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July  2021, 26(7): 3737-3765. doi: 10.3934/dcdsb.2020255

The coupled 1:2 resonance in a symmetric case and parametric amplification model

Reza Mazrooei-Sebdani , and  Zahra Yousefi

Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran

* Corresponding author: Reza Mazrooei-Sebdani

Received  November 2019 Revised  June 2020 Published  July 2021 Early access  August 2020

This paper deals with the coupled Hamiltonian $ 1 $:$ 2 $ resonance, i.e. the Hamiltonian $ 1 $:$ 2 $:$ 1 $:$ 2 $ resonance. This resonance is of the first order. We isolate several integrable cases. Our main focus is on two models. In the first part of the paper, we present a discrete symmetric normal form truncated to order three and we compute the relative equilibria for its corresponding system. In the second part, the paper is devoted to the study of the Hamiltonian describing the four-wave mixing (FWM) model. In addition to the Hamiltonian, the corresponding system possesses three more independent integrals. We use these integrals to obtain estimates for the phase space and total energy. Further, we compute the relative equilibria of the FWM system for the $ 1 $:$ 2 $:$ 1 $:$ 2 $ resonance. Finally, we carry out some numerical experiments for the detuned system.

Keywords: Coupled Hamiltonian 1:2 resonance, normal form and first integrals, relative equilibria, symmetry, four wave mixing (FWM) model, optimization of phase-sensitive double-pumped parametric amplifiers, maximum gain and energy.
Mathematics Subject Classification: Primary: 37J12, 37N20; Secondary: 37J40, 37J35.
Citation: Reza Mazrooei-Sebdani, Zahra Yousefi. The coupled 1:2 resonance in a symmetric case and parametric amplification model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3737-3765. doi: 10.3934/dcdsb.2020255
References:
[1]

V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, in Dynamical Systems III, Encyc. Math. Sciences, Springer-Verlag, Berlin, 2006.  Google Scholar

[2]

H. W. Broer, G. A. Lunter and G. Vegter, Equivariant singularity theory with distinguished parameters: Two case studies of resonant Hamiltonian systems, Phys. D, 112 (1998), 64-80.  doi: 10.1016/S0167-2789(97)00202-9.  Google Scholar

[3]

H. Broer and F. Takens, Dynamical Systems and Chaos, Appl. Math. Sciences, Vol. 172, Springer, New York, 2011. doi: 10.1007/978-1-4419-6870-8.  Google Scholar

[4]

R. Bruggeman and F. Verhulst, The inhomogeneous Fermi-Pasta-Ulam chain. A case study of the $1:2:3$ Resonance, Acta Appl. Math., 152 (2017), 111-145.  doi: 10.1007/s10440-017-0115-4.  Google Scholar

[5]

G. Cappellini and S. Trillo, Third-order three-wave mixxing in single-mode fibers: Exact solutions and spatial instability effects, J. Opt. Soc. Am. B., 8 (1991), 824-838.   Google Scholar

[6]

O. Christov, Non-integrability of first order resonances of Hamiltonian systems in three degrees of freedom, Celestial Mech. Dynam. Astronom., 112 (2012), 147-167.  doi: 10.1007/s10569-011-9389-4.  Google Scholar

[7]

C. De Angelis, M. Santagiustina and S. Trillo, Four-photon homoclinic instabilities in nonlinear highly birefringent media, Phys. Rev. A., 51 (1995), 774-791.  doi: 10.1103/PhysRevA.51.774.  Google Scholar

[8]

J. J. Duistermaat, Non-integrability of the $1$ : $2$ : $1$-resonance, Ergodic Theory Dynam. Systems, 4 (1984), 553-568.  doi: 10.1017/S0143385700002649.  Google Scholar

[9]

J. Egea, S. Ferrer and J. C. van der Meer, Bifurcations of the Hamiltonian fourfold $1$ : $1$ resonance with toroidal symmetry, J. Nonlinear Sci., 21 (2011), 835-874.  doi: 10.1007/s00332-011-9102-5.  Google Scholar

[10]

D. D. Holm and P. Lynch, Stepwise precession of the resonant swinging spring, SIAM J. Appl. Dyn. Syst., 1 (2002), 44-64.  doi: 10.1137/S1111111101388571.  Google Scholar

[11]

G. Haller and S. Wiggins, Geometry and chaos near resonant equilibria of 3-DOF Hamiltonian systems, Physica D, 90 (1996), 319-365.  doi: 10.1016/0167-2789(95)00247-2.  Google Scholar

[12]

H. Hanßmann, Local and Semi-Local Bifurcations in Hamiltonian Dynamical Systems. Results and Examples, Lecture Notes Math., 1893, Springer-Verlag, Berlin, Heidelberg, 2007. Google Scholar

[13]

H. Hanßmann, R. Mazrooei-Sebdani, F. Verhulst, The $1: 2: 4$ resonance in a particle chain, preprint, 2020, arXiv: 2002.01263. Google Scholar

[14]

G. Y. Kryuchkyan and K. V. Kheruntsyan, Four-wave mixing with non-degenerate pumps: Steady states and squeezing in the presence of phase modulation, Quantum Semiclass. Opt., 7 (1995), 529-539.  doi: 10.1088/1355-5111/7/4/010.  Google Scholar

[15]

M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices, Cambridge University, Cambridge, 2008. doi: 10.1017/CBO9780511600265.  Google Scholar

[16]

S. Medvedev and B. Bednyakova, Hamiltonian approach for optimization of phase-sensitive double-pumped parametric amplifiers, Opt. Express., 26 (2018), 15503. doi: 10.1364/OE.26.015503.  Google Scholar

[17]

H. Pourbeyram and A. Mafi, Four-wave mixing of a laser and its frequency-doubled version in a multimode optical fiber, Photonics, 2 (2015), 906-915.  doi: 10.3390/photonics2030906.  Google Scholar

[18]

J. R. Ott, H. Steffensen, K. Rottwitt and C. J. Mckinstrie, Geometric interpreation of four-wave mixing, Phys. Rev. A., 88 (2013), 043805. Google Scholar

[19]

A. A. Redyuk, A. E. Bednyakova, S. B. Medvedev, M. P. Fedoruk and S. K. Turitsyn, Simple Geometric interpreation of signal evolution in phase-sensitive fibre optic parametric amplifier, Opt. Express., 25 (2017), 223-231.   Google Scholar

[20]

D. A. Sadovski and B. I. Zhilinski, Hamiltonian systems with detuned $1$:$1$:$2$ resonance: Manifestation of bidromy, Ann. Physics, 322 (2007), 164-200.  doi: 10.1016/j.aop.2006.09.011.  Google Scholar

[21]

J. A. Sanders, F. Verhulst and J. Murdock, Averaging methods in nonlinear dynamical systems. Second Edition., Applied Mathematical Sciences, , Vol. 59, Springer, New York, 2007.  Google Scholar

[22]

S. Trillo and S. Wabnitz, Dynamics of the nonlinear modulational instability in optical fibers, Opt. Lett., 16 (1991), 986-988.  doi: 10.1364/OL.16.000986.  Google Scholar

[23]

E. van der Aa, First order resonances in three-degrees-of-freedom systems, Celestial Mech., 31 (1983), 163-191.  doi: 10.1007/BF01686817.  Google Scholar

[24]

E. van der Aa and J. A. Sanders, The $1$: $2$: $1$-resonance, its periodic orbits and integrals, in Asymptotic Analysis: From Theory to Application, Lecture Notes Math., Vol. 711, Springer, 1979,187–208. Google Scholar

[25]

E. van der Aa and F. Verhulst, Asymptotic integrability and periodic solutions of a Hamiltonian system in $1$ : $2$ : $2$-resonance, SIAM J. Math. Anal., 15 (1984), 890-911.  doi: 10.1137/0515067.  Google Scholar

[26]

F. Verhulst, Integrability and non-integrability of Hamiltonian normal forms, Acta Appl. Math., 137 (2015), 253-272.  doi: 10.1007/s10440-014-9998-5.  Google Scholar

[27] L. Vivien and L. Pavesi, Handbook of Silicon Photonics. First Edition, CRC Press, Taylor & Francis Group, 2013.   Google Scholar
[28]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos. Second Edition, in Texts in Appl. Math., Springer-Verlag, New York, 2003.  Google Scholar

show all references

References:
[1]

V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, in Dynamical Systems III, Encyc. Math. Sciences, Springer-Verlag, Berlin, 2006.  Google Scholar

[2]

H. W. Broer, G. A. Lunter and G. Vegter, Equivariant singularity theory with distinguished parameters: Two case studies of resonant Hamiltonian systems, Phys. D, 112 (1998), 64-80.  doi: 10.1016/S0167-2789(97)00202-9.  Google Scholar

[3]

H. Broer and F. Takens, Dynamical Systems and Chaos, Appl. Math. Sciences, Vol. 172, Springer, New York, 2011. doi: 10.1007/978-1-4419-6870-8.  Google Scholar

[4]

R. Bruggeman and F. Verhulst, The inhomogeneous Fermi-Pasta-Ulam chain. A case study of the $1:2:3$ Resonance, Acta Appl. Math., 152 (2017), 111-145.  doi: 10.1007/s10440-017-0115-4.  Google Scholar

[5]

G. Cappellini and S. Trillo, Third-order three-wave mixxing in single-mode fibers: Exact solutions and spatial instability effects, J. Opt. Soc. Am. B., 8 (1991), 824-838.   Google Scholar

[6]

O. Christov, Non-integrability of first order resonances of Hamiltonian systems in three degrees of freedom, Celestial Mech. Dynam. Astronom., 112 (2012), 147-167.  doi: 10.1007/s10569-011-9389-4.  Google Scholar

[7]

C. De Angelis, M. Santagiustina and S. Trillo, Four-photon homoclinic instabilities in nonlinear highly birefringent media, Phys. Rev. A., 51 (1995), 774-791.  doi: 10.1103/PhysRevA.51.774.  Google Scholar

[8]

J. J. Duistermaat, Non-integrability of the $1$ : $2$ : $1$-resonance, Ergodic Theory Dynam. Systems, 4 (1984), 553-568.  doi: 10.1017/S0143385700002649.  Google Scholar

[9]

J. Egea, S. Ferrer and J. C. van der Meer, Bifurcations of the Hamiltonian fourfold $1$ : $1$ resonance with toroidal symmetry, J. Nonlinear Sci., 21 (2011), 835-874.  doi: 10.1007/s00332-011-9102-5.  Google Scholar

[10]

D. D. Holm and P. Lynch, Stepwise precession of the resonant swinging spring, SIAM J. Appl. Dyn. Syst., 1 (2002), 44-64.  doi: 10.1137/S1111111101388571.  Google Scholar

[11]

G. Haller and S. Wiggins, Geometry and chaos near resonant equilibria of 3-DOF Hamiltonian systems, Physica D, 90 (1996), 319-365.  doi: 10.1016/0167-2789(95)00247-2.  Google Scholar

[12]

