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Qualitative analysis of a simple tumor-immune system with time delay of tumor action
On the Lorenz '96 model and some generalizations
Georgetown University, Washington, DC 20057, USA |
In 1996, Edward Lorenz introduced a system of ordinary differential equations that describes a scalar quantity evolving on a circular array of sites, undergoing forcing, dissipation, and rotation invariant advection. Lorenz constructed the system as a test problem for numerical weather prediction. Since then, the system has also found use as a test case in data assimilation. Mathematically, this is a dynamical system with a single bifurcation parameter (rescaled forcing) that undergoes multiple bifurcations and exhibits chaotic behavior for large forcing. In this paper, the main characteristics of the advection term in the model are identified and used to describe and classify possible generalizations of the system. A graphical method to study the bifurcation behavior of constant solutions is introduced, and it is shown how to use the rotation invariance to compute normal forms of the system analytically. Problems with site-dependent forcing, dissipation, or advection are considered and basic existence and stability results are proved for these extensions. We address some related topics in the appendices, wherein the Lorenz '96 system in Fourier space is considered, explicit solutions for some advection-only systems are found, and it is demonstrated how to use advection-only systems to assess numerical schemes.
References:
[1] |
R. V. Abramov and A. J. Majda,
Quantifying uncertainty for non-Gaussian ensembles in complex systems, SIAM Journal on Scientific Computing, 26 (2004), 411-447.
doi: 10.1137/S1064827503426310. |
[2] |
R. V. Abramov and A. J. Majda,
Blended response algorithms for linear fluctuation-dissipation for complex nonlinear dynamical systems, Nonlinearity, 20 (2007), 2793-2821.
doi: 10.1088/0951-7715/20/12/004. |
[3] |
R. V. Abramov and A. J. Majda,
New approximations and tests of linear fluctuation-response for chaotic nonlinear forced-dissipative dynamical systems, Journal of Nonlinear Science, 18 (2008), 303-341.
doi: 10.1007/s00332-007-9011-9. |
[4] |
E. L. Allgower and K. Georg, Numerical Continuation Methods: An Introduction, Springer Series in Computational Mathematics, 13. Springer-Verlag, Berlin, 1990.
doi: 10.1007/978-3-642-61257-2. |
[5] |
R. Blender, J. Wouters and V. Lucarini, Avalanches, breathers, and flow reversal in a continuous Lorenz-96 model, Physical Review E, 88 (2013), 013201, 5pp.
doi: 10.1103/PhysRevE.88.013201. |
[6] |
J. G. Charney and J. G. DeVore,
Multiple flow equilibria in the atmosphere and blocking, Journal of the Atmospheric Sciences, 36 (1979), 1205-1216.
doi: 10.1175/1520-0469(1979)036<1205:MFEITA>2.0.CO;2. |
[7] |
P. Chossat and R. Lauterbach, Methods in Equivariant Bifurcations and Dynamical Systems, World Scientific Publishing Co., Singapore, 2000.
doi: 10.1142/4062. |
[8] |
J. A. Dutton,
The nonlinear quasi-geostrophic equation. Part Ⅱ: Predictability, recurrence and limit properties of thermally-forced and unforced flows, Journal of the Atmospheric Sciences, 33 (1976), 1431-1453.
doi: 10.1175/1520-0469(1976)033<1431:TNQGEP>2.0.CO;2. |
[9] |
M. R. Frank, L. Mitchell, P. S. Dodds and C. M. Danforth, Standing swells surveyed showing surprisingly stable solutions for the Lorenz'96 model, International Journal of Bifurcation and Chaos, 24 (2014), 1430027, 14pp.
doi: 10.1142/S0218127414300274. |
[10] |
G. Gallavotti and V. Lucarini,
Equivalence of non-equilibrium ensembles and representation of friction in turbulent flows: the Lorenz 96 model, Journal of Statistical Physics, 156 (2014), 1027-1065.
doi: 10.1007/s10955-014-1051-6. |
[11] |
S. J. Jacobs,
A note on multiple flow equilibria, Pure and Applied Geophysics, 130 (1989), 743-749.
doi: 10.1007/BF00881609. |
[12] |
D. L. van Kekem and A. E. Sterk, Wave propagation in the Lorenz-96 model, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 29 (2019), 1950008, 18 pp.
doi: 10.1142/S0218127419500081. |
[13] |
D. L. van Kekem and A. E. Sterk,
Travelling waves and their bifurcations in the Lorenz-96 model, Physica D: Nonlinear Phenomena, 367 (2018), 38-60.
doi: 10.1016/j.physd.2017.11.008. |
[14] |
D. L. van Kekem and A. E. Sterk, Symmetries in the Lorenz-96 model, International Journal of Bifurcation and Chaos, 29 (2019), 195008, 18pp.
doi: 10.1142/S0218127419500081. |
[15] |
Y. A. Kuznetsov,
Numerical normalization techniques for all codim 2 bifurcations of equilibria in ODE's, SIAM Journal on Numerical Analysis, 36 (1999), 1104-1124.
doi: 10.1137/S0036142998335005. |
[16] |
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Second edition. Applied Mathematical Sciences, 112. Springer-Verlag, New York, 1998. |
[17] |
E. N. Lorenz,
Deterministic nonperiodic flow, Journal of the Atmospheric Sciences, 20 (1963), 130-141.
doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2. |
[18] |
E. N. Lorenz, Predictability: A problem partly solved,, in Predictability of Weather and Climate (eds. Tim Palmer and Renate Hagedorn), Cambridge University Press, (2006), 40–58.
doi: 10.1017/CBO9780511617652.004. |
[19] |
E. N. Lorenz and K. A. Emanuel,
Optimal sites for supplementary weather observations: Simulation with a small model, Journal of the Atmospheric Sciences, 55 (1998), 399-414.
doi: 10.1175/1520-0469(1998)055<0399:OSFSWO>2.0.CO;2. |
[20] |
E. N. Lorenz,
Designing chaotic models, Journal of the Atmospheric Sciences, 62 (2005), 1574-1587.
doi: 10.1175/JAS3430.1. |
[21] |
V. Lucarini and S. Sarno,
A statistical mechanical approach for the computation of the climatic response to general forcings, Nonlinear Processes in Geophysics, 18 (2011), 7-28.
doi: 10.5194/npg-18-7-2011. |
[22] |
S. A. Orszag and J. B. McLaughlin,
Evidence that random behavior is generic for nonlinear differential equations, Physica D: Nonlinear Phenomena, 1 (1980), 68-79.
doi: 10.1016/0167-2789(80)90005-6. |
[23] |
L. R. Petzold and A. C. Hindmarsh, Lsoda, Computing and Mathematics Research Division, Lawrence Livermore National Laboratory, Livermore, CA, 1997. Google Scholar |
[24] |
K. Soetaert, T. Petzoldt and R. W. Setzer,
Solving differential equations in R: Package deSolve, Journal of Statistical Software, 33 (2010), 1-25.
doi: 10.32614/RJ-2010-013. |
[25] |
A. E. Sterk and D. L. van Kekem, Predictability of extreme waves in the Lorenz-96 model near intermittency and quasi-periodicity, Complexity, 2017 (2017), Art. ID 9419024, 14 pp.
doi: 10.1155/2017/9419024. |
[26] |
D. S. Wilks,
Effects of stochastic parametrizations in the Lorenz'96 system, Quarterly Journal of the Royal Meteorological Society: A Journal of the Atmospheric Sciences, Applied Meteorology and Physical Oceanography, 131 (2005), 389-407.
doi: 10.1256/qj.04.03. |
show all references
References:
[1] |
R. V. Abramov and A. J. Majda,
Quantifying uncertainty for non-Gaussian ensembles in complex systems, SIAM Journal on Scientific Computing, 26 (2004), 411-447.
doi: 10.1137/S1064827503426310. |
[2] |
R. V. Abramov and A. J. Majda,
Blended response algorithms for linear fluctuation-dissipation for complex nonlinear dynamical systems, Nonlinearity, 20 (2007), 2793-2821.
