American Institute of Mathematical Sciences

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April  2021, 20(4): 1385-1412. doi: 10.3934/cpaa.2021025

Hopf bifurcations and transitions of two-dimensional Quasi-Geostrophic flows

Zhigang Pan 1, , Chanh Kieu 2, and  Quan Wang 3,,

1. 

School of Mathematics, Southwest Jiaotong University, Chengdu, 610031, China
2. 

Department of Earth and Atmospheric Sciences, Indiana University, Bloomington, IN 47405, USA
3. 

College of Mathematics, Sichuan University, Chengdu, Sichuan, 610065, China

* Corresponding author

Received  September 2020 Revised  December 2020 Published  March 2021

Fund Project: The research of the corresponding author was supported by National Science Foundation of China (NSFC) grant 11901408. The research of C. Kieu was supported by ONR Awards N000141812588 and N000142012411

This study examines the Hopf (double Hopf) bifurcations and transitions of two dimensional quasi-geostrophic (QG) flows that model various large-scale oceanic and atmospheric circulations. Using the Kolmogorov function to represent an external forcing in the tropical region, it is shown that the equilibrium of the QG model loses its stability if the combination of the Rossby number, the Ekman number, and the eddy viscosity satisfies a specific condition. Further use of the center manifold technique reveals two different types of the dynamical transition from either a pair of simple complex eigenvalues or a double pair of complex conjugate eigenvalues. These dynamical transitions are confirmed in the numerical analyses of the QG dynamics at the equilibrium, which capture Hopf (double Hopf) bifurcations due to the existence of a nonzero imaginary part of the first eigenvalue. The transition from a pair of simple complex eigenvalues is of particular interest, because it indicates the existence of a stable periodic pattern that is similar to atmospheric easterly waves and related tropical cyclone formation in the tropical atmosphere.

Keywords: Dynamical transition, ITCZ breakdown, instability, Hopf bifurcation, barotropic dynamics.
Mathematics Subject Classification: Primary: 37L10, 37L15; Secondary: 76D99, 76E20.
Citation: Zhigang Pan, Chanh Kieu, Quan Wang. Hopf bifurcations and transitions of two-dimensional Quasi-Geostrophic flows. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1385-1412. doi: 10.3934/cpaa.2021025
References:
[1]

P. Berloff and S. Meacham, On the stability of the wind-driven circulation, J. Mar. Res., 56 (1998), 937-993.  doi: 10.1357/002224098765173437.  Google Scholar

[2]

P. Cessi and G. R. Ierley, Symmetry-breaking multiple equilibria in quasi-geostrophic, wind-driven flows, J. Phys. Oceanogr., 25 (1995), 1196-1205.  doi: 10.1175/1520-0485(1995)025<1196:SBMEIQ>2.0.CO;2.  Google Scholar

[3]

J. Charney and D. Straus, Form-drag instability, multiple equilibria and propagating planetary waves in baroclinic, orographically forced, planetary wave systems, J. Atmos. Sci., 37 (1980), 1157-1176.  doi: 10.1175/1520-0469(1980)037<1157:FDIMEA>2.0.CO;2.  Google Scholar

[4]

J. G. Charney and J. DeVore, Multiple flow equilibria in the atmosphere and blocking, J. Atmos. Sci., 36 (1979), 1205-1216.  doi: 10.1175/1520-0469(1979)036<1205:mfeita>2.0.co;2.  Google Scholar

[5]

J. G. Charney, J. Shukla and K. C. Mo, Comparison of a barotropic blocking theory with observation, J. Atmos. Sci., 38 (1981), 762-779.  doi: 10.1175/1520-0469(1981)038<0762:COABBT>2.0.CO;2.  Google Scholar

[6]

Z. Chen, M. Ghil, E. Simonnet and S. Wang, Hopf bifurcation in quasi-geostrophic channel flow, SIAM J. Appl. Math., 64 (2003), 343-368.  doi: 10.1137/S0036139902406164.  Google Scholar

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Z. Chen and X. Xiong, Equilibrium states of the charney-devore quasi-geostrophic equation in mid-latitude atmosphere, J. Math. Anal. Appl., 444 (2016), 1403-1416.  doi: 10.1016/j.jmaa.2016.07.021.  Google Scholar

[8]

H. Dijkstra, T. Sengul, J. Shen and S. Wang, Dynamic transitions of quasi-geostrophic channel flow, SIAM J. Appl. Math., 75 (2015), 2361-2378.  doi: 10.1137/15M1008166.  Google Scholar

[9]

R. N. Ferreira and W. H. Schubert, Barotropic aspects of itcz breakdown, J. Atmos. Sci., 54 (19997), 261-285.  doi: 10.1175/1520-0469(1997)054<0261:BAOIB>2.0.CO;2.  Google Scholar

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M. Ghil, The wind-driven ocean circulation: applying dynamical systems theory to a climate problem, Discrete Contin. Dyn. Syst., 37 (2017), 189-228.  doi: 10.3934/dcds.2017008.  Google Scholar

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D. Han, M. Hernandez and Q. Wang, Dynamic bifurcation and transition in the Rayleigh-Bénard enard convection with internal heating and varying gravity, Commun. Math. Sci., 17 (2019), 175-192.  doi: 10.4310/CMS.2019.v17.n1.a7.  Google Scholar

[13]

D. Han, M. Hernandez and Q. Wang, On the instabilities and transitions of the western boundary current, Commun. Computa. Phys., 26 (2019), 35-56.  doi: 10.4208/cicp.oa-2018-0066.  Google Scholar

[14]

D. Han, M. Hernandez and Q. Wang, Dynamic Transitions and Bifurcations for a Class of Axisymmetric Geophysical Fluid Flow, SIAM J. Appl. Dyn. Syst., 20 (2020), 38-64.  doi: 10.1137/20M1321139.  Google Scholar

[15]

S. Jiang, F. F. Jin and M. Ghil, Multiple equilibria, periodic, and aperiodic solutions in a wind-driven, double-gyre, shallow-water model, J. Phys. Oceanogr., 25 (1995), 764-786.  doi: 10.1175/1520-0485(1995)025<0764:MEPAAS>2.0.CO;2.  Google Scholar

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C. Kieu, T. Sengul, Q. Wang and D. Yan, On the Hopf (double Hopf) bifurcations and transitions of two-layer western boundary currents, Commun. Nonlinear Sci. Numer. Simul., 65 (2018), 196-215.  doi: 10.1016/j.cnsns.2018.05.010.  Google Scholar

[18]

Y. A. Kuznetsov, Elements of applied bifurcation theory, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-3978-7.  Google Scholar

[19]

B. Legras and M. Ghil, Persistent anomalies, blocking and variations in atmospheric predictability, J. Atmos. Sci., 42 (1985), 433-471.  doi: 10.1175/1520-0469(1985)042<0433:PABAVI>2.0.CO;2.  Google Scholar

[20]

C. Lu, Y. Mao, Q. Wang and D. Yan, Hopf bifurcation and transition of three-dimensional wind-driven ocean circulation problem, J. Differ. Equ., 267 (2019), 2560-2593.  doi: 10.1016/j.jde.2019.03.021.  Google Scholar

[21]

C. Lu, Y. Mao, T. Sengul and Q. Wang, On the spectral instability and bifurcation of the 2d-quasi-geostrophic potential vorticity equation with a generalized kolmogorov forcing, Physica D, 43 (2020), 132296. doi: 10.1016/j.physd.2019.132296.  Google Scholar

[22]

T. Ma and A. Wang, Rayleigh-Bénard convection: dynamics and structure in the physical space, Commun. Math. Sci., 5 (2007), 553-574.   Google Scholar

[23]

T. Ma and S. Wang, Phase Transition Dynamics, Springer-Verlag, New York, 2014. doi: 10.1007/978-1-4614-8963-4.  Google Scholar

[24]

S. P. Meacham, Low-frequency variability in the wind-driven circulation, J. Phys. Oceanogr., 30 (2000), 269-293.  doi: 10.1175/1520-0485(2000)030<0269:LFVITW>2.0.CO;2.  Google Scholar

[25]

B. T. Nadiga and B. P. Luce, Global bifurcation of shilnikov type in a double-gyre ocean model, J. Phys. Oceanogr., 31 (2001), 2669-2690.  doi: 10.1175/1520-0485(2001)031<2669:GBOSTI>2.0.CO;2.  Google Scholar

[26]

J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-4650-3.  Google Scholar

[27]

S. Rambaldi and K. Mo, Forced stationary solutions in a barotropic channel: Multiple equilibria and theory of nonlinear resonance, J. Atmos. Sci., 41 (1984), 3135-3146.  doi: 10.1175/1520-0469(1984)041<3135:FSSIAB>2.0.CO;2.  Google Scholar

[28]

J. Shen, T. T. Medjo and S. Wang, On a wind-driven, double-gyre, quasi-geostrophic ocean model: numerical simulations and structural analysis, J. Comput. Phys., 155, 387–409. doi: 10.1006/jcph.1999.6344.  Google Scholar

[29]

J. Shen, T. Tang and L. L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer-Verlag, Berlin Heidelberg, 2011. doi: 10.1007/978-3-540-71041-7.  Google Scholar

[30]

V. A. Sheremet, G. R. Ierley and V. M. Kamenkovich, Eigenanalysis of the two-dimensional wind-driven ocean circulation problem, J. Mar. Res., 55 (1997), 57-92.  doi: 10.1357/0022240973224463.  Google Scholar

[31]

E. Simonnet, M. Ghi, K. Ide, R. Temam and S. Wang, Low-frequency variability in shallow-water models of the wind-driven ocean circulation. Part I: Steady-state solution, J. Phys. Oceanogr., 33 (2003), 712-728.  doi: 10.1175/1520-0485(2003)33<712:LVISMO>2.0.CO;2.  Google Scholar

[32]

E. Simonnet, M. Ghi, K. Ide, R. Temam and S. Wang, Low-frequency variability in shallow-water models of the wind-driven ocean circulation. Part II: time-dependent solutions, J. Phys. Oceanogr., 33 (2003), 729-751.  doi: 10.1175/1520-0485(2003)33<729:LVISMO>2.0.CO;2.  Google Scholar

[33]

E. Simonnet, M. Ghil and H. Dijkstra, Homoclinic bifurcations in the quasi-geostrophic double-gyre circulation, J. Mar. Res., 63 (2005), 931-956.  doi: 10.1357/002224005774464210.  Google Scholar

[34]

T. Ma and S. Wang, Stability and bifurcation of the taylor problem, Arch. Rational Mech. Anal., 181 (2006), 149-176.  doi: 10.1007/s00205-006-0415-8.  Google Scholar

[35]

G. Veronis, Wind-driven ocean circulation: Part 1. linear theory and perturbation analysis, Deep-Sea Research, 13 (1966), 17–29. doi: 10.1016/0011-7471(66)90003-9.  Google Scholar

[36]

G. Veronis, Wind-driven ocean circulation: Part 2. numerical solutions of the non-linear problem, Deep-Sea Research, 13 (1966), 31–55. doi: 10.1016/0011-7471(66)90004-0.  Google Scholar

[37]

C. C. Wang and G. Magnusdottir, The itcz in the central and eastern pacific on synoptic time scales, Mon. Wea. Rev., 134 (2006), 1405-1421.  doi: 10.1175/MWR3130.1.  Google Scholar

[38]

Q. Wang, C. Kieu and T. A. Vu, Large-scale dynamics of tropical cyclone formation associated with ITCZ breakdown, Atmos. Chem. Phys., 19 (2019), 8383-8397.  doi: 10.5194/acp-19-8383-2019.  Google Scholar

[39]

G. Wolansky, Existence, uniqueness, and stability of stationary barotropic flow with forcing and dissipation, Commun. Pure Appl. Math., 41 (1988), 19-46.  doi: 10.1002/cpa.3160410104.  Google Scholar

[40]

G. Wolansky, The barotropic vorticity equation under forcing and dissipation: Bifurcations of nonsymmetric responses and multiplicity of solutions, SIAM J. Appl. Math., 41 (1989), 1585-1607.  doi: 10.1137/0149096.  Google Scholar

show all references

References:
[1]

P. Berloff and S. Meacham, On the stability of the wind-driven circulation, J. Mar. Res., 56 (1998), 937-993.  doi: 10.1357/002224098765173437.  Google Scholar

[2]

P. Cessi and G. R. Ierley, Symmetry-breaking multiple equilibria in quasi-geostrophic, wind-driven flows, J. Phys. Oceanogr., 25 (1995), 1196-1205.  doi: 10.1175/1520-0485(1995)025<1196:SBMEIQ>2.0.CO;2.  Google Scholar

[3]

J. Charney and D. Straus, Form-drag instability, multiple equilibria and propagating planetary waves in baroclinic, orographically forced, planetary wave systems, J. Atmos. Sci., 37 (1980), 1157-1176.  doi: 10.1175/1520-0469(1980)037<1157:FDIMEA>2.0.CO;2.  Google Scholar

[4]

J. G. Charney and J. DeVore, Multiple flow equilibria in the atmosphere and blocking, J. Atmos. Sci., 36 (1979), 1205-1216.  doi: 10.1175/1520-0469(1979)036<1205:mfeita>2.0.co;2.  Google Scholar

[5]

J. G. Charney, J. Shukla and K. C. Mo, Comparison of a barotropic blocking theory with observation, J. Atmos. Sci., 38 (1981), 762-779.  doi: 10.1175/1520-0469(1981)038<0762:COABBT>2.0.CO;2.  Google Scholar

[6]

Z. Chen, M. Ghil, E. Simonnet and S. Wang, Hopf bifurcation in quasi-geostrophic channel flow, SIAM J. Appl. Math., 64 (2003), 343-368.  doi: 10.1137/S0036139902406164.  Google Scholar

[7]

Z. Chen and X. Xiong, Equilibrium states of the charney-devore quasi-geostrophic equation in mid-latitude atmosphere, J. Math. Anal. Appl., 444 (2016), 1403-1416.  doi: 10.1016/j.jmaa.2016.07.021.  Google Scholar

[8]

H. Dijkstra, T. Sengul, J. Shen and S. Wang, Dynamic transitions of quasi-geostrophic channel flow, SIAM J. Appl. Math., 75 (2015), 2361-2378.  doi: 10.1137/15M1008166.  Google Scholar

[9]

R. N. Ferreira and W. H. Schubert, Barotropic aspects of itcz breakdown, J. Atmos. Sci., 54 (19997), 261-285.  doi: 10.1175/1520-0469(1997)054<0261:BAOIB>2.0.CO;2.  Google Scholar

[10]

M. Ghil, The wind-driven ocean circulation: applying dynamical systems theory to a climate problem, Discrete Contin. Dyn. Syst., 37 (2017), 189-228.  doi: 10.3934/dcds.2017008.  Google Scholar

[11]

M. Ghil, M. D. Chekround and E. Simonnete, Climate dynamics and fluid mechanics: Natural variability and related uncertainties, Physica D, 237 (2008), 2111-2126.  doi: 10.1016/j.physd.2008.03.036.  Google Scholar

[12]

D. Han, M. Hernandez and Q. Wang, Dynamic bifurcation and transition in the Rayleigh-Bénard enard convection with internal heating and varying gravity, Commun. Math. Sci., 17 (2019), 175-192.  doi: 10.4310/CMS.2019.v17.n1.a7.  Google Scholar

[13]

D. Han, M. Hernandez and Q. Wang, On the instabilities and transitions of the western boundary current, Commun. Computa. Phys., 26 (2019), 35-56.  doi: 10.4208/cicp.oa-2018-0066.  Google Scholar

[14]

D. Han, M. Hernandez and Q. Wang, Dynamic Transitions and Bifurcations for a Class of Axisymmetric Geophysical Fluid Flow, SIAM J. Appl. Dyn. Syst., 20 (2020), 38-64.  doi: 10.1137/20M1321139.  Google Scholar

[15]

S. Jiang, F. F. Jin and M. Ghil, Multiple equilibria, periodic, and aperiodic solutions in a wind-driven, double-gyre, shallow-water model, J. Phys. Oceanogr., 25 (1995), 764-786.  doi: 10.1175/1520-0485(1995)025<0764:MEPAAS>2.0.CO;2.  Google Scholar

[16] E. Kalnay, Atmospheric Modeling, Data Assimilation and Predictability, Cambridge University Press, 2002.   Google Scholar
[17]

C. Kieu, T. Sengul, Q. Wang and D. Yan, On the Hopf (double Hopf) bifurcations and transitions of two-layer western boundary currents, Commun. Nonlinear Sci. Numer. Simul., 65 (2018), 196-215.  doi: 10.1016/j.cnsns.2018.05.010.  Google Scholar

[18]

Y. A. Kuznetsov, Elements of applied bifurcation theory, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-3978-7.  Google Scholar

[19]

B. Legras and M. Ghil, Persistent anomalies, blocking and variations in atmospheric predictability, J. Atmos. Sci., 42 (1985), 433-471.  doi: 10.1175/1520-0469(1985)042<0433:PABAVI>2.0.CO;2.  Google Scholar

[20]

C. Lu, Y. Mao, Q. Wang and D. Yan, Hopf bifurcation and transition of three-dimensional wind-driven ocean circulation problem, J. Differ. Equ., 267 (2019), 2560-2593.  doi: 10.1016/j.jde.2019.03.021.  Google Scholar

[21]

C. Lu, Y. Mao, T. Sengul and Q. Wang, On the spectral instability and bifurcation of the 2d-quasi-geostrophic potential vorticity equation with a generalized kolmogorov forcing, Physica D, 43 (2020), 132296. doi: 10.1016/j.physd.2019.132296.  Google Scholar

[22]

T. Ma and A. Wang, Rayleigh-Bénard convection: dynamics and structure in the physical space, Commun. Math. Sci., 5 (2007), 553-574.   Google Scholar

[23]

T. Ma and S. Wang, Phase Transition Dynamics, Springer-Verlag, New York, 2014. doi: 10.1007/978-1-4614-8963-4.  Google Scholar

[24]

S. P. Meacham, Low-frequency variability in the wind-driven circulation, J. Phys. Oceanogr., 30 (2000), 269-293.  doi: 10.1175/1520-0485(2000)030<0269:LFVITW>2.0.CO;2.  Google Scholar

[25]

B. T. Nadiga and B. P. Luce, Global bifurcation of shilnikov type in a double-gyre ocean model, J. Phys. Oceanogr., 31 (2001), 2669-2690.  doi: 10.1175/1520-0485(2001)031<2669:GBOSTI>2.0.CO;2.  Google Scholar

[26]

J. Pedlosky, Geophysical Fluid Dynamics, Springer-Verlag, New York, 1987. doi: 10.1007/978-1-4612-4650-3.  Google Scholar

[27]

S. Rambaldi and K. Mo, Forced stationary solutions in a barotropic channel: Multiple equilibria and theory of nonlinear resonance, J. Atmos. Sci., 41 (1984), 3135-3146.  doi: 10.1175/1520-0469(1984)041<3135:FSSIAB>2.0.CO;2.  Google Scholar

[28]

J. Shen, T. T. Medjo and S. Wang, On a wind-driven, double-gyre, quasi-geostrophic ocean model: numerical simulations and structural analysis, J. Comput. Phys., 155, 387–409. doi: 10.1006/jcph.1999.6344.  Google Scholar

[29]

J. Shen, T. Tang and L. L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer-Verlag, Berlin Heidelberg, 2011. doi: 10.1007/978-3-540-71041-7.  Google Scholar

[30]

V. A. Sheremet, G. R. Ierley and V. M. Kamenkovich, Eigenanalysis of the two-dimensional wind-driven ocean circulation problem, J. Mar. Res., 55 (1997), 57-92.  doi: 10.1357/0022240973224463.  Google Scholar

[31]

E. Simonnet, M. Ghi, K. Ide, R. Temam and S. Wang, Low-frequency variability in shallow-water models of the wind-driven ocean circulation. Part I: Steady-state solution, J. Phys. Oceanogr., 33 (2003), 712-728.  doi: 10.1175/1520-0485(2003)33<712:LVISMO>2.0.CO;2.  Google Scholar

[32]

E. Simonnet, M. Ghi, K. Ide, R. Temam and S. Wang, Low-frequency variability in shallow-water models of the wind-driven ocean circulation. Part II: time-dependent solutions, J. Phys. Oceanogr., 33 (2003), 729-751.  doi: 10.1175/1520-0485(2003)33<729:LVISMO>2.0.CO;2.  Google Scholar

[33]

E. Simonnet, M. Ghil and H. Dijkstra, Homoclinic bifurcations in the quasi-geostrophic double-gyre circulation, J. Mar. Res., 63 (2005), 931-956.  doi: 10.1357/002224005774464210.  Google Scholar

[34]

T. Ma and S. Wang, Stability and bifurcation of the taylor problem, Arch. Rational Mech. Anal., 181 (2006), 149-176.  doi: 10.1007/s00205-006-0415-8.  Google Scholar

[35]

G. Veronis, Wind-driven ocean circulation: Part 1. linear theory and perturbation analysis, Deep-Sea Research, 13 (1966), 17–29. doi: 10.1016/0011-7471(66)90003-9.  Google Scholar

[36]

G. Veronis, Wind-driven ocean circulation: Part 2. numerical solutions of the non-linear problem, Deep-Sea Research, 13 (1966), 31–55. doi: 10.1016/0011-7471(66)90004-0.  Google Scholar

[37]

C. C. Wang and G. Magnusdottir, The itcz in the central and eastern pacific on synoptic time scales, Mon. Wea. Rev., 134 (2006), 1405-1421.  doi: 10.1175/MWR3130.1.  Google Scholar

[38]

Q. Wang, C. Kieu and T. A. Vu, Large-scale dynamics of tropical cyclone formation associated with ITCZ breakdown, Atmos. Chem. Phys., 19 (2019), 8383-8397.  doi: 10.5194/acp-19-8383-2019.  Google Scholar

[39]

G. Wolansky, Existence, uniqueness, and stability of stationary barotropic flow with forcing and dissipation, Commun. Pure Appl. Math., 41 (1988), 19-46.  doi: 10.1002/cpa.3160410104.  Google Scholar

[40]

G. Wolansky, The barotropic vorticity equation under forcing and dissipation: Bifurcations of nonsymmetric responses and multiplicity of solutions, SIAM J. Appl. Math., 41 (1989), 1585-1607.  doi: 10.1137/0149096.  Google Scholar

Figure 1.  Illustration of the steady-state flow $ \psi_{S} $ driven by the Kolmogorov forcing, which is obtained from the QG model (2.1). The blue dashed curve represents the horizontal profile of the mean flow, while the black arrows represent the direction of the mean flow on the horizontal domain $ \Omega_a $
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Figure 2.  Marginal stability curves $ \text{R}_{m}^*(a) $ determined by equation $ \text{Re}\rho_{m, 1}( \text{R}) = 0 $ for $ 0.1\leq a\leq0.35 $
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Figure 3.  Marginal stability curves $ \text{R}^*(a) $ determined by $ \text{R}^*(a) = \min\{ \text{R}: \text{Re}\rho_{m, 1}( \text{R}) = 0, m = 1, 2, \cdots, k\} $ for $ 0.1\leq a\leq0.35 $
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Figure 4.  Illustration of the first critical wave number $ n $ for a range of the aspect ratio parameter $ 0.1\leq a\leq0.35 $ and $ E = 0.05 $
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Figure 5.  The topological structure of the continuous transition
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Figure 6.  The topological structure of the catastrophic transition and a separation of periodic orbits
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Figure 7.  The case $ \delta \text{Re}(\rho_{n, 1}) < \text{Re}(\rho_{n+1, 1}) $ in Theorem 4.4
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Figure 8.  The case $ \delta \text{Re}(\rho_{n, 1}) > \text{Re}(\rho_{n+1, 1}) $ in Theorem 4.4
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Figure 9.  The regions in the $ \text{Re}(\rho_{n, 1}) $--$ \text{Re}(\rho_{n+1, 1}) $ plane with different dynamical behaviours according to Theorem 4.4. In region $ \texttt{IV} $, the basic steady state is locally asymptotically stable. In regions $ \texttt{I}, \texttt{III} $ and $\texttt{V}$, the system undergoes a supercritical Hopf bifurcation. In region $ \texttt{II} $, the system will tend to a double time periodic solution. This solution corresponds to an invariant 2D torus
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Figure 10.  Transition number with $ 0.1\leq a\leq0.35 $ and $ \epsilon = 0.3 $
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Figure 11.  The observed periodic solution and $ \epsilon = 0.3 $ with $ E = 0.05 $ at $ \text{R} = 3.8717> \text{R}^* = 3.8517 $, whose period is T = 2.776. The real period is 3.2116 days
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Table 1.  Coefficients in equations (4.6) with various $ E $ and $ \epsilon = 0.3 $ at the critical $ \text{R}^* $
$ a $ $ E $ $ n $ $ \text{R}^* $ $ A_{1} $ $ B_{1} $ $ C_{1} $ $ D_{1} $
0.11934208 0.01 4(5) 2.9824 -0.5545 -1.6917 -1.8423 -1.1788
0.11935935 0.03 4(5) 3.4699 -0.2779 -0.8562 -0.8955 -0.5052
0.09768743 0.01 5(6) 2.9518 -0.6002 -1.7218 -1.8647 -1.1083
0.09771521 0.03 5(6) 3.4344 -0.2983 -0.8626 -0.8986 -0.4841
0.08267975 0.01 6(7) 2.9345 -0.6330 -1.7385 -1.8711 -1.0615
0.08271008 0.03 6(7) 3.4141 -0.3127 -0.8655 -0.8982 -0.4697
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$ a $ $ E $ $ n $ $ \text{R}^* $ $ A_{1} $ $ B_{1} $ $ C_{1} $ $ D_{1} $
0.11934208 0.01 4(5) 2.9824 -0.5545 -1.6917 -1.8423 -1.1788
0.11935935 0.03 4(5) 3.4699 -0.2779 -0.8562 -0.8955 -0.5052
0.09768743 0.01 5(6) 2.9518 -0.6002 -1.7218 -1.8647 -1.1083
0.09771521 0.03 5(6) 3.4344 -0.2983 -0.8626 -0.8986 -0.4841
0.08267975 0.01 6(7) 2.9345 -0.6330 -1.7385 -1.8711 -1.0615
0.08271008 0.03 6(7) 3.4141 -0.3127 -0.8655 -0.8982 -0.4697
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