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Generalized Lorenz models and their routes to chaos. II. Energy-conserving horizontal mode truncations

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  • Roy, D.
  • Musielak, Z.E.

Abstract

All attempts to generalize the three-dimensional Lorenz model by selecting higher-order Fourier modes can be divided into three categories, namely: vertical, horizontal and vertical–horizontal mode truncations. The previous study showed that the first method allowed only construction of a nine-dimensional system when the selected modes were energy-conserving. The results presented in this paper demonstrate that a five-dimensional model is the lowest-order generalized Lorenz model that can be constructed by the second method and that its route to chaos is the same as that observed in the original Lorenz model. It is shown that the onset of chaos in both systems is determined by a number of modes that describe the vertical temperature difference in a convection roll. In addition, a simple rule that allows selecting modes that conserve energy for each method is derived.

Suggested Citation

  • Roy, D. & Musielak, Z.E., 2007. "Generalized Lorenz models and their routes to chaos. II. Energy-conserving horizontal mode truncations," Chaos, Solitons & Fractals, Elsevier, vol. 31(3), pages 747-756.
    • Handle: RePEc:eee:chsofr:v:31:y:2007:i:3:p:747-756
    DOI: 10.1016/j.chaos.2006.03.082
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    Cited by:

    1. Khodakaram-Tafti, Amin & Emdad, Homayoun & Mahzoon, Mojtaba, 2024. "Periodicity and chaos of thermal convective flows in annular cylindrical domains using the method of isolation by spectral expansions," Chaos, Solitons & Fractals, Elsevier, vol. 179(C).
    2. Duan, Zhisheng & Wang, Jinzhi & Yang, Ying & Huang, Lin, 2009. "Frequency-domain and time-domain methods for feedback nonlinear systems and applications to chaos control," Chaos, Solitons & Fractals, Elsevier, vol. 40(2), pages 848-861.
    3. Reyes, Tiffany & Shen, Bo-Wen, 2019. "A recurrence analysis of chaotic and non-chaotic solutions within a generalized nine-dimensional Lorenz model," Chaos, Solitons & Fractals, Elsevier, vol. 125(C), pages 1-12.
    4. Cui, Jialin & Shen, Bo-Wen, 2021. "A kernel principal component analysis of coexisting attractors within a generalized Lorenz model," Chaos, Solitons & Fractals, Elsevier, vol. 146(C).
    5. Khodakaram-Tafti, Amin & Emdad, Homayoun & Mahzoon, Mojtaba, 2022. "Dynamical and chaotic behaviors of natural convection flow in semi-annular cylindrical domains using energy-conserving low-order spectral models," Applied Mathematics and Computation, Elsevier, vol. 433(C).
    6. Garay, B.M. & Indig, B., 2015. "Chaos in Vallis’ asymmetric Lorenz model for El Niño," Chaos, Solitons & Fractals, Elsevier, vol. 75(C), pages 253-262.

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