov I. I., Simo C., Turaev D. Three-dimensional Hénon-like maps and wild Lorenz-like attar tors. International Journal of Bifurcation and Chaos, 2005, vol. 15, no. 11, pp. 3493–3508. DOI: https://doi.org/10.1142/S0218127405014180
  5. Gonchenko A. S., Gonchenko S. V., Shilnikov L. P. Towards scenarios of chaos appearance in three-dimensional maps. Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 1, pp. 3–28.
  6. Gonchenko A. S., Gonchenko S. V., Kazakov A., Turaev D. Simple scenarios of onset of chaos in three-dimensional maps. International Journal of Bifurcation and Chaos, 2014, vol. 24, no. 8, pp. 1440005. DOI: https://doi.org/10.1142/S0218127414400057
  7. Gonchenko A. S., Gonchenko S. V. Variety of strange pseudohyperbolic attractors in three-dimensional generalized Hénon maps. Physica D: Nonlinear Phenomena, 2016, vol. 337, pp. 43–57. DOI: https://doi.org/10.1016/j.physd.2016.07.006
  8. Pikovsky A., Politi A. Lyapunov exponents: a tool to explore complex dynamics. Cambridge University Press, 2016. 295 p.
  9. Arrowsmith D. K., Cartwright J. H. E., Lansbury A. N., Place C. M. The Bogdanov map: bifurcations, mode locking, and chaos in a dissipative system. International Journal of Bifurcation and Chaos, 1993, vol. 3, no. 4, pp. 803. DOI: https://doi.org/10.1142/S021812749300074X
  10. Zaslavsky G. M. The Physics of Chaos in Hamiltonian Systems. World Scientifi c, 2007. DOI: https://doi.org/10.1142/p507
  11. Kuznetsov A. P., Savin A. V., Sedova Y. V. BogdanovTakens bifurcation: from fl ows to discrete systems. Izvestiya VUZ. Applied Nonlinear Dynamics, 2009, vol.17, no. 6, pp. 139–158. DOI: https://doi.org/10.18500/0869-6632-2009-17-6-139-158
  12. Adilova A. B., Kuznetsov A. P., Savin A. V. Complex dynamics in the system of two couple discrete Rossler oscillators. Izvestiya VUZ. Applied Nonlinear Dynamics, 2013, vol. 21, no. 5, pp. 108–119. DOI: https://doi.org/10.18500/0869-6632-2013-21-5-108-119.
  13. Kuznetsov A. P., Sedova Y. V. Maps with quasi-periodicity of different dimension and quasi-periodic bifurcations. Izvestiya VUZ. Applied Nonlinear Dynamics, 2017, vol. 25, no. 4, pp. 33–50. DOI: https://doi.org/10.18500/0869-6632-2017-25-4-33-50
  14. Anishchenko V. S., Nikolaev S. M. Generator of quasi-periodic oscillations featuring two-dimensional torus doubling bifurcations. Technical Physics Letters, 2005, vol. 31, no. 10, pp. 853–855. DOI: https://doi.org/10.1134/1.2121837
  15. Kuznetsov A. P., Stankevich N. V. Autonomous systems with quasiperiodic dynamics. Examples and their properties: Review. Izvestiya VUZ. Applied Nonlinear Dynamics, 2015, vol. 23, no. 3, pp. 71–93. DOI: https://doi.org/10.18500/0869-6632-2015-23-3-71-93
  16. Broer H., Simó C., Vitolo R. Bifurcations and strange attractors in the Lorenz-84 climate model with seasonal forcing. Nonlinearity, 2002, vol. 15, no. 4, pp. 1205. DOI: https://doi.org/10.1088/0951-7715/15/4/312
  17. Broer H. W., Vitolo R., Simó C. Quasi-periodic Hénonlike attractors in the Lorenz-84 climate model with seasonal forcing. EQUADIFF 2003, World Scientific, 2005, pp. 601–606. DOI: https://doi.org/10.1142/9789812702067_0100
  18. Broer H. W., Simó C., Vitolo R. Chaos and quasi-periodicity in diffeomorphisms of the solid torus. Discrete Contin. Dyn. Syst. Ser. B, 2010, vol. 14, no. 3, pp. 871–905. DOI: https://doi.org/10.3934/dcdsb.2010.14.871
