Articles | Volume 23, issue 4
https://doi.org/10.5194/npg-23-189-2016
© Author(s) 2016. This work is distributed under
the Creative Commons Attribution 3.0 License.
the Creative Commons Attribution 3.0 License.
https://doi.org/10.5194/npg-23-189-2016
© Author(s) 2016. This work is distributed under
the Creative Commons Attribution 3.0 License.
the Creative Commons Attribution 3.0 License.
Research article
| 
08 Jul 2016
Research article |
| 08 Jul 2016
Hierarchical scale dependence associated with the extension of the nonlinear feedback loop in a seven-dimensional Lorenz model
Bo-Wen Shen
CORRESPONDING AUTHOR
Department of Mathematics and Statistics, San Diego State University, 5500 Campanile Drive, San Diego, CA 92182-7720, USA
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Cited
16 citations as recorded by crossref.
- On periodic solutions in the non-dissipative Lorenz model: the role of the nonlinear feedback loop B. Shen https://doi.org/10.1080/16000870.2018.1471912
- The Dual Nature of Chaos and Order in the Atmosphere B. Shen et al. https://doi.org/10.3390/atmos13111892
- Aggregated Negative Feedback in a Generalized Lorenz Model B. Shen https://doi.org/10.1142/S0218127419500378
- On the Predictability of 30-Day Global Mesoscale Simulations of African Easterly Waves during Summer 2006: A View with the Generalized Lorenz Model B. Shen https://doi.org/10.3390/geosciences9070281
- The 50th Anniversary of the Metaphorical Butterfly Effect since Lorenz (1972): Multistability, Multiscale Predictability, and Sensitivity in Numerical Models B. Shen et al. https://doi.org/10.3390/atmos14081279
- Homoclinic Orbits and Solitary Waves within the Nondissipative Lorenz Model and KdV Equation B. Shen https://doi.org/10.1142/S0218127420502570
- Quasi-Periodic Orbits in the Five-Dimensional Nondissipative Lorenz Model: The Role of the Extended Nonlinear Feedback Loop S. Faghih-Naini & B. Shen https://doi.org/10.1142/S0218127418500724
- Exploring Downscaling in High-Dimensional Lorenz Models Using the Transformer Decoder B. Shen https://doi.org/10.3390/make6040107
- A recurrence analysis of chaotic and non-chaotic solutions within a generalized nine-dimensional Lorenz model T. Reyes & B. Shen https://doi.org/10.1016/j.chaos.2019.05.003
- Dimensionality reduction of chaos by feedbacks and periodic forcing is a source of natural climate change P. Salmon https://doi.org/10.1007/s00382-024-07191-5
- One Saddle Point and Two Types of Sensitivities within the Lorenz 1963 and 1969 Models B. Shen et al. https://doi.org/10.3390/atmos13050753
- A Review of Lorenz’s Models from 1960 to 2008 B. Shen https://doi.org/10.1142/S0218127423300240
- A kernel principal component analysis of coexisting attractors within a generalized Lorenz model J. Cui & B. Shen https://doi.org/10.1016/j.chaos.2021.110865
- A KdV–SIR Equation and Its Analytical Solutions for Solitary Epidemic Waves W. Paxson & B. Shen https://doi.org/10.1142/S0218127422501991
- Multiscale Processes of Hurricane Sandy (2012) as Revealed by the Parallel Ensemble Empirical Mode Decomposition and Advanced Visualization Technology B. Shen et al. https://doi.org/10.1142/S2424922X16500054
- Dynamical analysis and boundedness for a generalized chaotic Lorenz model X. Mao et al. https://doi.org/10.3934/math.20231005
16 citations as recorded by crossref.
- On periodic solutions in the non-dissipative Lorenz model: the role of the nonlinear feedback loop B. Shen https://doi.org/10.1080/16000870.2018.1471912
- The Dual Nature of Chaos and Order in the Atmosphere B. Shen et al. https://doi.org/10.3390/atmos13111892
- Aggregated Negative Feedback in a Generalized Lorenz Model B. Shen https://doi.org/10.1142/S0218127419500378
- On the Predictability of 30-Day Global Mesoscale Simulations of African Easterly Waves during Summer 2006: A View with the Generalized Lorenz Model B. Shen https://doi.org/10.3390/geosciences9070281
- The 50th Anniversary of the Metaphorical Butterfly Effect since Lorenz (1972): Multistability, Multiscale Predictability, and Sensitivity in Numerical Models B. Shen et al. https://doi.org/10.3390/atmos14081279
- Homoclinic Orbits and Solitary Waves within the Nondissipative Lorenz Model and KdV Equation B. Shen https://doi.org/10.1142/S0218127420502570
- Quasi-Periodic Orbits in the Five-Dimensional Nondissipative Lorenz Model: The Role of the Extended Nonlinear Feedback Loop S. Faghih-Naini & B. Shen https://doi.org/10.1142/S0218127418500724
- Exploring Downscaling in High-Dimensional Lorenz Models Using the Transformer Decoder B. Shen https://doi.org/10.3390/make6040107
- A recurrence analysis of chaotic and non-chaotic solutions within a generalized nine-dimensional Lorenz model T. Reyes & B. Shen https://doi.org/10.1016/j.chaos.2019.05.003
- Dimensionality reduction of chaos by feedbacks and periodic forcing is a source of natural climate change P. Salmon https://doi.org/10.1007/s00382-024-07191-5
- One Saddle Point and Two Types of Sensitivities within the Lorenz 1963 and 1969 Models B. Shen et al. https://doi.org/10.3390/atmos13050753
- A Review of Lorenz’s Models from 1960 to 2008 B. Shen https://doi.org/10.1142/S0218127423300240
- A kernel principal component analysis of coexisting attractors within a generalized Lorenz model J. Cui & B. Shen https://doi.org/10.1016/j.chaos.2021.110865
- A KdV–SIR Equation and Its Analytical Solutions for Solitary Epidemic Waves W. Paxson & B. Shen https://doi.org/10.1142/S0218127422501991
- Multiscale Processes of Hurricane Sandy (2012) as Revealed by the Parallel Ensemble Empirical Mode Decomposition and Advanced Visualization Technology B. Shen et al. https://doi.org/10.1142/S2424922X16500054
- Dynamical analysis and boundedness for a generalized chaotic Lorenz model X. Mao et al. https://doi.org/10.3934/math.20231005
Saved (final revised paper)
Latest update: 12 Sep 2026
Short summary
We construct a seven-dimensional Lorenz model (7DLM) to discuss the impact of an extended nonlinear feedback loop on solutions' stability and illustrate the hierarchical scale dependence of chaotic solutions. The 7DLM requires a much larger critical value for the Rayleigh parameter (rc ∼ 116.9) for the onset of chaos. For chaotic solutions with r = 120, high correlation coefficients among the modes at different scales indicate hierarchical scale dependence.
We construct a seven-dimensional Lorenz model (7DLM) to discuss the impact of an extended...
