Advertisement

Coexistence of Chaotic and Non-chaotic Orbits in a New Nine-Dimensional Lorenz Model

  • B.-W.  ShenEmail author
  • T. Reyes
  • S. Faghih-Naini
Conference paper
  • 224 Downloads
Part of the Springer Proceedings in Complexity book series (SPCOM)

Abstract

In this study, we present a new nine-dimensional Lorenz model (9DLM) that requires a larger critical value for the Rayleigh parameter (a rc of 679.8) for the onset of chaos, as compared to a rc of 24.74 for the 3DLM, a rc of 42.9 for the 5DLM, and a rc 116.9 for the 7DLM. Major features within the 9DLM include: (1) the coexistence of chaotic and non-chaotic orbits with moderate Rayleigh parameters, and (2) the coexistence of limit cycle/torus orbits and spiral sinks with large Rayleigh parameters. Version 2 of the 9DLM, referred to as the 9DLM-V2, is derived to show that: (i) based on a linear stability analysis, two non-trivial critical points are stable for all Rayleigh parameters greater than one; (ii) under non-dissipative and linear conditions, the extended nonlinear feedback loop produces four incommensurate frequencies; and (iii) for a stable orbit, small deviations away from equilibrium (e.g., the stable critical point) do not have a significant impact on orbital stability. Based on our results, we suggest that the entirety of weather is a superset that consists of both chaotic and non-chaotic processes.

Keywords

Lorenz model Limit cycle Nonlinear feedback loop Coexistence Incommensurate frequencies Aggregated negative feedback 

Notes

Acknowledgements

We are grateful for support from the College of Science at San Diego State University.

References

  1. 1.
    R. Anthes, Turning the tables on chaos: is the atmosphere more predictable than we assume?, UCAR Magazine, spring/summer, available at: https://news.ucar.edu/4505/turning-tables-chaos-atmosphere-more-predictable-we-assume (Last access: 2 April 2019), 2011
  2. 2.
    J.H. Curry, J.R. Herring, J. Loncaric, S.A. Orszag, Order and disorder in two- and three-dimensional Benard convection. J. Fluid. Mech. 147, 1–38 (1984)ADSCrossRefGoogle Scholar
  3. 3.
    S. Faghih-Naini, B.-W. Shen, Quasi-periodic in the five-dimensional non-dissipative lorenz model: the role of the extended nonlinear feedback loop. Int. J. Bifurc. Chaos 28(6), 1850072 (20 pages) (2018). https://doi.org/10.1142/S0218127418500724
  4. 4.
    J. Gleick, Chaos: Making a New Science (Penguin, New York, 1987), p. 360zbMATHGoogle Scholar
  5. 5.
    J. Guckenheimer, R.F. Williams, Structural stability of lorenz attractors. Publ. Math. IHES. 50, 59 (1979)Google Scholar
  6. 6.
    E.N. Lorenz, Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141 (1963)Google Scholar
  7. 7.
    E.N. Lorenz, The predictability of a flow which possesses many scales of motion. Tellus, 21, 289–307 (1969)Google Scholar
  8. 8.
    E.N. Lorenz, Predictability: does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?, in American Association for the Advancement of Science, 139th Meeting, 29 December 1972, Boston, Mass., AAAS Section on Environmental Sciences, New Approaches to Global Weather, GARP. Available at: http://eaps4.mit.edu/research/Lorenz/Butterfly_1972.pdf, Last access: 14 Dec 2015 (1972)
  9. 9.
    D. Roy and Z.E. Musielak.: Generalized Lorenz models and their routes to chaos, I. Energy-conserving vertical mode truncations, Chaos Soliton. Fract., 32, 1038–1052 (2007)Google Scholar
  10. 10.
    B.-W. Shen, Nonlinear feedback in a five-dimensional Lorenz model. J. Atmos. Sci. 71, 1701–1723 (2014).  https://doi.org/10.1175/JAS-D-13-0223.1ADSCrossRefGoogle Scholar
  11. 11.
    B.-W. Shen, Nonlinear feedback in a six dimensional Lorenz model: impact of an additional heating term. Nonlin. Processes Geophys. 22, 749–764 (2015).  https://doi.org/10.5194/npg-22-749-2015ADSCrossRefGoogle Scholar
  12. 12.
    B.-W. Shen, Hierarchical scale dependence associated with the extension of the nonlinear feedback loop in a seven-dimensional Lorenz model. Nonlin. Processes Geophys. 23, 189–203 (2016).  https://doi.org/10.5194/npg-23-189-2016ADSCrossRefGoogle Scholar
  13. 13.
    B.-W. Shen, On an extension of the nonlinear feedback loop in a nine-dimensional Lorenz model. Chaotic Model. Simul. (CMSIM) 2, 147–157 (2017)Google Scholar
  14. 14.
    B.-W. Shen, On periodic solutions in the non-dissipative lorenz model: the role of the nonlinear feedback loop. Tellus A 70, 1471912 (2018).  https://doi.org/10.1080/16000870.2018.1471912
  15. 15.
    B.-W. Shen, Aggregated Negative Feedback in a Generalized Lorenz Model. Int. J. Bifurc. Chaos 29(3), 1950037, (20 pages) (2019).  https://doi.org/10.1142/S0218127419500378
  16. 16.
    B.-W. Shen, R.A. Pielke Sr., X. Zeng, S. Faghih-Naini, C.-L. Shie, R. Atlas, J.-J. Baik, T. A. L. Reyes: Butterfly effects of the first and second kinds: new insights revealed by high-dimensional lorenz models, in The 11th Chaos International Conference (CHAOS2018), Rome, Italy, June 5–8, 2018Google Scholar
  17. 17.
    B.-W. Shen, S. Faghih-Naini, On recurrent solutions within high-dimensional non-dissipative Lorenz models: the role of the nonlinear feedback loop, in The 10th Chaos Modeling and Simulation International Conference (CHAOS2017), Barcelona, Spain, 30 May–2 June, 2017Google Scholar
  18. 18.
    C. Sparrow, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors (Springer, New York. Appl. Math. Sci., 41, 1982)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsSan Diego State UniversitySan DiegoUSA

Personalised recommendations