H. Hanßmann, Local and Semi-Local Bifurcations in Hamiltonian Dynamical Systems. Results and Examples, Lecture Notes Math., 1893, Springer-Verlag, Berlin, Heidelberg, 2007. Google Scholar

[13]

H. Hanßmann, R. Mazrooei-Sebdani, F. Verhulst, The $1: 2: 4$ resonance in a particle chain, preprint, 2020, arXiv: 2002.01263. Google Scholar

[14]

G. Y. Kryuchkyan and K. V. Kheruntsyan, Four-wave mixing with non-degenerate pumps: Steady states and squeezing in the presence of phase modulation, Quantum Semiclass. Opt., 7 (1995), 529-539.  doi: 10.1088/1355-5111/7/4/010.  Google Scholar

[15]

M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices, Cambridge University, Cambridge, 2008. doi: 10.1017/CBO9780511600265.  Google Scholar

[16]

S. Medvedev and B. Bednyakova, Hamiltonian approach for optimization of phase-sensitive double-pumped parametric amplifiers, Opt. Express., 26 (2018), 15503. doi: 10.1364/OE.26.015503.  Google Scholar

[17]

H. Pourbeyram and A. Mafi, Four-wave mixing of a laser and its frequency-doubled version in a multimode optical fiber, Photonics, 2 (2015), 906-915.  doi: 10.3390/photonics2030906.  Google Scholar

[18]

J. R. Ott, H. Steffensen, K. Rottwitt and C. J. Mckinstrie, Geometric interpreation of four-wave mixing, Phys. Rev. A., 88 (2013), 043805. Google Scholar

[19]

A. A. Redyuk, A. E. Bednyakova, S. B. Medvedev, M. P. Fedoruk and S. K. Turitsyn, Simple Geometric interpreation of signal evolution in phase-sensitive fibre optic parametric amplifier, Opt. Express., 25 (2017), 223-231.   Google Scholar

[20]

D. A. Sadovski and B. I. Zhilinski, Hamiltonian systems with detuned $1$:$1$:$2$ resonance: Manifestation of bidromy, Ann. Physics, 322 (2007), 164-200.  doi: 10.1016/j.aop.2006.09.011.  Google Scholar

[21]

J. A. Sanders, F. Verhulst and J. Murdock, Averaging methods in nonlinear dynamical systems. Second Edition., Applied Mathematical Sciences, , Vol. 59, Springer, New York, 2007.  Google Scholar

[22]

S. Trillo and S. Wabnitz, Dynamics of the nonlinear modulational instability in optical fibers, Opt. Lett., 16 (1991), 986-988.  doi: 10.1364/OL.16.000986.  Google Scholar

[23]

E. van der Aa, First order resonances in three-degrees-of-freedom systems, Celestial Mech., 31 (1983), 163-191.  doi: 10.1007/BF01686817.  Google Scholar

[24]

E. van der Aa and J. A. Sanders, The $1$: $2$: $1$-resonance, its periodic orbits and integrals, in Asymptotic Analysis: From Theory to Application, Lecture Notes Math., Vol. 711, Springer, 1979,187–208. Google Scholar

[25]

E. van der Aa and F. Verhulst, Asymptotic integrability and periodic solutions of a Hamiltonian system in $1$ : $2$ : $2$-resonance, SIAM J. Math. Anal., 15 (1984), 890-911.  doi: 10.1137/0515067.  Google Scholar

[26]

F. Verhulst, Integrability and non-integrability of Hamiltonian normal forms, Acta Appl. Math., 137 (2015), 253-272.  doi: 10.1007/s10440-014-9998-5.  Google Scholar

[27] L. Vivien and L. Pavesi, Handbook of Silicon Photonics. First Edition, CRC Press, Taylor & Francis Group, 2013.   Google Scholar
[28]

S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos. Second Edition, in Texts in Appl. Math., Springer-Verlag, New York, 2003.  Google Scholar

Figure 1.  Sketch of the FWM model where a photon at $ \omega_1 $ and $ \omega_4 $ is annihilated while a photon at $ \omega_2 $ and $ \omega_3 $ is created
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Figure 2.  Changes of $ \tilde{H} $ respect to initial conditions
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Figure 3.  Changes of $ \tilde{H} $ respect to distance
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Figure 4.  $ \pi_j(T) $ for $ j = 1,k_ge 5,k_ge 6,k_ge 9,k_ge 10,k_ge 11,k_ge 12 $ respect to $ T $
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Figure 5.  $ (\pi_j(T),\pi_{j+1}(T)) $ for all $ j = 5,k_ge 9,k_ge 11 $
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Figure 6.  $ (\pi_1(T),\pi_j(T)) $ for all $ j = 5,k_ge 6,k_ge 9,k_ge 10,k_ge 11,k_ge 12 $
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Figure 7.  Changes of $ \tilde{H} $ respect to $ \pi_1(T) $
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Figure 8.  $ (\pi_j(T),\pi_{j+1}(T)) $ for all $ j = 5,k_ge 9,k_ge 11 $ near some relative equilibria for $ \nu_1 = 1+\frac{1}{8},k_ge \nu_2 = 2+\frac{1}{2},k_ge \nu_3 = 1+\frac{1}{8},k_ge \nu_4 = 2+\frac{1}{2} $ and $ \eta_1 = \eta_2 = \frac{3}{2}(\frac{\nu_2-2\nu_3}{9\gamma}) = \frac{1}{24} $, $ \eta = \eta_1+\frac{1}{3}\eta_2 = \frac{1}{18} $
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Figure 9.  $ (\pi_j(T),\pi_{j+1}(T)) $ for all $ j = 5,k_ge 9,k_ge 11 $ near some relative equilibria for $ \nu_1 = 1,k_ge \nu_2 = 2+\frac{1}{10},k_ge \nu_3 = 1,k_ge \nu_4 = 2 $ and $ \eta = 0.1118423612,k_ge \eta_1 = 0.07710541672,k_ge \eta_2 = 0.1042108334 $
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Figure 10.  $ (\pi_j(T),\pi_{j+1}(T)) $ for all $ j = 5,k_ge 9,k_ge 11 $ near some relative equilibria for $ \nu_1 = 1,k_ge \nu_2 = 2+\frac{1}{10},k_ge \nu_3 = 1,k_ge \nu_4 = 2 $ and $ \eta = 0.06174877145,k_ge \eta_1 = \frac{1}{20},k_ge \eta_2 = \frac{1}{40} $
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Figure 11.  $ (\pi_j(T),\pi_{j+1}(T)) $ for all $ j = 5,k_ge 9,k_ge 11 $ near some relative equilibria for $ \nu_1 = 1,k_ge \nu_2 = 2+\frac{1}{10},k_ge \nu_3 = 1,k_ge \nu_4 = 2 $ and $ \eta = \frac{7}{160},k_ge \eta_1 = \frac{1}{60},k_ge \eta_2 = \frac{1}{20} $
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Table 1.  First order genuine resonances table with $ |\omega_j|<10, \; j = 1, \; 2, \; 3, \; 4 $
$ 1:2:3:4 $ $ 1:2:3:5 $ $ 1:2:3:6 $ $ 1:2:3:7 $ $ 1:2:3:8 $ $ 1:2:3:9 $ $ 1:2:4:5 $
$ 1:2:4:6 $ $ 1:2:4:7 $ $ 1:2:4:8 $ $ 1:2:4:9 $ $ 1:2:5:6 $ $ 1:2:5:7 $ $ 1:2:6:7 $
$ 1:2:6:8 $ $ 1:2:7:8 $ $ 1:2:7:9 $ $ 1:2:8:9 $ $ 1:3:4:5 $ $ 1:3:4:6 $ $ 1:3:4:7 $
$ 1:3:4:8 $ $ 1:3:5:6 $ $ 1:3:6:7 $ $ 1:3:6:9 $ $ 1:4:5:6 $ $ 1:4:5:8 $ $ 1:4:5:9 $
$ 1:4:7:8 $ $ 1:4:8:9 $ $ 1:5:6:7 $ $ 1:6:7:8 $ $ 1:7:8:9 $ $ 1:2:2:3 $ $ 1:2:2:4 $
$ 1:2:2:5 $ $ 1:2:2:6 $ $ 1:2:2:7 $ $ 1:2:2:8 $ $ 1:2:2:9 $ $ 1:3:3:2 $ $ 1:3:3:4 $
$ 1:3:3:6 $ $ 1:4:4:2 $ $ 1:4:4:3 $ $ 1:4:4:5 $ $ 1:4:4:8 $ $ 1:5:5:4 $ $ 1:5:5:6 $
$ 1:6:6:3 $ $ 1:6:6:5 $ $ 1:6:6:7 $ $ 1:7:7:6 $ $ 1:7:7:8 $ $ 1:8:8:4 $ $ 1:8:8:7 $
$ 1:8:8:9 $ $ 1:9:9:8 $ $ 1:2:2:2 $ $ 1:1:2:3 $ $ 1:1:2:4 $ $ 1:1:2:5 $ $ 1:1:2:6 $
$ 1:1:2:7 $ $ 1:1:2:8 $ $ 1:1:2:9 $ $ 1:1:3:4 $ $ 1:1:4:5 $ $ 1:1:5:6 $ $ 1:1:6:7 $
$ 1:1:7:8 $ $ 1:1:8:9 $ $ 1:2:1:2 $ $ 1:1:1:2 $ $ 2:3:4:5 $ $ 2:3:4:6 $ $ 2:3:4:7 $
$ 2:3:4:8 $ $ 2:3:5:6 $ $ 2:3:5:7 $ $ 2:3:5:8 $ $ 2:3:6:8 $ $ 2:3:6:9 $ $ 2:4:5:6 $
$ 2:4:5:7 $ $ 2:4:5:8 $ $ 2:4:5:9 $ $ 2:4:6:7 $ $ 2:4:6:9 $ $ 2:4:7:8 $ $ 2:4:7:9 $
$ 2:4:8:9 $ $ 2:5:7:9 $ $ 2:3:3:5 $ $ 2:3:3:6 $ $ 2:4:4:3 $ $ 2:4:4:5 $ $ 2:4:4:7 $
$ 2:4:4:9 $ $ 2:5:5:3 $ $ 2:5:5:7 $ $ 2:6:6:3 $ $ 2:7:7:5 $ $ 2:7:7:9 $ $ 2:2:3:4 $
$ 2:2:3:5 $ $ 2:2:4:5 $ $ 2:2:4:7 $ $ 2:2:4:9 $ $ 2:2:5:7 $ $ 2:2:7:9 $ $ 3:4:6:7 $
$ 3:4:6:8 $ $ 3:4:6:9 $ $ 3:4:7:8 $ $ 3:5:6:8 $ $ 3:5:6:9 $ $ 3:6:7:9 $ $ 3:6:8:9 $
$ 3:4:4:7 $ $ 3:4:4:8 $ $ 3:5:5:8 $ $ 3:6:6:7 $ $ 3:6:6:8 $ $ 3:3:4:6 $ $ 3:3:4:7 $
$ 3:3:5:6 $ $ 3:3:5:8 $ $ 3:3:6:7 $ $ 3:3:6:8 $ $ 4:5:8:9 $ $ 4:5:5:9 $ $ 4:8:8:9 $
$ 4:4:5:8 $ $ 4:4:5:9 $ $ 4:4:7:8 $ $ 4:4:8:9 $
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$ 1:2:3:4 $ $ 1:2:3:5 $ $ 1:2:3:6 $ $ 1:2:3:7 $ $ 1:2:3:8 $ $ 1:2:3:9 $ $ 1:2:4:5 $
$ 1:2:4:6 $ $ 1:2:4:7 $ $ 1:2:4:8 $ $ 1:2:4:9 $ $ 1:2:5:6 $ $ 1:2:5:7 $ $ 1:2:6:7 $
$ 1:2:6:8 $ $ 1:2:7:8 $ $ 1:2:7:9 $ $ 1:2:8:9 $ $ 1:3:4:5 $ $ 1:3:4:6 $ $ 1:3:4:7 $
$ 1:3:4:8 $ $ 1:3:5:6 $ $ 1:3:6:7 $ $ 1:3:6:9 $ $ 1:4:5:6 $ $ 1:4:5:8 $ $ 1:4:5:9 $
$ 1:4:7:8 $ $ 1:4:8:9 $ $ 1:5:6:7 $ $ 1:6:7:8 $ $ 1:7:8:9 $ $ 1:2:2:3 $ $ 1:2:2:4 $
$ 1:2:2:5 $ $ 1:2:2:6 $ $ 1:2:2:7 $ $ 1:2:2:8 $ $ 1:2:2:9 $ $ 1:3:3:2 $ $ 1:3:3:4 $
$ 1:3:3:6 $ $ 1:4:4:2 $ $ 1:4:4:3 $ $ 1:4:4:5 $ $ 1:4:4:8 $ $ 1:5:5:4 $ $ 1:5:5:6 $
$ 1:6:6:3 $ $ 1:6:6:5 $ $ 1:6:6:7 $ $ 1:7:7:6 $ $ 1:7:7:8 $ $ 1:8:8:4 $ $ 1:8:8:7 $
$ 1:8:8:9 $ $ 1:9:9:8 $ $ 1:2:2:2 $ $ 1:1:2:3 $ $ 1:1:2:4 $ $ 1:1:2:5 $ $ 1:1:2:6 $
$ 1:1:2:7 $ $ 1:1:2:8 $ $ 1:1:2:9 $ $ 1:1:3:4 $ $ 1:1:4:5 $ $ 1:1:5:6 $ $ 1:1:6:7 $
$ 1:1:7:8 $ $ 1:1:8:9 $ $ 1:2:1:2 $ $ 1:1:1:2 $ $ 2:3:4:5 $ $ 2:3:4:6 $ $ 2:3:4:7 $
$ 2:3:4:8 $ $ 2:3:5:6 $ $ 2:3:5:7 $ $ 2:3:5:8 $ $ 2:3:6:8 $ $ 2:3:6:9 $ $ 2:4:5:6 $
$ 2:4:5:7 $ $ 2:4:5:8 $ $ 2:4:5:9 $ $ 2:4:6:7 $ $ 2:4:6:9 $ $ 2:4:7:8 $ $ 2:4:7:9 $
$ 2:4:8:9 $ $ 2:5:7:9 $ $ 2:3:3:5 $ $ 2:3:3:6 $ $ 2:4:4:3 $ $ 2:4:4:5 $ $ 2:4:4:7 $
$ 2:4:4:9 $ $ 2:5:5:3 $ $ 2:5:5:7 $ $ 2:6:6:3 $ $ 2:7:7:5 $ $ 2:7:7:9 $ $ 2:2:3:4 $
$ 2:2:3:5 $ $ 2:2:4:5 $ $ 2:2:4:7 $ $ 2:2:4:9 $ $ 2:2:5:7 $ $ 2:2:7:9 $ $ 3:4:6:7 $
$ 3:4:6:8 $ $ 3:4:6:9 $ $ 3:4:7:8 $ $ 3:5:6:8 $ $ 3:5:6:9 $ $ 3:6:7:9 $ $ 3:6:8:9 $
$ 3:4:4:7 $ $ 3:4:4:8 $ $ 3:5:5:8 $ $ 3:6:6:7 $ $ 3:6:6:8 $ $ 3:3:4:6 $ $ 3:3:4:7 $
$ 3:3:5:6 $ $ 3:3:5:8 $ $ 3:3:6:7 $ $ 3:3:6:8 $ $ 4:5:8:9 $ $ 4:5:5:9 $ $ 4:8:8:9 $
$ 4:4:5:8 $ $ 4:4:5:9 $ $ 4:4:7:8 $ $ 4:4:8:9 $