doi: 10.1088/0951-7715/20/12/004. |
[3] |
R. V. Abramov and A. J. Majda,
New approximations and tests of linear fluctuation-response for chaotic nonlinear forced-dissipative dynamical systems, Journal of Nonlinear Science, 18 (2008), 303-341.
doi: 10.1007/s00332-007-9011-9. |
[4] |
E. L. Allgower and K. Georg, Numerical Continuation Methods: An Introduction, Springer Series in Computational Mathematics, 13. Springer-Verlag, Berlin, 1990.
doi: 10.1007/978-3-642-61257-2. |
[5] |
R. Blender, J. Wouters and V. Lucarini, Avalanches, breathers, and flow reversal in a continuous Lorenz-96 model, Physical Review E, 88 (2013), 013201, 5pp.
doi: 10.1103/PhysRevE.88.013201. |
[6] |
J. G. Charney and J. G. DeVore,
Multiple flow equilibria in the atmosphere and blocking, Journal of the Atmospheric Sciences, 36 (1979), 1205-1216.
doi: 10.1175/1520-0469(1979)036<1205:MFEITA>2.0.CO;2. |
[7] |
P. Chossat and R. Lauterbach, Methods in Equivariant Bifurcations and Dynamical Systems, World Scientific Publishing Co., Singapore, 2000.
doi: 10.1142/4062. |
[8] |
J. A. Dutton,
The nonlinear quasi-geostrophic equation. Part Ⅱ: Predictability, recurrence and limit properties of thermally-forced and unforced flows, Journal of the Atmospheric Sciences, 33 (1976), 1431-1453.
doi: 10.1175/1520-0469(1976)033<1431:TNQGEP>2.0.CO;2. |
[9] |
M. R. Frank, L. Mitchell, P. S. Dodds and C. M. Danforth, Standing swells surveyed showing surprisingly stable solutions for the Lorenz'96 model, International Journal of Bifurcation and Chaos, 24 (2014), 1430027, 14pp.
doi: 10.1142/S0218127414300274. |
[10] |
G. Gallavotti and V. Lucarini,
Equivalence of non-equilibrium ensembles and representation of friction in turbulent flows: the Lorenz 96 model, Journal of Statistical Physics, 156 (2014), 1027-1065.
doi: 10.1007/s10955-014-1051-6. |
[11] |
S. J. Jacobs,
A note on multiple flow equilibria, Pure and Applied Geophysics, 130 (1989), 743-749.
doi: 10.1007/BF00881609. |
[12] |
D. L. van Kekem and A. E. Sterk, Wave propagation in the Lorenz-96 model, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 29 (2019), 1950008, 18 pp.
doi: 10.1142/S0218127419500081. |
[13] |
D. L. van Kekem and A. E. Sterk,
Travelling waves and their bifurcations in the Lorenz-96 model, Physica D: Nonlinear Phenomena, 367 (2018), 38-60.
doi: 10.1016/j.physd.2017.11.008. |
[14] |
D. L. van Kekem and A. E. Sterk, Symmetries in the Lorenz-96 model, International Journal of Bifurcation and Chaos, 29 (2019), 195008, 18pp.
doi: 10.1142/S0218127419500081. |
[15] |
Y. A. Kuznetsov,
Numerical normalization techniques for all codim 2 bifurcations of equilibria in ODE's, SIAM Journal on Numerical Analysis, 36 (1999), 1104-1124.
doi: 10.1137/S0036142998335005. |
[16] |
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Second edition. Applied Mathematical Sciences, 112. Springer-Verlag, New York, 1998. |
[17] |
E. N. Lorenz,
Deterministic nonperiodic flow, Journal of the Atmospheric Sciences, 20 (1963), 130-141.
doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2. |
[18] |
E. N. Lorenz, Predictability: A problem partly solved,, in Predictability of Weather and Climate (eds. Tim Palmer and Renate Hagedorn), Cambridge University Press, (2006), 40–58.
