American Institute of Mathematical Sciences

Advanced
November  2006, 6(6): 1403-1416. doi: 10.3934/dcdsb.2006.6.1403

Revisiting the slow manifold of the Lorenz-Krishnamurthy quintet

M. Phani Sudheer 1, , Ravi S. Nanjundiah 2, and  A. S. Vasudeva Murthy 3,

1. 

Department of Mechanical Engineering, National Institute of Technology Karnataka, Surathkal, India
2. 

Center for Atmospheric and Oceanic Sciences, Indian Institute of Science, Bangalore 560 012, India
3. 

TIFR Centre, Indian Institute of Science, Bangalore 560 012, India

Received  August 2005 Revised  March 2006 Published  August 2006

The slow-manifold for the Lorenz-Krishnamurthy model has been studied. By minimizing the evolution rate we find that the analytical functions for the fast variables are devoid of high frequency oscillations. However upon solving this model with initial values of the fast variables obtained from the analytical functions, the LK model exhibits high frequency oscillations. Upon using the time derivatives of the analytic functions for computing the evolution of fast variables, we find a slow-manifold in the neighbourhood of the LK model.
    Minimization of evolution rate does not guarantee the invariance of the manifold. Using a locally linear approximate reduction scheme, the invariance can be maintained. However, the solutions so obtained do develop high frequency oscillations. The onset of these high frequency oscillations is delayed vis-a-vis other previous studies. These methods have potential to be used in improving the predictions of weather systems.
Keywords: predictability., minimization of evolution rate, invariant slow-manifold.
Mathematics Subject Classification: Primary: 37D10; Secondary: 35B4.
Citation: M. Phani Sudheer, Ravi S. Nanjundiah, A. S. Vasudeva Murthy. Revisiting the slow manifold of the Lorenz-Krishnamurthy quintet. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1403-1416. doi: 10.3934/dcdsb.2006.6.1403
[1]

Parker Childs, James P. Keener. Slow manifold reduction of a stochastic chemical reaction: Exploring Keizer's paradox. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1775-1794. doi: 10.3934/dcdsb.2012.17.1775
[2]

George Osipenko. Linearization near a locally nonunique invariant manifold. Discrete & Continuous Dynamical Systems, 1997, 3 (2) : 189-205. doi: 10.3934/dcds.1997.3.189
[3]

Ronny Bergmann, Raymond H. Chan, Ralf Hielscher, Johannes Persch, Gabriele Steidl. Restoration of manifold-valued images by half-quadratic minimization. Inverse Problems & Imaging, 2016, 10 (2) : 281-304. doi: 10.3934/ipi.2016001
[4]

Riccarda Rossi, Ulisse Stefanelli, Marita Thomas. Rate-independent evolution of sets. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 89-119. doi: 10.3934/dcdss.2020304
[5]

Antonios Zagaris, Christophe Vandekerckhove, C. William Gear, Tasso J. Kaper, Ioannis G. Kevrekidis. Stability and stabilization of the constrained runs schemes for equation-free projection to a slow manifold. Discrete & Continuous Dynamical Systems, 2012, 32 (8) : 2759-2803. doi: 10.3934/dcds.2012.32.2759
[6]

M. Montaz Ali. A recursive topographical differential evolution algorithm for potential energy minimization. Journal of Industrial & Management Optimization, 2010, 6 (1) : 29-46. doi: 10.3934/jimo.2010.6.29
[7]

J. B. van den Berg, J. D. Mireles James. Parameterization of slow-stable manifolds and their invariant vector bundles: Theory and numerical implementation. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4637-4664. doi: 10.3934/dcds.2016002
[8]

Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166
[9]

Amina Amassad, Mircea Sofonea. Analysis of some nonlinear evolution systems arising in rate-type viscoplasticity. Conference Publications, 1998, 1998 (Special) : 58-71. doi: 10.3934/proc.1998.1998.58
[10]

Ulisse Stefanelli, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of a rate-independent evolution equation via viscous regularization. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1467-1485. doi: 10.3934/dcdss.2017076
[11]

Jiying Liu, Jubo Zhu, Fengxia Yan, Zenghui Zhang. Compressive sampling and $l_1$ minimization for SAR imaging with low sampling rate. Inverse Problems & Imaging, 2013, 7 (4) : 1295-1305. doi: 10.3934/ipi.2013.7.1295
[12]

Ziang Long, Penghang Yin, Jack Xin. Global convergence and geometric characterization of slow to fast weight evolution in neural network training for classifying linearly non-separable data. Inverse Problems & Imaging, 2021, 15 (1) : 41-62. doi: 10.3934/ipi.2020077
[13]

E. Camouzis, H. Kollias, I. Leventides. Stable manifold market sequences. Journal of Dynamics & Games, 2018, 5 (2) : 165-185. doi: 10.3934/jdg.2018010
[14]

Camillo De Lellis, Emanuele Spadaro. Center manifold: A case study. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1249-1272. doi: 10.3934/dcds.2011.31.1249
[15]

Zhiguo Feng, Ka-Fai Cedric Yiu. Manifold relaxations for integer programming. Journal of Industrial & Management Optimization, 2014, 10 (2) : 557-566. doi: 10.3934/jimo.2014.10.557
[16]

Weinan E, Weiguo Gao. Orbital minimization with localization. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 249-264. doi: 10.3934/dcds.2009.23.249
[17]

Meijuan Shang, Yanan Liu, Lingchen Kong, Xianchao Xiu, Ying Yang. Nonconvex mixed matrix minimization. Mathematical Foundations of Computing, 2019, 2 (2) : 107-126. doi: 10.3934/mfc.2019009
[18]

Justin Holmer, Maciej Zworski. Slow soliton interaction with delta impurities. Journal of Modern Dynamics, 2007, 1 (4) : 689-718. doi: 10.3934/jmd.2007.1.689
[19]

Debora Amadori, Wen Shen. Front tracking approximations for slow erosion. Discrete & Continuous Dynamical Systems, 2012, 32 (5) : 1481-1502. doi: 10.3934/dcds.2012.32.1481
[20]

Franz W. Kamber and Peter W. Michor. The flow completion of a manifold with vector field. Electronic Research Announcements, 2000, 6: 95-97.

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences