American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdsb.2021064

On the Lorenz '96 model and some generalizations

John Kerin and  Hans Engler ,

Georgetown University, Washington, DC 20057, USA

* Corresponding author: Hans Engler

Received  November 2020 Revised  December 2020 Early access  February 2021

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.

Keywords: Lorenz-96 model, equivariant dynamical system, Hopf bifurcation, normal form, Neimark-Sacker bifurcation.
Mathematics Subject Classification: Primary: 34C23, 86A10; Secondary: 37C81, 37G05.
Citation: John Kerin, Hans Engler. On the Lorenz '96 model and some generalizations. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021064
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

P. Chossat and R. Lauterbach, Methods in Equivariant Bifurcations and Dynamical Systems, World Scientific Publishing Co., Singapore, 2000. doi: 10.1142/4062.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

S. J. Jacobs, A note on multiple flow equilibria, Pure and Applied Geophysics, 130 (1989), 743-749.  doi: 10.1007/BF00881609.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Second edition. Applied Mathematical Sciences, 112. Springer-Verlag, New York, 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

E. N. Lorenz, Designing chaotic models, Journal of the Atmospheric Sciences, 62 (2005), 1574-1587.  doi: 10.1175/JAS3430.1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

P. Chossat and R. Lauterbach, Methods in Equivariant Bifurcations and Dynamical Systems, World Scientific Publishing Co., Singapore, 2000. doi: 10.1142/4062.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

S. J. Jacobs, A note on multiple flow equilibria, Pure and Applied Geophysics, 130 (1989), 743-749.  doi: 10.1007/BF00881609.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Second edition. Applied Mathematical Sciences, 112. Springer-Verlag, New York, 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