doi: 10.1017/CBO9780511617652.004. |
[19] |
E. N. Lorenz and K. A. Emanuel,
Optimal sites for supplementary weather observations: Simulation with a small model, Journal of the Atmospheric Sciences, 55 (1998), 399-414.
doi: 10.1175/1520-0469(1998)055<0399:OSFSWO>2.0.CO;2. |
[20] |
E. N. Lorenz,
Designing chaotic models, Journal of the Atmospheric Sciences, 62 (2005), 1574-1587.
doi: 10.1175/JAS3430.1. |
[21] |
V. Lucarini and S. Sarno,
A statistical mechanical approach for the computation of the climatic response to general forcings, Nonlinear Processes in Geophysics, 18 (2011), 7-28.
doi: 10.5194/npg-18-7-2011. |
[22] |
S. A. Orszag and J. B. McLaughlin,
Evidence that random behavior is generic for nonlinear differential equations, Physica D: Nonlinear Phenomena, 1 (1980), 68-79.
doi: 10.1016/0167-2789(80)90005-6. |
[23] |
L. R. Petzold and A. C. Hindmarsh, Lsoda, Computing and Mathematics Research Division, Lawrence Livermore National Laboratory, Livermore, CA, 1997. Google Scholar |
[24] |
K. Soetaert, T. Petzoldt and R. W. Setzer,
Solving differential equations in R: Package deSolve, Journal of Statistical Software, 33 (2010), 1-25.
doi: 10.32614/RJ-2010-013. |
[25] |
A. E. Sterk and D. L. van Kekem, Predictability of extreme waves in the Lorenz-96 model near intermittency and quasi-periodicity, Complexity, 2017 (2017), Art. ID 9419024, 14 pp.
doi: 10.1155/2017/9419024. |
[26] |
D. S. Wilks,
Effects of stochastic parametrizations in the Lorenz'96 system, Quarterly Journal of the Royal Meteorological Society: A Journal of the Atmospheric Sciences, Applied Meteorology and Physical Oceanography, 131 (2005), 389-407.
doi: 10.1256/qj.04.03. |









Laurent polynomial |
Shape of |
|||
ellipse | none | pitchfork/Hopf | ||
ellipse | none | pitchfork/Hopf |
||
trefoil | Hopf | pitchfork/Hopf | ||
ellipse | none | pitchfork/Hopf |
||
butterfly | Hopf | Hopf | ||
kidney | Hopf | pitchfork/Hopf | ||
vertical line | none | none | ||
bee | pitchfork/Hopf | Hopf |
Laurent polynomial |
Shape of |
|||
ellipse | none | pitchfork/Hopf | ||
ellipse | none | pitchfork/Hopf |
||
trefoil | Hopf | pitchfork/Hopf | ||
ellipse | none | pitchfork/Hopf |
||
butterfly | Hopf | Hopf | ||
kidney | Hopf | pitchfork/Hopf | ||
vertical line | none | none | ||
bee | pitchfork/Hopf | Hopf |
12 | 1 | 4 | 1 | 6 | 1 | 1 | |
14 | .8901 | 7 | 1.1820 | 14 | 1.5206 | not observed | not observed |
18 | .8982 | 9 | 1 | 6 | 1.1892 | not observed | not observed |
22 | .9076 | 22 | .9343 | 11 | .9915 | .996 | |
28 | .8901 | 14 | .9457 | 28 | 1.0293 | 1.072 | |
36 | .8982 | 9 | .9025 | 36 | .9094 | .904 |
12 | 1 | 4 | 1 | 6 | 1 | 1 | |
14 | .8901 | 7 | 1.1820 | 14 | 1.5206 | not observed | not observed |
18 | .8982 | 9 | 1 | 6 | 1.1892 | not observed | not observed |
22 | .9076 | 22 | .9343 | 11 | .9915 | .996 | |
28 | .8901 | 14 | .9457 | 28 | 1.0293 | 1.072 | |
36 | .8982 | 9 | .9025 | 36 | .9094 | .904 |
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