Table 2.  Second order genuine resonances table with $ |\omega_j|<10, \; j = 1, \; 2, \; 3, \; 4 $
$ 1:2:5:8 $ $ 1:2:5:9 $ $ 1:3:5:7 $ $ 1:3:5:9 $ $ 1:3:7:9 $ $ 1:4:5:7 $ $ 1:4:6:7 $
$ 1:4:6:8 $ $ 1:4:6:9 $ $ 1:4:7:9 $ $ 1:5:7:9 $ $ 1:3:3:5 $ $ 1:3:3:7 $ $ 1:4:4:6 $
$ 1:4:4:7 $ $ 1:4:4:9 $ $ 1:5:5:2 $ $ 1:5:5:3 $ $ 1:5:5:7 $ $ 1:5:5:9 $ $ 1:6:6:4 $
$ 1:6:6:8 $ $ 1:7:7:3 $ $ 1:7:7:4 $ $ 1:7:7:5 $ $ 1:7:7:9 $ $ 1:8:8:6 $ $ 1:9:9:4 $
$ 1:9:9:5 $ $ 1:9:9:7 $ $ 1:3:3:3 $ $ 1:4:4:4 $ $ 1:5:5:5 $ $ 1:6:6:6 $ $ 1:7:7:7 $
$ 1:8:8:8 $ $ 1:9:9:9 $ $ 1:1:3:5 $ $ 1:1:3:6 $ $ 1:1:3:7 $ $ 1:1:3:8 $ $ 1:1:3:9 $
$ 1:1:4:6 $ $ 1:1:4:7 $ $ 1:1:4:9 $ $ 1:1:5:7 $ $ 1:1:5:9 $ $ 1:1:6:8 $ $ 1:1:7:9 $
$ 1:3:1:3 $ $ 1:4:1:4 $ $ 1:5:1:5 $ $ 1:6:1:6 $ $ 1:7:1:7 $ $ 1:8:1:8 $ $ 1:9:1:9 $
$ 1:1:1:3 $ $ 1:1:1:4 $ $ 1:1:1:5 $ $ 1:1:1:6 $ $ 1:1:1:7 $ $ 1:1:1:8 $ $ 1:1:1:9 $
$ 1:1:1:1 $ $ 2:3:7:8 $ $ 2:3:3:4 $ $ 2:3:3:7 $ $ 2:3:3:8 $ $ 2:5:5:8 $ $ 2:5:5:9 $
$ 2:7:7:3 $ $ 2:8:8:3 $ $ 2:3:3:3 $ $ 2:5:5:5 $ $ 2:7:7:7 $ $ 2:9:9:9 $ $ 2:2:3:7 $
$ 2:2:3:8 $ $ 2:2:5:8 $ $ 2:2:5:9 $ $ 2:3:2:3 $ $ 2:5:2:5 $ $ 2:7:2:7 $ $ 2:9:2:9 $
$ 2:2:2:3 $ $ 2:2:2:5 $ $ 2:2:2:9 $ $ 3:4:5:6 $ $ 3:5:6:7 $ $ 3:5:7:9 $ $ 3:4:4:5 $
$ 3:5:5:7 $ $ 3:4:4:4 $ $ 3:5:5:5 $ $ 3:7:7:7 $ $ 3:8:8:8 $ $ 3:3:4:5 $ $ 3:3:5:7 $
$ 3:4:3:4 $ $ 3:5:3:5 $ $ 3:7:3:7 $ $ 3:8:3:8 $ $ 3:3:3:4 $ $ 3:3:3:5 $ $ 3:3:3:7 $
$ 3:3:3:8 $ $ 4:5:6:7 $ $ 4:5:6:8 $ $ 4:6:7:8 $ $ 4:5:5:6 $ $ 4:5:5:5 $ $ 4:7:7:7 $
$ 4:9:9:9 $ $ 4:4:5:6 $ $ 4:5:4:5 $ $ 4:7:4:7 $ $ 4:9:4:9 $ $ 4:4:4:5 $ $ 4:4:4:7 $
$ 4:4:4:9 $ $ 5:6:7:8 $ $ 5:6:6:7 $ $ 5:7:7:9 $ $ 5:6:6:6 $ $ 5:7:7:7 $ $ 5:8:8:8 $
$ 5:9:9:9 $ $ 5:5:6:7 $ $ 5:5:7:9 $ $ 5:6:5:6 $ $ 5:7:5:7 $ $ 5:8:5:8 $ $ 5:9:5:9 $
$ 5:5:5:6 $ $ 5:5:5:7 $ $ 5:5:5:8 $ $ 5:5:5:9 $ $ 6:7:8:9 $ $ 6:7:7:8 $ $ 6:7:7:7 $
$ 6:6:7:8 $ $ 6:7:6:7 $ $ 6:6:6:7 $ $ 7:8:8:9 $ $ 7:8:8:8 $ $ 7:9:9:9 $ $ 7:7:8:9 $
$ 7:8:7:8 $ $ 7:9:7:9 $ $ 7:7:7:8 $ $ 7:7:7:9 $ $ 8:9:9:9 $ $ 8:9:8:9 $ $ 8:8:8:9 $
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$ 1:2:5:8 $ $ 1:2:5:9 $ $ 1:3:5:7 $ $ 1:3:5:9 $ $ 1:3:7:9 $ $ 1:4:5:7 $ $ 1:4:6:7 $
$ 1:4:6:8 $ $ 1:4:6:9 $ $ 1:4:7:9 $ $ 1:5:7:9 $ $ 1:3:3:5 $ $ 1:3:3:7 $ $ 1:4:4:6 $
$ 1:4:4:7 $ $ 1:4:4:9 $ $ 1:5:5:2 $ $ 1:5:5:3 $ $ 1:5:5:7 $ $ 1:5:5:9 $ $ 1:6:6:4 $
$ 1:6:6:8 $ $ 1:7:7:3 $ $ 1:7:7:4 $ $ 1:7:7:5 $ $ 1:7:7:9 $ $ 1:8:8:6 $ $ 1:9:9:4 $
$ 1:9:9:5 $ $ 1:9:9:7 $ $ 1:3:3:3 $ $ 1:4:4:4 $ $ 1:5:5:5 $ $ 1:6:6:6 $ $ 1:7:7:7 $
$ 1:8:8:8 $ $ 1:9:9:9 $ $ 1:1:3:5 $ $ 1:1:3:6 $ $ 1:1:3:7 $ $ 1:1:3:8 $ $ 1:1:3:9 $
$ 1:1:4:6 $ $ 1:1:4:7 $ $ 1:1:4:9 $ $ 1:1:5:7 $ $ 1:1:5:9 $ $ 1:1:6:8 $ $ 1:1:7:9 $
$ 1:3:1:3 $ $ 1:4:1:4 $ $ 1:5:1:5 $ $ 1:6:1:6 $ $ 1:7:1:7 $ $ 1:8:1:8 $ $ 1:9:1:9 $
$ 1:1:1:3 $ $ 1:1:1:4 $ $ 1:1:1:5 $ $ 1:1:1:6 $ $ 1:1:1:7 $ $ 1:1:1:8 $ $ 1:1:1:9 $
$ 1:1:1:1 $ $ 2:3:7:8 $ $ 2:3:3:4 $ $ 2:3:3:7 $ $ 2:3:3:8 $ $ 2:5:5:8 $ $ 2:5:5:9 $
$ 2:7:7:3 $ $ 2:8:8:3 $ $ 2:3:3:3 $ $ 2:5:5:5 $ $ 2:7:7:7 $ $ 2:9:9:9 $ $ 2:2:3:7 $
$ 2:2:3:8 $ $ 2:2:5:8 $ $ 2:2:5:9 $ $ 2:3:2:3 $ $ 2:5:2:5 $ $ 2:7:2:7 $ $ 2:9:2:9 $
$ 2:2:2:3 $ $ 2:2:2:5 $ $ 2:2:2:9 $ $ 3:4:5:6 $ $ 3:5:6:7 $ $ 3:5:7:9 $ $ 3:4:4:5 $
$ 3:5:5:7 $ $ 3:4:4:4 $ $ 3:5:5:5 $ $ 3:7:7:7 $ $ 3:8:8:8 $ $ 3:3:4:5 $ $ 3:3:5:7 $
$ 3:4:3:4 $ $ 3:5:3:5 $ $ 3:7:3:7 $ $ 3:8:3:8 $ $ 3:3:3:4 $ $ 3:3:3:5 $ $ 3:3:3:7 $
$ 3:3:3:8 $ $ 4:5:6:7 $ $ 4:5:6:8 $ $ 4:6:7:8 $ $ 4:5:5:6 $ $ 4:5:5:5 $ $ 4:7:7:7 $
$ 4:9:9:9 $ $ 4:4:5:6 $ $ 4:5:4:5 $ $ 4:7:4:7 $ $ 4:9:4:9 $ $ 4:4:4:5 $ $ 4:4:4:7 $
$ 4:4:4:9 $ $ 5:6:7:8 $ $ 5:6:6:7 $ $ 5:7:7:9 $ $ 5:6:6:6 $ $ 5:7:7:7 $ $ 5:8:8:8 $
$ 5:9:9:9 $ $ 5:5:6:7 $ $ 5:5:7:9 $ $ 5:6:5:6 $ $ 5:7:5:7 $ $ 5:8:5:8 $ $ 5:9:5:9 $
$ 5:5:5:6 $ $ 5:5:5:7 $ $ 5:5:5:8 $ $ 5:5:5:9 $ $ 6:7:8:9 $ $ 6:7:7:8 $ $ 6:7:7:7 $
$ 6:6:7:8 $ $ 6:7:6:7 $ $ 6:6:6:7 $ $ 7:8:8:9 $ $ 7:8:8:8 $ $ 7:9:9:9 $ $ 7:7:8:9 $
$ 7:8:7:8 $ $ 7:9:7:9 $ $ 7:7:7:8 $ $ 7:7:7:9 $ $ 8:9:9:9 $ $ 8:9:8:9 $ $ 8:8:8:9 $
Table 3.  The bracket relations
$\{ \downarrow , \rightarrow \}$ $\pi_1$ $\pi_2$ $\pi_3$ $\pi_4$ $\pi_5$ $\pi_6$ $\pi_7$ $\pi_8$
$\pi_1$ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $\pi_6$ $-\pi_5$ $0$ $0$
$\pi_2$ $0$ $0$ $0$ $0$ $0$ $0$ $\pi_8$ $-\pi_7$
$\pi_3$ $0$ $0$ $0$ $0$ $-\pi_6$ $\pi_5$ $0$ $0$
$\pi_4$ $0$ $0$ $0$ $0$ $0$ $0$ $-\pi_8$ $\pi_7$
$\pi_5$ $-\pi_6$ $0$ $\pi_6$ $0$ $0$ $\frac{1}{2}(\pi_1-\pi_3)$ $0$ $0$
$\pi_6$ $\pi_5$ $0$ $-\pi_5$ $0$ $-\frac{1}{2}(\pi_1-\pi_3)$ $0$ $0$ $0$
$\pi_7$ $0$ $-\pi_8$ $0$ $\pi_8$ $0$ $0$ $0$ $\frac{1}{2}(\pi_2-\pi_4)$
$\pi_8$ $0$ $\pi_7$ $0$ $-\pi_7$ $0$ $0$ $-\frac{1}{2}(\pi_2-\pi_4)$ $0$
$\pi_{9}$ $-2\pi_{10}$ $\pi_{10}$ $0$ $0$ $-\frac{1}{2}\pi_{18}$ $-\frac{1}{2}\pi_{17}$ $\frac{1}{2}\pi_{12}$ $-\frac{1}{2}\pi_{11}$
$\pi_{10}$ $2\pi_9$ $-\pi_9$ $0$ $0$ $\frac{1}{2}\pi_{17}$ $-\frac{1}{2}\pi_{18}$ $-\frac{1}{2}\pi_{11}$ $-\frac{1}{2}\pi_{12}$
$\pi_{11}$ $-2\pi_{12}$ $0$ $0$ $\pi_{12}$ $-\frac{1}{2}\pi_{20}$ $-\frac{1}{2}\pi_{19}$ $\frac{1}{2}\pi_{10}$ $\frac{1}{2}\pi_{9}$
$\pi_{12}$ $2\pi_{11}$ $0$ $0$ $-\pi_{11}$ $\frac{1}{2}\pi_{19}$ $-\frac{1}{2}\pi_{20}$ $-\frac{1}{2}\pi_{9}$ $\frac{1}{2}\pi_{10}$
$\pi_{13}$ $0$ $\pi_{14}$ $-2\pi_{14}$ $0$ $-\frac{1}{2}\pi_{18}$ $\frac{1}{2}\pi_{17}$ $\frac{1}{2}\pi_{16}$ $-\frac{1}{2}\pi_{15}$
$\pi_{14}$ $0$ $-\pi_{13}$ $2\pi_{13}$ $0$ $\frac{1}{2}\pi_{17}$ $\frac{1}{2}\pi_{18}$ $-\frac{1}{2}\pi_{15}$ $-\frac{1}{2}\pi_{16}$
$\pi_{15}$ $0$ $0$ $-2\pi_{16}$ $\pi_{16}$ $-\frac{1}{2}\pi_{20}$ $\frac{1}{2}\pi_{19}$ $\frac{1}{2}\pi_{14}$ $\frac{1}{2}\pi_{13}$
$\pi_{16}$ $0$ $0$ $2\pi_{15}$ $-\pi_{15}$ $\frac{1}{2}\pi_{19}$ $\frac{1}{2}\pi_{20}$ $-\frac{1}{2}\pi_{13}$ $\frac{1}{2}\pi_{14}$
$\{ \downarrow , \rightarrow \}$ $\pi_9$ $\pi_{10}$ $\pi_{11}$ $\pi_{12}$ $\pi_{13}$ $\pi_{14}$ $\pi_{15}$ $\pi_{16}$
$\pi_1$ $2\pi_{10}$ $-2\pi_9$ $2\pi_{12}$ $-2\pi_{11}$ $0$ $0$ $0$ $0$
$\pi_2$ $-\pi_{10}$ $\pi_9$ $0$ $0$ $-\pi_{14}$ $\pi_{13}$ $0$ $0$
$\pi_3$ $0$ $0$ $0$ $0$ $2\pi_{14}$ $-2\pi_{13}$ $2\pi_{16}$ $-2\pi_{15}$
$\pi_4$ $0$ $0$ $-\pi_{12}$ $\pi_{11}$ $0$ $0$ $-\pi_{16}$ $\pi_{15}$
$\pi_5$ $\frac{1}{2}\pi_{18}$ $-\frac{1}{2}\pi_{17}$ $\frac{1}{2}\pi_{20}$ $-\frac{1}{2}\pi_{19}$ $\frac{1}{2}\pi_{18}$ $-\frac{1}{2}\pi_{17}$ $\frac{1}{2}\pi_{20}$ $-\frac{1}{2}\pi_{19}$
$\pi_6$ $\frac{1}{2}\pi_{17}$ $\frac{1}{2}\pi_{18}$ $\frac{1}{2}\pi_{19}$ $\frac{1}{2}\pi_{20}$ $-\frac{1}{2}\pi_{17}$ $-\frac{1}{2}\pi_{18}$ $-\frac{1}{2}\pi_{19}$ $-\frac{1}{2}\pi_{20}$
$\pi_7$ $-\frac{1}{2}\pi_{12}$ $\frac{1}{2}\pi_{11}$ $-\frac{1}{2}\pi_{10}$ $\frac{1}{2}\pi_{9}$ $-\frac{1}{2}\pi_{16}$ $\frac{1}{2}\pi_{17}$ $-\frac{1}{2}\pi_{14}$ $\frac{1}{2}\pi_{13}$
$\pi_8$ $\frac{1}{2}\pi_{11}$ $\frac{1}{2}\pi_{12}$ $-\frac{1}{2}\pi_{9}$ $-\frac{1}{2}\pi_{10}$ $\frac{1}{2}\pi_{15}$ $\frac{1}{2}\pi_{16}$ $-\frac{1}{2}\pi_{13}$ $-\frac{1}{2}\pi_{14}$
$\pi_{9}$ $0$ $\pi_1(\pi_1-4\pi_2)$ $4 \pi_1 \pi_8$ $-4 \pi_1 \pi_7$ $2 \pi_5 \pi_6$ $(\pi_5^{2}-\pi_6^{2})$ $0$ $0$
$\pi_{10}$ $-\pi_1(\pi_1-4\pi_2)$ $0$ $4 \pi_1 \pi_7$ $4 \pi_1 \pi_8$ $-(\pi_5^{2}-\pi_6^{2})$ $2 \pi_5 \pi_6$ $0$ $0$
$\pi_{11}$ $-4\pi_1 \pi_8$ $-4\pi_1 \pi_7$ $0$ $\pi_1(\pi_1-4\pi_4)$ $0$ $0$ $2 \pi_5 \pi_6$ $(\pi_5^{2}-\pi_6^{2})$
$\pi_{12}$ $4\pi_1 \pi_7$ $-4\pi_1 \pi_8$ $-\pi_1(\pi_1-4\pi_4)$ $0$ $0$ $0$ $-(\pi_5^{2}-\pi_6^{2})$ $2 \pi_5 \pi_6$
$\pi_{13}$ $-2 \pi_5 \pi_6$ $(\pi_5^{2}-\pi_6^{2})$ $0$ $0$ $0$ $-\pi_3(4\pi_2-\pi_3)$ $4\pi_3 \pi_8$ $-4\pi_3 \pi_7$
$\pi_{14}$ $-(\pi_5^{2}-\pi_6^{2})$ $-2 \pi_5 \pi_6$ $0$ $0$ $\pi_3(4\pi_2-\pi_3)$ $0$ $4\pi_3 \pi_7$ $4\pi_3 \pi_8$
$\pi_{15}$ $0$ $0$ $-2 \pi_5 \pi_6$ $(\pi_5^{2}-\pi_6^{2})$ $-4\pi_3 \pi_8$ $-4\pi_3 \pi_7$ $0$ $-\pi_3(4\pi_4-\pi_3)$
$\pi_{16}$ $0$ $0$ $-(\pi_5^{2}-\pi_6^{2})$ $-2 \pi_5 \pi_6$ $4\pi_3 \pi_7$ $-4\pi_3 \pi_8$ $\pi_3(4\pi_4-\pi_3)$ $0$
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$\{ \downarrow , \rightarrow \}$ $\pi_1$ $\pi_2$ $\pi_3$ $\pi_4$ $\pi_5$ $\pi_6$ $\pi_7$ $\pi_8$
$\pi_1$ $ 0 $ $ 0 $ $ 0 $ $ 0 $ $\pi_6$ $-\pi_5$ $0$ $0$
$\pi_2$ $0$ $0$ $0$ $0$ $0$ $0$ $\pi_8$ $-\pi_7$
$\pi_3$ $0$ $0$ $0$ $0$ $-\pi_6$ $\pi_5$ $0$ $0$
$\pi_4$ $0$ $0$ $0$ $0$ $0$ $0$ $-\pi_8$ $\pi_7$
$\pi_5$ $-\pi_6$ $0$ $\pi_6$ $0$ $0$ $\frac{1}{2}(\pi_1-\pi_3)$ $0$ $0$
$\pi_6$ $\pi_5$ $0$ $-\pi_5$ $0$ $-\frac{1}{2}(\pi_1-\pi_3)$ $0$ $0$ $0$
$\pi_7$ $0$ $-\pi_8$ $0$ $\pi_8$ $0$ $0$ $0$ $\frac{1}{2}(\pi_2-\pi_4)$
$\pi_8$ $0$ $\pi_7$ $0$ $-\pi_7$ $0$ $0$ $-\frac{1}{2}(\pi_2-\pi_4)$ $0$
$\pi_{9}$ $-2\pi_{10}$ $\pi_{10}$ $0$ $0$ $-\frac{1}{2}\pi_{18}$ $-\frac{1}{2}\pi_{17}$ $\frac{1}{2}\pi_{12}$ $-\frac{1}{2}\pi_{11}$
$\pi_{10}$ $2\pi_9$ $-\pi_9$ $0$ $0$ $\frac{1}{2}\pi_{17}$ $-\frac{1}{2}\pi_{18}$ $-\frac{1}{2}\pi_{11}$ $-\frac{1}{2}\pi_{12}$
$\pi_{11}$ $-2\pi_{12}$ $0$ $0$ $\pi_{12}$ $-\frac{1}{2}\pi_{20}$ $-\frac{1}{2}\pi_{19}$ $\frac{1}{2}\pi_{10}$ $\frac{1}{2}\pi_{9}$
$\pi_{12}$ $2\pi_{11}$ $0$ $0$ $-\pi_{11}$ $\frac{1}{2}\pi_{19}$ $-\frac{1}{2}\pi_{20}$ $-\frac{1}{2}\pi_{9}$ $\frac{1}{2}\pi_{10}$
$\pi_{13}$ $0$ $\pi_{14}$ $-2\pi_{14}$ $0$ $-\frac{1}{2}\pi_{18}$ $\frac{1}{2}\pi_{17}$ $\frac{1}{2}\pi_{16}$ $-\frac{1}{2}\pi_{15}$
$\pi_{14}$ $0$ $-\pi_{13}$ $2\pi_{13}$ $0$ $\frac{1}{2}\pi_{17}$ $\frac{1}{2}\pi_{18}$ $-\frac{1}{2}\pi_{15}$ $-\frac{1}{2}\pi_{16}$
$\pi_{15}$ $0$ $0$ $-2\pi_{16}$ $\pi_{16}$ $-\frac{1}{2}\pi_{20}$ $\frac{1}{2}\pi_{19}$ $\frac{1}{2}\pi_{14}$ $\frac{1}{2}\pi_{13}$
$\pi_{16}$ $0$ $0$ $2\pi_{15}$ $-\pi_{15}$ $\frac{1}{2}\pi_{19}$ $\frac{1}{2}\pi_{20}$ $-\frac{1}{2}\pi_{13}$ $\frac{1}{2}\pi_{14}$
$\{ \downarrow , \rightarrow \}$ $\pi_9$ $\pi_{10}$ $\pi_{11}$ $\pi_{12}$ $\pi_{13}$ $\pi_{14}$ $\pi_{15}$ $\pi_{16}$
$\pi_1$ $2\pi_{10}$ $-2\pi_9$ $2\pi_{12}$ $-2\pi_{11}$ $0$ $0$ $0$ $0$
$\pi_2$ $-\pi_{10}$ $\pi_9$ $0$ $0$ $-\pi_{14}$ $\pi_{13}$ $0$ $0$
$\pi_3$ $0$ $0$ $0$ $0$ $2\pi_{14}$ $-2\pi_{13}$ $2\pi_{16}$ $-2\pi_{15}$
$\pi_4$ $0$ $0$ $-\pi_{12}$ $\pi_{11}$ $0$ $0$ $-\pi_{16}$ $\pi_{15}$
$\pi_5$ $\frac{1}{2}\pi_{18}$ $-\frac{1}{2}\pi_{17}$ $\frac{1}{2}\pi_{20}$ $-\frac{1}{2}\pi_{19}$ $\frac{1}{2}\pi_{18}$ $-\frac{1}{2}\pi_{17}$ $\frac{1}{2}\pi_{20}$ $-\frac{1}{2}\pi_{19}$
$\pi_6$ $\frac{1}{2}\pi_{17}$ $\frac{1}{2}\pi_{18}$ $\frac{1}{2}\pi_{19}$ $\frac{1}{2}\pi_{20}$ $-\frac{1}{2}\pi_{17}$ $-\frac{1}{2}\pi_{18}$ $-\frac{1}{2}\pi_{19}$ $-\frac{1}{2}\pi_{20}$
$\pi_7$ $-\frac{1}{2}\pi_{12}$ $\frac{1}{2}\pi_{11}$ $-\frac{1}{2}\pi_{10}$ $\frac{1}{2}\pi_{9}$ $-\frac{1}{2}\pi_{16}$ $\frac{1}{2}\pi_{17}$ $-\frac{1}{2}\pi_{14}$ $\frac{1}{2}\pi_{13}$
$\pi_8$ $\frac{1}{2}\pi_{11}$ $\frac{1}{2}\pi_{12}$ $-\frac{1}{2}\pi_{9}$ $-\frac{1}{2}\pi_{10}$ $\frac{1}{2}\pi_{15}$ $\frac{1}{2}\pi_{16}$ $-\frac{1}{2}\pi_{13}$ $-\frac{1}{2}\pi_{14}$
$\pi_{9}$ $0$ $\pi_1(\pi_1-4\pi_2)$ $4 \pi_1 \pi_8$ $-4 \pi_1 \pi_7$ $2 \pi_5 \pi_6$ $(\pi_5^{2}-\pi_6^{2})$ $0$ $0$
$\pi_{10}$ $-\pi_1(\pi_1-4\pi_2)$ $0$ $4 \pi_1 \pi_7$ $4 \pi_1 \pi_8$ $-(\pi_5^{2}-\pi_6^{2})$ $2 \pi_5 \pi_6$ $0$ $0$
$\pi_{11}$ $-4\pi_1 \pi_8$ $-4\pi_1 \pi_7$ $0$ $\pi_1(\pi_1-4\pi_4)$ $0$ $0$ $2 \pi_5 \pi_6$ $(\pi_5^{2}-\pi_6^{2})$
$\pi_{12}$ $4\pi_1 \pi_7$ $-4\pi_1 \pi_8$ $-\pi_1(\pi_1-4\pi_4)$ $0$ $0$ $0$ $-(\pi_5^{2}-\pi_6^{2})$ $2 \pi_5 \pi_6$
$\pi_{13}$ $-2 \pi_5 \pi_6$ $(\pi_5^{2}-\pi_6^{2})$ $0$ $0$ $0$ $-\pi_3(4\pi_2-\pi_3)$ $4\pi_3 \pi_8$ $-4\pi_3 \pi_7$
$\pi_{14}$ $-(\pi_5^{2}-\pi_6^{2})$ $-2 \pi_5 \pi_6$ $0$ $0$ $\pi_3(4\pi_2-\pi_3)$ $0$ $4\pi_3 \pi_7$ $4\pi_3 \pi_8$
$\pi_{15}$ $0$ $0$ $-2 \pi_5 \pi_6$ $(\pi_5^{2}-\pi_6^{2})$ $-4\pi_3 \pi_8$ $-4\pi_3 \pi_7$ $0$ $-\pi_3(4\pi_4-\pi_3)$
$\pi_{16}$ $0$ $0$ $-(\pi_5^{2}-\pi_6^{2})$ $-2 \pi_5 \pi_6$ $4\pi_3 \pi_7$ $-4\pi_3 \pi_8$ $\pi_3(4\pi_4-\pi_3)$ $0$
Table 4.  The manifolds of equilibria of type OEE
No. Relative Equilibria Features Conditions and Parameters
ⅰ $\begin{array}{l}(\alpha, 0, \frac{\sqrt{a_3a_7}}{a_7}\alpha, \mp \frac{a_7}{a_5} \gamma, -\frac{a_7}{a_5}\beta, \pm \gamma, \beta), \\(\alpha, 0, - \frac{\sqrt{a_3a_7}}{a_7}\alpha, \mp \frac{a_7}{a_5} \gamma, -\frac{a_7}{a_5}\beta, \pm \gamma, \beta)\end{array}$ $\begin{array}{l}\pi_{10}\pi_{12}\neq0, \\ \pi_5=0, \\ \pi_6 \neq 0\end{array}$ $\begin{array}{l}a_3a_7>0, \\{(a_3a_5^2\alpha+2a_7^3)[2a_5^2a_7\alpha^2(2\eta-\alpha)-(a_3a_5^2\alpha+2a_7^3)\beta^2]}\geq0, \\ \gamma=\frac{\sqrt{(a_3a_5^2\alpha+2a_7^3)[2a_5^2a_7\alpha^2(2\eta-\alpha)-(a_3a_5^2\alpha+2a_7^3)\beta^2]}}{a_3a_5^2\alpha+2a_7^3}, \\ \forall~\alpha, ~\beta\end{array}$
ⅱ $\begin{array}{l}(\alpha, \frac{\sqrt{-a_3a_7}}{a_7}\alpha, 0, \mp \frac{a_7}{a_5} \gamma, -\frac{a_7}{a_5}\beta, \pm \gamma, \beta), \\(\alpha, - \frac{\sqrt{-a_3a_7}}{a_7}\alpha, 0, \mp \frac{a_7}{a_5} \gamma, -\frac{a_7}{a_5}\beta, \pm \gamma, \beta)\end{array}$ $\begin{array}{l}\pi_{10}\pi_{12}\neq0, \\ \pi_5 \neq 0, \\ \pi_6=0, \\ \pi_1 \neq \frac{-2a_7^3}{a_3a_5^2}, \\{(a_3a_5^2\alpha+2a_7^3)[2a_5^2a_7\alpha^2(2\eta-\alpha)-(a_3a_5^2\alpha+2a_7^3)\beta^2]}\geq0\end{array}$ $\begin{array}{l}a_3a_7<0, \\ \gamma=\frac{\sqrt{(a_3a_5^2\alpha+2a_7^3)[2a_5^2a_7\alpha^2(2\eta-\alpha)-(a_3a_5^2\alpha+2a_7^3)\beta^2]}}{a_3a_5^2\alpha+2a_7^3}, \\ \forall~\alpha, ~\beta\end{array}$
ⅲ $(\frac{-2a_7^3}{a_3a_5^2}, \pm 2a_7^2 \frac{\sqrt{-a_3a_7}}{a_3a_5^2}, 0, 0, -\frac{a_7}{a_5}\beta, 0, \beta)$ $\begin{array}{l}\pi_{10}\pi_{12}\neq0, \\ \pi_5 \neq 0, \\ \pi_6=0, \\ \pi_1 = \frac{-2a_7^3}{a_3a_5^2}\end{array}$ $\begin{array}{l}a_3a_7<0, \\ \forall~\beta\end{array}$
ⅵ $(\varrho, \gamma_1, \alpha, \gamma_2, -\frac{a_3}{a_1}\gamma_3, \gamma_4, \gamma_3)$ $\begin{array}{l}\pi_{10}\pi_{12}\neq0, \\ \pi_5 \neq 0, \\ \pi_6 \neq 0\end{array}$ $\begin{array}{l}|a_5\alpha|\leq|a_3|\varrho, \\ \varrho=\frac{4}{3}\frac{\eta a_5^2}{a_5^2+a_3^2}, \\ \gamma_1 = \pm \frac{\sqrt{a_3^2\varrho^2-a_5^2\alpha^2}}{a_5}, \\ \gamma_2 = \frac{2 a_5 \gamma_4 \pm \sqrt{2\varrho^3(a_5^2+a_7^2)}}{2 a_7}, \\ \gamma_3 = \mp \frac{a_5a_7 \sqrt{2} \gamma_1 \alpha}{a_3 \sqrt{\varrho(a_5^2+a_7^2)}}, \\ \gamma_4 = \pm \frac{2a_5a_7\alpha^2-a_3(a_5-a_1)\varrho^2}{a_3\sqrt{2\varrho(a_5^2+a_7^2)}}, \\ \forall~\alpha\end{array}$
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No. Relative Equilibria Features Conditions and Parameters
ⅰ $\begin{array}{l}(\alpha, 0, \frac{\sqrt{a_3a_7}}{a_7}\alpha, \mp \frac{a_7}{a_5} \gamma, -\frac{a_7}{a_5}\beta, \pm \gamma, \beta), \\(\alpha, 0, - \frac{\sqrt{a_3a_7}}{a_7}\alpha, \mp \frac{a_7}{a_5} \gamma, -\frac{a_7}{a_5}\beta, \pm \gamma, \beta)\end{array}$ $\begin{array}{l}\pi_{10}\pi_{12}\neq0, \\ \pi_5=0, \\ \pi_6 \neq 0\end{array}$ $\begin{array}{l}a_3a_7>0, \\{(a_3a_5^2\alpha+2a_7^3)[2a_5^2a_7\alpha^2(2\eta-\alpha)-(a_3a_5^2\alpha+2a_7^3)\beta^2]}\geq0, \\ \gamma=\frac{\sqrt{(a_3a_5^2\alpha+2a_7^3)[2a_5^2a_7\alpha^2(2\eta-\alpha)-(a_3a_5^2\alpha+2a_7^3)\beta^2]}}{a_3a_5^2\alpha+2a_7^3}, \\ \forall~\alpha, ~\beta\end{array}$
ⅱ $\begin{array}{l}(\alpha, \frac{\sqrt{-a_3a_7}}{a_7}\alpha, 0, \mp \frac{a_7}{a_5} \gamma, -\frac{a_7}{a_5}\beta, \pm \gamma, \beta), \\(\alpha, - \frac{\sqrt{-a_3a_7}}{a_7}\alpha, 0, \mp \frac{a_7}{a_5} \gamma, -\frac{a_7}{a_5}\beta, \pm \gamma, \beta)\end{array}$ $\begin{array}{l}\pi_{10}\pi_{12}\neq0, \\ \pi_5 \neq 0, \\ \pi_6=0, \\ \pi_1 \neq \frac{-2a_7^3}{a_3a_5^2}, \\{(a_3a_5^2\alpha+2a_7^3)[2a_5^2a_7\alpha^2(2\eta-\alpha)-(a_3a_5^2\alpha+2a_7^3)\beta^2]}\geq0\end{array}$ $\begin{array}{l}a_3a_7<0, \\ \gamma=\frac{\sqrt{(a_3a_5^2\alpha+2a_7^3)[2a_5^2a_7\alpha^2(2\eta-\alpha)-(a_3a_5^2\alpha+2a_7^3)\beta^2]}}{a_3a_5^2\alpha+2a_7^3}, \\ \forall~\alpha, ~\beta\end{array}$
ⅲ $(\frac{-2a_7^3}{a_3a_5^2}, \pm 2a_7^2 \frac{\sqrt{-a_3a_7}}{a_3a_5^2}, 0, 0, -\frac{a_7}{a_5}\beta, 0, \beta)$ $\begin{array}{l}\pi_{10}\pi_{12}\neq0, \\ \pi_5 \neq 0, \\ \pi_6=0, \\ \pi_1 = \frac{-2a_7^3}{a_3a_5^2}\end{array}$ $\begin{array}{l}a_3a_7<0, \\ \forall~\beta\end{array}$
ⅵ $(\varrho, \gamma_1, \alpha, \gamma_2, -\frac{a_3}{a_1}\gamma_3, \gamma_4, \gamma_3)$ $\begin{array}{l}\pi_{10}\pi_{12}\neq0, \\ \pi_5 \neq 0, \\ \pi_6 \neq 0\end{array}$ $\begin{array}{l}|a_5\alpha|\leq|a_3|\varrho, \\ \varrho=\frac{4}{3}\frac{\eta a_5^2}{a_5^2+a_3^2}, \\ \gamma_1 = \pm \frac{\sqrt{a_3^2\varrho^2-a_5^2\alpha^2}}{a_5}, \\ \gamma_2 = \frac{2 a_5 \gamma_4 \pm \sqrt{2\varrho^3(a_5^2+a_7^2)}}{2 a_7}, \\ \gamma_3 = \mp \frac{a_5a_7 \sqrt{2} \gamma_1 \alpha}{a_3 \sqrt{\varrho(a_5^2+a_7^2)}}, \\ \gamma_4 = \pm \frac{2a_5a_7\alpha^2-a_3(a_5-a_1)\varrho^2}{a_3\sqrt{2\varrho(a_5^2+a_7^2)}}, \\ \forall~\alpha\end{array}$
Table 5.  Equilibria with π10 = π12 = 0
No. Relative Equilibria Features Conditions and Parameters Types
ⅰ $(\frac{4}{3}\eta, 0, 0, \pm\frac{4a_1 \eta\sqrt{6(a_1^2+a_3^2)\eta}}{9(a_1^2+a_3^2)}, 0, \pm\frac{4a_3 \sqrt{6} \eta^{2}}{9 \sqrt{(a_1^2+a_3^2) \eta}}, 0)$ $\begin{array}{l}\pi_5=\pi_6=0, \\\pi_{10}= \pi_{12}=0\end{array}$ − $\begin{array}{l}EEE, ~EEH, \\EEO\end{array}$
ⅱ $\begin{array}{l}e_j= \big( \frac{4}{3}\frac{\eta \alpha_1}{\beta_3}, \pm\frac{4}{3} \eta\frac{\sqrt{\alpha_1\alpha_2}}{\alpha_3} , 0, +\frac{4 \sqrt{6}}{9}\frac{(a_3-a_7)\alpha_1 \eta }{\alpha_3\sqrt{\alpha_3}}, 0, -\frac{4 \sqrt{6}}{9}\frac{(a_1-a_5)\alpha_1 \eta }{\alpha_3\sqrt{\alpha_3}}, 0\big), ~~ j = 1, 2\\e_j= \big( \frac{4}{3}\frac{\eta \alpha_1}{\alpha_3}, \pm\frac{4}{3} \eta\frac{\sqrt{\alpha_1\alpha_2}}{\alpha_3} , 0, -\frac{4 \sqrt{6}}{9}\frac{(a_3-a_7)\alpha_1 \eta }{\alpha_3\sqrt{\alpha_3}}, 0, +\frac{4 \sqrt{6}}{9}\frac{(a_1-a_5)\alpha_1 \eta }{\alpha_3\sqrt{\alpha_3}}, 0\big), ~~ j = 3, 4 \end{array}$ $\begin{array}{l}\pi_{10}= \pi_{12}=0, \\\pi_6=0, \\\pi_9(a_1-a_5)+\pi_{11}(a_3-a_7)=0\end{array}$ $\begin{array}{l}a_1 \neq a_5, \\a_3 \neq a_7, \\\alpha_1 = a_5^2+a_7^2\\-a_1a_5-a_3a_7\geq0, \\\alpha_2 = a_1^2+a_3^2\\-a_1a_5-a_3a_7\geq0, \\\alpha_3 = (a_1-a_5)^2\\+(a_3-a_7)^2>0\end{array}$ $\begin{array}{l}EHH, ~EEE, \\EHE, ~EOH, \\EOE, ~EOO, \\OOO\end{array}$
ⅲ $(\frac{4}{3}\frac{\eta a_7}{(a_7-a_3)}, \pm\frac{4}{3} \frac{\sqrt{-a_3a_7}\eta}{(a_7-a_3)}, 0, \pm\frac{4}{9}\frac{\sqrt{6 \eta a_7^2 }\eta}{ (a_7-a_3)}, 0, 0, 0)$ $\begin{array}{l}\pi_{10}= \pi_{12}=0, \\\pi_6=0, \\~\pi_9(a_1-a_5)+\pi_{11}(a_3-a_7)=0\end{array}$ $\begin{array}{l}a_1 = a_5, \\a_3 \neq a_7, \\a_3a_7<0\end{array}$ $EEH, ~EHE$
ⅳ $(\frac{4}{3}\frac{\eta a_5}{(a_5-a_1)}, \pm\frac{4}{3} \frac{\sqrt{-a_1a_5}\eta}{(a_5-a_1)}, 0, \pm\frac{4}{9}\frac{\sqrt{6 \eta a_5^2 }\eta}{ (a_5-a_1)}, 0, 0, 0)$ $\begin{array}{l}\pi_{10}= \pi_{12}=0, \\\pi_6=0, \\\pi_9(a_1-a_5)+\pi_{11}(a_3-a_7)=0\end{array}$ $\begin{array}{l}a_1 \neq a_5, \\a_3 =a_7, \\a_1a_5<0\end{array}$ $EEE, ~EHE$
ⅴ $\begin{array}{l}e_j= \big( \frac{4}{3}\frac{\eta \alpha_1}{\beta_3}, \pm\frac{4}{3} \eta\frac{\sqrt{\alpha_1\alpha_2}}{\alpha_3} , 0, +\frac{4 \sqrt{6}}{9}\frac{(a_3+a_7)\alpha_1 \eta }{\alpha_3\sqrt{\alpha_3}}, 0, -\frac{4 \sqrt{6}}{9}\frac{(a_1+a_5)\alpha_1 \eta }{\alpha_3\sqrt{\alpha_3}}, 0\big), ~~ j = 1, 2\\e_j= \big( \frac{4}{3}\frac{\eta \alpha_1}{\alpha_3}, \pm\frac{4}{3} \eta\frac{\sqrt{\alpha_1\alpha_2}}{\alpha_3} , 0, -\frac{4 \sqrt{6}}{9}\frac{(a_3+a_7)\alpha_1 \eta }{\alpha_3\sqrt{\alpha_3}}, 0, +\frac{4 \sqrt{6}}{9}\frac{(a_1+a_5)\alpha_1 \eta }{\alpha_3\sqrt{\alpha_3}}, 0\big), ~~ j = 3, 4\end{array}$ $\begin{array}{l}\pi_{10}= \pi_{12}=0, \\\pi_5=0, \\\pi_9(a_1+a_5)+\pi_{11}(a_3+a_7)=0\end{array}$ $\begin{array}{l}a_1 \neq -a_5, \\a_3 \neq -a_7, \\\alpha_1 = a_5^2+a_7^2\\+a_1a_5+a_3a_7 \geq 0, \\\alpha_2 = a_1^2+a_3^2\\+a_1a_5+a_3a_7 \geq 0, \\\alpha_3 = (a_1+a_5)^2\\+(a_3+a_7)^2 >0\end{array}$ $\begin{array}{l}EHH, ~EEE, \\EHE, ~EOH, \\EOE, ~EOO, \\OOO\end{array}$
ⅵ $(\frac{4}{3}\frac{\eta a_7}{(a_7+a_3)}, 0, \pm\frac{4}{3} \frac{\sqrt{a_3a_7}\eta}{(a_7+a_3)}, \pm\frac{4}{9}\frac{\sqrt{6 \eta a_7^2 }\eta}{ (a_7+a_3)}, 0, 0, 0)$ $\begin{array}{l}\pi_{10}= \pi_{12}=0, \\\pi_5=0, \\\pi_9(a_1+a_5)+\pi_{11}(a_3+a_7)=0\end{array}$ $\begin{array}{l}a_1 = -a_5, \\a_3 \neq -a_7, \\a_3a_7>0\end{array}$ $EHE, ~EEE$
ⅶ $(\frac{4}{3}\frac{\eta a_5}{(a_1+a_5)}, 0, \pm\frac{4}{3} \frac{\sqrt{a_1a_5}\eta}{(a_1+a_5)}, \pm\frac{4}{9}\frac{\sqrt{6 \eta a_5^2 }\eta}{ (a_5+a_1)}, 0, 0, 0)$ $\begin{array}{l}\pi_{10}= \pi_{12}=0, \\\pi_5=0, \\\pi_9(a_1+a_5)+\pi_{11}(a_3+a_7)=0\end{array}$ $\begin{array}{l}a_1 \neq -a_5, \\a_3=-a_7, \\a_1a_5>0\end{array}$ $EHE, ~EEE$
ⅷ $\begin{array}{l}(\frac{2\eta a_5}{(a_1+a_5)}, 0, \pm \frac{2\eta \sqrt{a_1a_5}}{(a_1+a_5)}, 0, 0, 0, 0), \\(-\frac{2\eta a_5}{(a_1-a_5)}, \pm \frac{2\eta \sqrt{-a_5a_1}}{(a_1-a_5)}, 0, 0, 0, 0, 0)\end{array}$ $\begin{array}{l}\pi_{10}= \pi_{12}=0, \\\pi_9(a_1+a_5)+\pi_{11}(a_3+a_7)=0, \\\pi_9(a_1-a_5)+\pi_{11}(a_3-a_7)=0\end{array}$ − $OEE$
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No. Relative Equilibria Features Conditions and Parameters Types
ⅰ $(\frac{4}{3}\eta, 0, 0, \pm\frac{4a_1 \eta\sqrt{6(a_1^2+a_3^2)\eta}}{9(a_1^2+a_3^2)}, 0, \pm\frac{4a_3 \sqrt{6} \eta^{2}}{9 \sqrt{(a_1^2+a_3^2) \eta}}, 0)$ $\begin{array}{l}\pi_5=\pi_6=0, \\\pi_{10}= \pi_{12}=0\end{array}$ − $\begin{array}{l}EEE, ~EEH, \\EEO\end{array}$
ⅱ $\begin{array}{l}e_j= \big( \frac{4}{3}\frac{\eta \alpha_1}{\beta_3}, \pm\frac{4}{3} \eta\frac{\sqrt{\alpha_1\alpha_2}}{\alpha_3} , 0, +\frac{4 \sqrt{6}}{9}\frac{(a_3-a_7)\alpha_1 \eta }{\alpha_3\sqrt{\alpha_3}}, 0, -\frac{4 \sqrt{6}}{9}\frac{(a_1-a_5)\alpha_1 \eta }{\alpha_3\sqrt{\alpha_3}}, 0\big), ~~ j = 1, 2\\e_j= \big( \frac{4}{3}\frac{\eta \alpha_1}{\alpha_3}, \pm\frac{4}{3} \eta\frac{\sqrt{\alpha_1\alpha_2}}{\alpha_3} , 0, -\frac{4 \sqrt{6}}{9}\frac{(a_3-a_7)\alpha_1 \eta }{\alpha_3\sqrt{\alpha_3}}, 0, +\frac{4 \sqrt{6}}{9}\frac{(a_1-a_5)\alpha_1 \eta }{\alpha_3\sqrt{\alpha_3}}, 0\big), ~~ j = 3, 4 \end{array}$ $\begin{array}{l}\pi_{10}= \pi_{12}=0, \\\pi_6=0, \\\pi_9(a_1-a_5)+\pi_{11}(a_3-a_7)=0\end{array}$ $\begin{array}{l}a_1 \neq a_5, \\a_3 \neq a_7, \\\alpha_1 = a_5^2+a_7^2\\-a_1a_5-a_3a_7\geq0, \\\alpha_2 = a_1^2+a_3^2\\-a_1a_5-a_3a_7\geq0, \\\alpha_3 = (a_1-a_5)^2\\+(a_3-a_7)^2>0\end{array}$ $\begin{array}{l}EHH, ~EEE, \\EHE, ~EOH, \\EOE, ~EOO, \\OOO\end{array}$
ⅲ $(\frac{4}{3}\frac{\eta a_7}{(a_7-a_3)}, \pm\frac{4}{3} \frac{\sqrt{-a_3a_7}\eta}{(a_7-a_3)}, 0, \pm\frac{4}{9}\frac{\sqrt{6 \eta a_7^2 }\eta}{ (a_7-a_3)}, 0, 0, 0)$ $\begin{array}{l}\pi_{10}= \pi_{12}=0, \\\pi_6=0, \\~\pi_9(a_1-a_5)+\pi_{11}(a_3-a_7)=0\end{array}$ $\begin{array}{l}a_1 = a_5, \\a_3 \neq a_7, \\a_3a_7<0\end{array}$ $EEH, ~EHE$
ⅳ $(\frac{4}{3}\frac{\eta a_5}{(a_5-a_1)}, \pm\frac{4}{3} \frac{\sqrt{-a_1a_5}\eta}{(a_5-a_1)}, 0, \pm\frac{4}{9}\frac{\sqrt{6 \eta a_5^2 }\eta}{ (a_5-a_1)}, 0, 0, 0)$ $\begin{array}{l}\pi_{10}= \pi_{12}=0, \\\pi_6=0, \\\pi_9(a_1-a_5)+\pi_{11}(a_3-a_7)=0\end{array}$ $\begin{array}{l}a_1 \neq a_5, \\a_3 =a_7, \\a_1a_5<0\end{array}$ $EEE, ~EHE$
ⅴ $\begin{array}{l}e_j= \big( \frac{4}{3}\frac{\eta \alpha_1}{\beta_3}, \pm\frac{4}{3} \eta\frac{\sqrt{\alpha_1\alpha_2}}{\alpha_3} , 0, +\frac{4 \sqrt{6}}{9}\frac{(a_3+a_7)\alpha_1 \eta }{\alpha_3\sqrt{\alpha_3}}, 0, -\frac{4 \sqrt{6}}{9}\frac{(a_1+a_5)\alpha_1 \eta }{\alpha_3\sqrt{\alpha_3}}, 0\big), ~~ j = 1, 2\\e_j= \big( \frac{4}{3}\frac{\eta \alpha_1}{\alpha_3}, \pm\frac{4}{3} \eta\frac{\sqrt{\alpha_1\alpha_2}}{\alpha_3} , 0, -\frac{4 \sqrt{6}}{9}\frac{(a_3+a_7)\alpha_1 \eta }{\alpha_3\sqrt{\alpha_3}}, 0, +\frac{4 \sqrt{6}}{9}\frac{(a_1+a_5)\alpha_1 \eta }{\alpha_3\sqrt{\alpha_3}}, 0\big), ~~ j = 3, 4\end{array}$ $\begin{array}{l}\pi_{10}= \pi_{12}=0, \\\pi_5=0, \\\pi_9(a_1+a_5)+\pi_{11}(a_3+a_7)=0\end{array}$ $\begin{array}{l}a_1 \neq -a_5, \\a_3 \neq -a_7, \\\alpha_1 = a_5^2+a_7^2\\+a_1a_5+a_3a_7 \geq 0, \\\alpha_2 = a_1^2+a_3^2\\+a_1a_5+a_3a_7 \geq 0, \\\alpha_3 = (a_1+a_5)^2\\+(a_3+a_7)^2 >0\end{array}$ $\begin{array}{l}EHH, ~EEE, \\EHE, ~EOH, \\EOE, ~EOO, \\OOO\end{array}$