E. N. Lorenz, Designing chaotic models, Journal of the Atmospheric Sciences, 62 (2005), 1574-1587.  doi: 10.1175/JAS3430.1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  A solution of the L96 system with $ F = 2 $ starting from random initial data. (a) All 36 sites at $ t = 500 $. (b) First site for $ 500 \leq t \leq 510 $. (c) Hovmoeller plot for $ 500 \leq t \leq 510 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  A solution of the L96 system with $ F = 8 $ starting from random initial data. (a) All 36 sites at $ t = 500 $. (b) First site for $ 500 \leq t \leq 510 $. (c) Hovmoeller plot for $ 500 \leq t \leq 510 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Left: Eigenvalue curve of the linearization of the L96 system about the constant vector $ \mathbb{e} $ and eigenvalues on this curve for $ N = 36 $ sites. Right: Eigenvalue curves of $ FA - I $ for the L96 system. The curve in the left panel is stretched by $ F $ and shifted to the left by 1 unit. Black: $ F = -0.8 $. For even $ N $, a pitchfork bifurcation has occurred for $ F = -1/2 $. For odd $ N $, a Hopf bifurcation has occurred for some $ F < -1/2 $. Red: $ F = 0.5 $. The constant solution is stable. Blue: $ F = 1.1 $. A Hopf bifurcation has occurred for $ F \approx 8/9 $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Images of the complex unit circle under the Laurent polynomials given in Table 1. (a): Ellipse ($ G_1, \, G_2, \, G_4 $). (b): Trefoil ($ G_3 $, L96 system). (c): Butterfly ($ G_5 $). (d): Kidney ($ G_6 $). (e): Vertical line ($ G_7 $). (f): Bee ($ G_8 $)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Eigenvalue curves of advection terms in $ \mathcal{G}_2 $ for various values of $ F $. (a) $ G_1 $, no bifurcation occurs for positive $ F $. (b) $ G_5 $, a supercritical Hopf bifurcation occurs for positive $ F $. (c) $ -G_1 + \frac{1}{2} G_5 $, a supercritical Hopf bifurcation occurs for positive $ F $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Left: Eigenvalues of $ F_1A - I $ (black and red) and of $ F_1A - I + \alpha_0 C_\ell $ (black and green), for $ N = 14 $. Right: Bifurcation diagram of system (38) in the $ (F,\alpha) $ plane. Blue line: Hopf bifurcation ($ \Re \, \lambda_k = 0 $). Blue stipples: A stable limit cycle exists. Green line: Hopf bifurcation ($ \Re \, \tilde \lambda_\ell = 0 $). Green stipples: A second periodic orbit exists (unstable). Red curves: Neimark-Sacker (N-S) bifurcation. Red stipples: Two stable limit cycles coexist. Magenta line: Linear approximation of N-S bifurcation curve. Red triangle: Hopf-Hopf bifurcation at $ (F_1, \, \alpha_0) $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  Stationary solutions of Eq. (41) with $ G = G_L, \, C = B = I $ and inhomogeneous forcing $ {\bf{F}} $ for $ N = 120 $ sites. Here $ F_i = 1 $ for $ 0 \le i < N/2 $ and $ F_i = M $ for $ i \ge N/2) $
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 8.  Hovmoeller plots showing inhomogeneous advection and dissipation as described in Eq. (46), for $ N = 100 $. Both panels use the same parameters $ (1,1,2) $ in the left half, but solutions have very different behavior. Left: Sites in the right half have parameters $ (0.5,1,1) $. Smaller advection in the right half leads to smaller spatial amplitudes. Perturbations are seen to travel to the right. Right: Sites in the right half have parameters $ (1,1.5,1) $. Larger dissipation in the right half leads to nearly constant solutions over a substantial range of sites
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 9.  Relative energy loss $ \Delta E(t)/E(0) $ for RK4 and scaled relative energy loss $ 10^3 \times \Delta E(t)/E(0) $ for lsoda, for $ N = 36 $, $ \Delta t = 0.05 $, and $ E(0) = 400 $
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  Description of the eigenvalue curves of the eight simplest 3-localized $ \mathcal{G} $-maps identified in Section 2.2. The two rightmost columns give the types of the first expected bifurcation for $ F>0 $ and $ F<0 $ as the magnitude of $ F $ increases. Asterisks indicate exceptions for certain site numbers
$ \mathcal{G} $-map Laurent polynomial $ p_A(z) $ Shape of $ p_A(\mathbb{S}^1) $ $ F>0 $ $ F<0 $
$ G_1 $ $ - z^{-1} - 1 + 2z $ ellipse none pitchfork/Hopf
$ G_2 $ $ -z^{-2} - 1 + 2z^2 $ ellipse none pitchfork/Hopf $ ^\ast $
$ G_3 $ $ - z^{-2} + z $ trefoil Hopf pitchfork/Hopf
$ G_4 $ $ - z^{-3} -1 + 2 z^3 $ ellipse none pitchfork/Hopf $ ^{\ast} $
$ G_5 $ $ -z^{-2} - z + z^2 + z^3 $ butterfly Hopf Hopf
$ G_6 $ $ -z^{-1} + z - z^2 + z^3 $ kidney Hopf pitchfork/Hopf
$ G_7 $ $ -z^{-2} - z^{-1} + z + z^2 $ vertical line none none
$ G_8 $ $ -z^{-3} - z^{-1} + z^2 + z^3 $ bee pitchfork/Hopf Hopf
Table Options

Download as excel

$ \mathcal{G} $-map Laurent polynomial $ p_A(z) $ Shape of $ p_A(\mathbb{S}^1) $ $ F>0 $ $ F<0 $
$ G_1 $ $ - z^{-1} - 1 + 2z $ ellipse none pitchfork/Hopf
$ G_2 $ $ -z^{-2} - 1 + 2z^2 $ ellipse none pitchfork/Hopf $ ^\ast $
$ G_3 $ $ - z^{-2} + z $ trefoil Hopf pitchfork/Hopf
$ G_4 $ $ - z^{-3} -1 + 2 z^3 $ ellipse none pitchfork/Hopf $ ^{\ast} $
$ G_5 $ $ -z^{-2} - z + z^2 + z^3 $ butterfly Hopf Hopf
$ G_6 $ $ -z^{-1} + z - z^2 + z^3 $ kidney Hopf pitchfork/Hopf
$ G_7 $ $ -z^{-2} - z^{-1} + z + z^2 $ vertical line none none
$ G_8 $ $ -z^{-3} - z^{-1} + z^2 + z^3 $ bee pitchfork/Hopf Hopf
Table 2.  Multiple stable limit cycles are expected if approximately $ F \ge F_3^\ast $ and are found numerically for $ \tilde F_3 \le F \le \tilde F_4 $. Limit cycles may be characterized by their spatial periods $ m_1, \, m_2 $
$ N $ $ F_1 $ $ m_1 $ $ F_2 $ $ m_2 $ $ F_3^\ast $ $ \tilde F_3 $ $ \tilde F_4 $
12 1 4 1 6 1 1 $>2 $
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 $>4 $
28 .8901 14 .9457 28 1.0293 1.072 $>3 $
36 .8982 9 .9025 36 .9094 .904 $>2 $
Table Options

