Articles | Volume 22, issue 6
https://doi.org/10.5194/npg-22-749-2015
© Author(s) 2015. 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-22-749-2015
© Author(s) 2015. This work is distributed under
the Creative Commons Attribution 3.0 License.
the Creative Commons Attribution 3.0 License.
Research article
| 
21 Dec 2015
Research article |
| 21 Dec 2015
Nonlinear feedback in a six-dimensional Lorenz model: impact of an additional heating term
B.-W. 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
31 citations as recorded by crossref.
- 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
- 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
- On the dynamics of a high-order Lorenz–Stenflo system P. Rech https://doi.org/10.1088/0031-8949/91/12/125201
- 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
- The Dual Nature of Chaos and Order in the Atmosphere B. Shen et al. https://doi.org/10.3390/atmos13111892
- Periodicity and Chaos of High-Order Lorenz Systems S. Moon et al. https://doi.org/10.1142/S0218127417501760
- Hierarchical scale dependence associated with the extension of the nonlinear feedback loop in a seven-dimensional Lorenz model B. Shen https://doi.org/10.5194/npg-23-189-2016
- Ultimate Boundedness and Finite Time Stability for a High Dimensional Fractional-Order Lorenz Model M. Huang et al. https://doi.org/10.3390/fractalfract6110630
- Study of Rayleigh–Bénard convection in a chemically reactive fluid using a generalized Lorenz model and the cubic–quintic Ginzburg–Landau equation C. Kanchana et al. https://doi.org/10.1063/5.0081060
- 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
- Aggregated Negative Feedback in a Generalized Lorenz Model B. Shen https://doi.org/10.1142/S0218127419500378
- Dynamical analysis and boundedness for a generalized chaotic Lorenz model X. Mao et al. https://doi.org/10.3934/math.20231005
- An Evaluation of the Parallel Ensemble Empirical Mode Decomposition Method in Revealing the Role of Downscaling Processes Associated with African Easterly Waves in Tropical Cyclone Genesis Y. Wu & B. Shen https://doi.org/10.1175/JTECH-D-15-0257.1
- Dynamical and chaotic behaviors of natural convection flow in semi-annular cylindrical domains using energy-conserving low-order spectral models A. Khodakaram-Tafti et al. https://doi.org/10.1016/j.amc.2022.127415
- Using the (3N)-Dimensional Generalized Lorenz Systems as a Testbed for Data Assimilation: The Ensemble Kalman Filter S. Moon & J. Baik https://doi.org/10.1175/MWR-D-21-0078.1
- High-dimensional generalizations of the Lorenz system and implications for predictability S. Moon et al. https://doi.org/10.1088/1402-4896/ab9d3e
- Dynamics of a High-Order Generalized Lorenz Model for Magnetoconvection N. Pati & P. Rech https://doi.org/10.1142/S0218127420501874
- 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
- Chaos synchronization in generalized Lorenz systems and an application to image encryption S. Moon et al. https://doi.org/10.1016/j.cnsns.2021.105708
- Stability and periodicity of high-order Lorenz–Stenflo equations J. Park et al. https://doi.org/10.1088/0031-8949/91/6/065202
- 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
- Homoclinic Orbits and Solitary Waves within the Nondissipative Lorenz Model and KdV Equation B. Shen https://doi.org/10.1142/S0218127420502570
- Frenetic steering: Nonequilibrium-enabled navigation B. Lefebvre & C. Maes https://doi.org/10.1063/5.0177223
- Time Averages and Periodic Attractors at High Rayleigh Number for Lorenz-like Models I. Ovsyannikov et al. https://doi.org/10.1007/s00332-023-09933-x
- Periodicity and chaos of thermal convective flows in annular cylindrical domains using the method of isolation by spectral expansions A. Khodakaram-Tafti et al. https://doi.org/10.1016/j.chaos.2023.114383
- On the dynamics of five- and six-dimensional Lorenz models C. Felicio & P. Rech https://doi.org/10.1088/2399-6528/aaa955
- 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
- Period-Bubbling Transition to Chaos in Thermo-Viscoelastic Fluid Systems G. Layek & N. Pati https://doi.org/10.1142/S021812742030013X
- 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
- A KdV–SIR Equation and Its Analytical Solutions for Solitary Epidemic Waves W. Paxson & B. Shen https://doi.org/10.1142/S0218127422501991
- A Review of Lorenz’s Models from 1960 to 2008 B. Shen https://doi.org/10.1142/S0218127423300240
31 citations as recorded by crossref.
- 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
- 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
- On the dynamics of a high-order Lorenz–Stenflo system P. Rech https://doi.org/10.1088/0031-8949/91/12/125201
- 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
- The Dual Nature of Chaos and Order in the Atmosphere B. Shen et al. https://doi.org/10.3390/atmos13111892
- Periodicity and Chaos of High-Order Lorenz Systems S. Moon et al. https://doi.org/10.1142/S0218127417501760
- Hierarchical scale dependence associated with the extension of the nonlinear feedback loop in a seven-dimensional Lorenz model B. Shen https://doi.org/10.5194/npg-23-189-2016
- Ultimate Boundedness and Finite Time Stability for a High Dimensional Fractional-Order Lorenz Model M. Huang et al. https://doi.org/10.3390/fractalfract6110630
- Study of Rayleigh–Bénard convection in a chemically reactive fluid using a generalized Lorenz model and the cubic–quintic Ginzburg–Landau equation C. Kanchana et al. https://doi.org/10.1063/5.0081060
- 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
- Aggregated Negative Feedback in a Generalized Lorenz Model B. Shen https://doi.org/10.1142/S0218127419500378
- Dynamical analysis and boundedness for a generalized chaotic Lorenz model X. Mao et al. https://doi.org/10.3934/math.20231005
- An Evaluation of the Parallel Ensemble Empirical Mode Decomposition Method in Revealing the Role of Downscaling Processes Associated with African Easterly Waves in Tropical Cyclone Genesis Y. Wu & B. Shen https://doi.org/10.1175/JTECH-D-15-0257.1
- Dynamical and chaotic behaviors of natural convection flow in semi-annular cylindrical domains using energy-conserving low-order spectral models A. Khodakaram-Tafti et al. https://doi.org/10.1016/j.amc.2022.127415
- Using the (3N)-Dimensional Generalized Lorenz Systems as a Testbed for Data Assimilation: The Ensemble Kalman Filter S. Moon & J. Baik https://doi.org/10.1175/MWR-D-21-0078.1
- High-dimensional generalizations of the Lorenz system and implications for predictability S. Moon et al. https://doi.org/10.1088/1402-4896/ab9d3e
- Dynamics of a High-Order Generalized Lorenz Model for Magnetoconvection N. Pati & P. Rech https://doi.org/10.1142/S0218127420501874
- 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
- Chaos synchronization in generalized Lorenz systems and an application to image encryption S. Moon et al. https://doi.org/10.1016/j.cnsns.2021.105708
- Stability and periodicity of high-order Lorenz–Stenflo equations J. Park et al. https://doi.org/10.1088/0031-8949/91/6/065202
- 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
- Homoclinic Orbits and Solitary Waves within the Nondissipative Lorenz Model and KdV Equation B. Shen https://doi.org/10.1142/S0218127420502570
- Frenetic steering: Nonequilibrium-enabled navigation B. Lefebvre & C. Maes https://doi.org/10.1063/5.0177223
- Time Averages and Periodic Attractors at High Rayleigh Number for Lorenz-like Models I. Ovsyannikov et al. https://doi.org/10.1007/s00332-023-09933-x
- Periodicity and chaos of thermal convective flows in annular cylindrical domains using the method of isolation by spectral expansions A. Khodakaram-Tafti et al. https://doi.org/10.1016/j.chaos.2023.114383
- On the dynamics of five- and six-dimensional Lorenz models C. Felicio & P. Rech https://doi.org/10.1088/2399-6528/aaa955
- 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
- Period-Bubbling Transition to Chaos in Thermo-Viscoelastic Fluid Systems G. Layek & N. Pati https://doi.org/10.1142/S021812742030013X
- 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
- A KdV–SIR Equation and Its Analytical Solutions for Solitary Epidemic Waves W. Paxson & B. Shen https://doi.org/10.1142/S0218127422501991
- A Review of Lorenz’s Models from 1960 to 2008 B. Shen https://doi.org/10.1142/S0218127423300240
Saved (final revised paper)
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Latest update: 24 Jun 2026
Short summary
While the negative nonlinear feedback associated with two new modes in the 5DLM can stabilize solutions, additional resolved heating processes by a third mode in the 6DLM can destabilize solutions. The findings support the view of Lorenz (1972) on the role of small-scale processes: if the flap of a butterfly’s wings can be instrumental in generating a tornado, it can equally well be instrumental in preventing a tornado.
While the negative nonlinear feedback associated with two new modes in the 5DLM can stabilize...