  19. Kuznetsov A. P, Kuznetsov S. P., Shchegoleva N. A., Stankevich N. V. Dynamics of coupled generators of quasiperiodic oscillations: Different types of synchronization and other phenomena. Physica D, 2019, vol. 398, pp. 1–12. DOI: https://doi.org/10.1016/j.physd.2019.05.014
  20. Lorenz E. N. Irregularity: a fundamental property of the atmosphere. Tellus A, 1984, vol. 36, no. 2, pp. 98–110. DOI: https://doi.org/10.1111/j.1600-0870.1984.tb00230.x
  21. Shil’nikov A., Nicolis G., Nicolis C. Bifurcation and predictability analysis of a low-order atmospheric circulation model. International Journal of Bifurcation and Chaos, 1995, vol. 5, no. 6, pp. 1701–1711. DOI: https://doi.org/10.1142/S0218127495001253
  22. Van Veen L. Baroclinic fl ow and the Lorenz-84 model. International Journal of Bifurcation and Chaos, 2003, vol. 13, no. 08, pp. 2117–2139. DOI: https://doi.org/10.1142/S0218127403007904
  23. Freire J. G., Bonatto C., DaCamara C. C., Gallas J. A. C. Multistability, phase diagrams, and intransitivity in the Lorenz-84 low-order atmospheric circulation model. Chaos: An Interdisciplinary Journal of Nonlinear Science, 2008, vol. 18, no. 3, pp. 033121. DOI: https://doi.org/10.1063/1.2953589
  24. Wang H., Yu Y., Wen G. Dynamical analysis of the Lorenz-84 atmospheric circulation model. Journal of Applied Mathematics, 2014, vol. 2014, article ID 296279. 15 p. DOI: https://doi.org/10.1155/2014/296279
  25. Benettin G., Galgani L., Giorgilli A., Strelcyn J. M. Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 1: Theory. Meccanica, 1980, vol. 15, no. 1, pp. 9–20. DOI: https://doi.org/10.1007/BF02128236
  26. Carcasses J. P., Mira C., Bosch M., Simo C., Tatjer J. C. CrossRoad Area–Spring Area. Transition (I) foliated parametric representation. International Journal of Bifurcation and Chaos, 1991, vol. 1, no. 1, pp. 183–196. DOI: https://doi.org/10.1142/S0218127491000117
  27. Mira C., Carcasses J. P., Bosch M., Simo C., Tatjer J. C. CrossRoad Area–Spring Area. Transition (II) foliated parametric representation. International Journal of Bifurcation and Chaos, 1991, vol. 1, no. 2, pp. 339–348. DOI: https://doi.org/10.1142/S0218127491000269
  28. Franceschini V. Bifurcations of tori and phase locking in a dissipative system of differential equations. Physica D: Nonlinear Phenomena, 1983, vol. 6, no. 3, pp. 285–304. DOI: https://doi.org/10.1109/ISCAS.2000.8571960
  29. Kaneko K. Doubling of torus. Progress of Theoretical Physics, 1983, vol. 69, no. 6. pp. 1806–1810. DOI: https://doi.org/10.1143/PTP.69.1806
  30. Kaneko K. Oscillation and doubling of torus. Progress of Theoretical Physics, 1984, vol. 72, no. 2, pp. 202–215. DOI: https://doi.org/10.1143/PTP.72.202
  31. Anishchenko V. S., Vadivasova T. E., Sosnovtseva O. Mechanisms of ergodic torus destruction and appearance of strange nonchaotic attractors. Physical Review E., 1996, vol. 53, no. 5. pp. 4451. DOI: https://doi.org/10.1142/S0218127401002195
  32. Kuznetsov S., Feudel U., Pikovsky A. Renormalization group for scaling at the torus-doubling terminal point. Physical Review E, 1998, vol. 57, no. 2, pp. 1585. DOI: https://doi.org/10.1103/PhysRevE.57.1585
  33. Sekikawa M., Inaba N., Yoshinaga T., Tsubouchi T. Bifurcation structure of successive torus doubling. Physics Letters A, 2006, vol. 348, no. 3–6. pp. 187–194. DOI: https://doi.org/10.1016/j.physleta.2005.08.089
Full text:
(downloads: 12)