Download as excel

$ N $ $ F_1 $ $ m_1 $ $ F_2 $ $ m_2 $ $ F_3^\ast $ $ \tilde F_3 $ $ \tilde F_4 $
12 1 4 1 6 1 1 $>2 $
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 $>4 $
28 .8901 14 .9457 28 1.0293 1.072 $>3 $
36 .8982 9 .9025 36 .9094 .904 $>2 $
[1]

Yunshyong Chow, Sophia Jang. Neimark-Sacker bifurcations in a host-parasitoid system with a host refuge. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1713-1728. doi: 10.3934/dcdsb.2016019
[2]

Kazuyuki Yagasaki. Existence of finite time blow-up solutions in a normal form of the subcritical Hopf bifurcation with time-delayed feedback for small initial functions. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021151
[3]

Jaume Llibre, Ernesto Pérez-Chavela. Zero-Hopf bifurcation for a class of Lorenz-type systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1731-1736. doi: 10.3934/dcdsb.2014.19.1731
[4]

Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete & Continuous Dynamical Systems, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997
[5]

Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045
[6]

John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete & Continuous Dynamical Systems, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805
[7]

Xiaoling Zou, Dejun Fan, Ke Wang. Stationary distribution and stochastic Hopf bifurcation for a predator-prey system with noises. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1507-1519. doi: 10.3934/dcdsb.2013.18.1507
[8]

Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026
[9]

Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325
[10]

Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735
[11]

Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121
[12]

Pengmiao Hao, Xuechen Wang, Junjie Wei. Global Hopf bifurcation of a population model with stage structure and strong Allee effect. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 973-993. doi: 10.3934/dcdss.2017051
[13]

Stephen Pankavich, Nathan Neri, Deborah Shutt. Bistable dynamics and Hopf bifurcation in a refined model of early stage HIV infection. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 2867-2893. doi: 10.3934/dcdsb.2020044
[14]

Jixun Chu, Pierre Magal. Hopf bifurcation for a size-structured model with resting phase. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 4891-4921. doi: 10.3934/dcds.2013.33.4891
[15]

Jisun Lim, Seongwon Lee, Yangjin Kim. Hopf bifurcation in a model of TGF-$\beta$ in regulation of the Th 17 phenotype. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3575-3602. doi: 10.3934/dcdsb.2016111
[16]

Qingyan Shi, Junping Shi, Yongli Song. Hopf bifurcation and pattern formation in a delayed diffusive logistic model with spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 467-486. doi: 10.3934/dcdsb.2018182
[17]

Hossein Mohebbi, Azim Aminataei, Cameron J. Browne, Mohammad Reza Razvan. Hopf bifurcation of an age-structured virus infection model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 861-885. doi: 10.3934/dcdsb.2018046
[18]

Xianlong Fu, Zhihua Liu, Pierre Magal. Hopf bifurcation in an age-structured population model with two delays. Communications on Pure & Applied Analysis, 2015, 14 (2) : 657-676. doi: 10.3934/cpaa.2015.14.657
[19]

Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152
[20]

Qigang Yuan, Jingli Ren. Periodic forcing on degenerate Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2857-2877. doi: 10.3934/dcdsb.2020208

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (52)
  • HTML views (248)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences