For citation:
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. Physics , 2020, vol. 20, iss. 3, pp. 222-232. DOI: 10.18500/1817-3020-2020-20-3-222-232
Cascade of Invariant Curve Doubling Bifurcations and Quasi-Periodic Hénon Attractor in the Discrete Lorenz-84 Model
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.
- 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
- Schuster H. G., Just W. Deterministic chaos: an introduction. 4th ed. John Wiley & Sons, 2006. 312 p.
- 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).
- 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
- 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.
- 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
- 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
- Pikovsky A., Politi A. Lyapunov exponents: a tool to explore complex dynamics. Cambridge University Press, 2016. 295 p.
- 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
- Zaslavsky G. M. The Physics of Chaos in Hamiltonian Systems. World Scientifi c, 2007. DOI: https://doi.org/10.1142/p507
- 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
- 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.
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 1362 reads