ⅵ $(\frac{4}{3}\frac{\eta a_7}{(a_7+a_3)}, 0, \pm\frac{4}{3} \frac{\sqrt{a_3a_7}\eta}{(a_7+a_3)}, \pm\frac{4}{9}\frac{\sqrt{6 \eta a_7^2 }\eta}{ (a_7+a_3)}, 0, 0, 0)$ $\begin{array}{l}\pi_{10}= \pi_{12}=0, \\\pi_5=0, \\\pi_9(a_1+a_5)+\pi_{11}(a_3+a_7)=0\end{array}$ $\begin{array}{l}a_1 = -a_5, \\a_3 \neq -a_7, \\a_3a_7>0\end{array}$ $EHE, ~EEE$
ⅶ $(\frac{4}{3}\frac{\eta a_5}{(a_1+a_5)}, 0, \pm\frac{4}{3} \frac{\sqrt{a_1a_5}\eta}{(a_1+a_5)}, \pm\frac{4}{9}\frac{\sqrt{6 \eta a_5^2 }\eta}{ (a_5+a_1)}, 0, 0, 0)$ $\begin{array}{l}\pi_{10}= \pi_{12}=0, \\\pi_5=0, \\\pi_9(a_1+a_5)+\pi_{11}(a_3+a_7)=0\end{array}$ $\begin{array}{l}a_1 \neq -a_5, \\a_3=-a_7, \\a_1a_5>0\end{array}$ $EHE, ~EEE$
ⅷ $\begin{array}{l}(\frac{2\eta a_5}{(a_1+a_5)}, 0, \pm \frac{2\eta \sqrt{a_1a_5}}{(a_1+a_5)}, 0, 0, 0, 0), \\(-\frac{2\eta a_5}{(a_1-a_5)}, \pm \frac{2\eta \sqrt{-a_5a_1}}{(a_1-a_5)}, 0, 0, 0, 0, 0)\end{array}$ $\begin{array}{l}\pi_{10}= \pi_{12}=0, \\\pi_9(a_1+a_5)+\pi_{11}(a_3+a_7)=0, \\\pi_9(a_1-a_5)+\pi_{11}(a_3-a_7)=0\end{array}$ − $OEE$
Table 6.  Equilibria of the reduced system
Equilibria Conditions and Features
$E_1=(\pi_1, 0, 0, 0, 0, 0, 0)$ $\begin{array}{l}\forall~\pi_1\end{array}$
$E_2=(\pi_1, 0, 0, 0, 0, \pi_{11}, \pi_{12})$ $\begin{array}{l}\forall~\pi_1, ~\pi_{11}, ~\pi_{12}~with\\3\gamma\tau+(2\nu_1-\nu_4)\pi_1^2=0\end{array}$
$E_3=(\pi_1, 0, 0, \pi_9, \pi_{10}, 0, 0)$ $\begin{array}{l}\forall~\pi_1, ~\pi_9, ~\pi_{10}~with\\3\gamma\sigma+(2\nu_1-\nu_2)\pi_1^2=0\end{array}$
$E_4=(\pi_1, \pi_5, \pi_6, 0, 0, 0, 0)$ $\begin{array}{l}\forall~\pi_1, ~\pi_5, ~\pi_6~with\\\sigma=0~and~\rho\neq0, ~2\gamma\rho+(\nu_1-\nu_3)\pi_1-2\gamma\pi_1^2=0\end{array}$
$E_5=(\frac{\nu_2-2\nu_3}{9\gamma}, \pi_5, \pi_6, \pi_9, \pi_{10}, \pi_{11}, \pi_{12})$ $\begin{array}{l} \forall~\pi_5, \pi_6, \pi_9, \pi_{10}, \pi_{11}, \pi_{12}~with\\\rho=\pi_1^2\neq0, ~\sigma\neq0~and\\ ~\pi_5\pi_9\pi_{12}-\pi_5\pi_{10}\pi_{11}+\pi_6\pi_9\pi_{11}+\pi_6\pi_{10}\pi_{12}=0\end{array}$
$E_6=(\pi_1, \pi_5, \pi_6, \pi_9, \pi_{10}, \pi_{11}, \pi_{12})$ $\begin{array}{*{20}{l}} {\forall {\pi _1},{\pi _9},{\pi _{10}},{\pi _{11}},{\pi _{12}}with}\\ {\rho = \pi _1^2 \ne 0 , \sigma \ne 0,2\gamma \tau - ({\nu _2} - 2{\nu _3})\pi _1^2 + \gamma \pi _1^3 = 0\;and}\\ {{\pi _5} = \frac{{{\pi _1}({\pi _9}{\pi _{11}} + {\pi _{10}}{\pi _{12}})[3\gamma \tau - ({\nu _4} - 2{\nu _3})\pi _1^2 + 6\gamma \pi _1^3]}}{{4\gamma \sigma (\pi _1^3 - \tau )}},}\\ {{\pi _6} = \frac{{{\pi _1}({\pi _9}{\pi _{12}} - {\pi _{10}}{\pi _{11}})[3\gamma \tau - ({\nu _4} - 2{\nu _3})\pi _1^2 + 6\gamma \pi _1^3]}}{{4\gamma \sigma (\pi _1^3 - \tau )}}} \end{array}$
$E_7=(\pi_1, \pi_5, \pi_6, \pi_9, \pi_{10}, \pi_{11}, \pi_{12})$ $\begin{array}{l}\forall~\pi_1, ~\pi_9, ~\pi_{10}, ~\pi_{11}, ~\pi_{12}~with~\rho=\pi_1^2\neq0, ~\sigma\neq0~and\\ \gamma(3\pi_1^3-\tau^2)\pi_9^4+[4\gamma\pi_1^6-(\nu_2-2\nu_3)\pi_1^5+6\gamma\pi_{10}^2\pi_1^3\\ +\tau(\nu_2-2\nu_3)\pi_1^2]\pi_9^2+4\gamma(\pi_{10}^2-\tau)\pi_1^6+[-(\nu_2-2\nu_3)\pi_{10}^2\\ -(2\nu_3-\nu_4)\tau]\pi_1^5+3\gamma(\pi_{10}^2-\tau)(\pi_{10}^2+\tau)\pi_1^3\\ +\pi_{10}^2\tau(\nu_2-\nu_4)\pi_1^2-\gamma\pi_{10}^2\tau(\pi_{10}^2-\tau)=0~and\\ \pi_5 = \frac{\pi_1(\pi_9\pi_{11}+\pi_{10}\pi_{12})[3\gamma\sigma+2\gamma\tau-(\nu_2-2\nu_3)\pi_1^2+4\gamma\pi_1^3]}{4\gamma\tau(\pi_1^3-\sigma)}, \\ \pi_6 = \frac{\pi_1(\pi_9\pi_{12}-\pi_{10}\pi_{12})[3\gamma\sigma+2\gamma\tau-(\nu_2-2\nu_3)\pi_1^2+4\gamma\pi_1^3]}{4\gamma(\pi_{11}^2+\pi_{12}^2)(\pi_1^3-\sigma)}\end{array}$
$E_8=(\pi_1, \pi_5, \pi_6, \pi_9, \pi_{10}, \pi_{11}, \pi_{12})$ $\begin{array}{l}\forall~\pi_1, ~\pi_9, ~\pi_{10}, ~\pi_{11}, ~\pi_{12}~with~\rho\neq\pi_1^2~and~\sigma\neq0~where\\12\rho\gamma^2\pi_1^8-8\gamma\rho(\nu_1-\nu_3)\pi_1^7+[-24\gamma^2\rho^2+2\gamma\sigma(2\nu_1-\nu_2)\\+\rho(\nu_1-\nu_3)^2]\pi_1^6+8\gamma\rho^2(\nu_1-\nu_3)\pi_1^5+[6(2\rho^3+\sigma^2)\\-4\gamma\rho\sigma(\nu_1-\nu_2+\nu_3)]\pi_1^4-[12\gamma^2\rho\sigma^2+2\gamma\rho^2\sigma(\nu_2-2\nu_3)]\pi_1^2\\+6\gamma^2\rho^2\sigma^2=0, \\72\gamma\rho^3\pi_1^{10}-108\gamma^2\rho(\nu_1-\nu_3)\pi_1^9+[-216\gamma^3\rho^2\\+24\gamma^2\sigma(2\nu_1-\nu_4)+54\gamma\rho(\nu_1-\nu_3)^2]\pi_1^8\\+[216\gamma^2\rho^2(\nu_1-\nu_3)-12\gamma\sigma(2\nu_1-\nu_4)]\pi_1^7\\+[72\gamma^3(3\rho^3+2\sigma^2)-24\gamma^2\rho\sigma(4\nu_1+2\nu_3-3\nu_4)\\-54\gamma\rho^2(\nu_1-\nu_3)^2]\pi_1^6+[-12\gamma^2(9\rho^3+2\sigma^2)\\+24\gamma\rho\sigma(\nu_1+\nu_3-\nu_4)](\nu_1-\nu_3)\pi_1^5\\+[-72\gamma^3(\rho^4+6\rho\sigma^2)-24\gamma^2\rho^2\sigma(2\nu_1+4\nu_3-3\nu_4)]\pi_1^4\\+[48\gamma^2\rho\sigma^2-12\gamma\rho^2\sigma(2\nu_3-\nu_4)](\nu_1-\nu_3)\pi_1^3\\+[432\gamma^3\rho^2\sigma^2-24\gamma^2\rho^3\sigma(2\nu_3-\nu_4)]\pi_1^2\\-24\gamma^2\rho^2\sigma^2(\nu_1-\nu_3)\pi_1-144\gamma^3\rho^3\sigma^2=0~and\\\pi_{11} = -\frac{\pi_1^2(\pi_5\pi_9+\pi_6\pi_{10})[2\gamma\pi_1^2-2\gamma\rho-(\nu_1-\nu_3)\pi_1]}{\gamma(\pi_1^2-\rho)\sigma}, \\\pi_{12} = -\frac{\pi_1^2(\pi_5\pi_{10}-\pi_6\pi_9)[2\gamma\pi_1^2-2\gamma\rho-(\nu_1-\nu_3)\pi_1]}{\gamma(\pi_1^2-\rho)\sigma} \end{array}$
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Equilibria Conditions and Features
$E_1=(\pi_1, 0, 0, 0, 0, 0, 0)$ $\begin{array}{l}\forall~\pi_1\end{array}$
$E_2=(\pi_1, 0, 0, 0, 0, \pi_{11}, \pi_{12})$ $\begin{array}{l}\forall~\pi_1, ~\pi_{11}, ~\pi_{12}~with\\3\gamma\tau+(2\nu_1-\nu_4)\pi_1^2=0\end{array}$
$E_3=(\pi_1, 0, 0, \pi_9, \pi_{10}, 0, 0)$ $\begin{array}{l}\forall~\pi_1, ~\pi_9, ~\pi_{10}~with\\3\gamma\sigma+(2\nu_1-\nu_2)\pi_1^2=0\end{array}$
$E_4=(\pi_1, \pi_5, \pi_6, 0, 0, 0, 0)$ $\begin{array}{l}\forall~\pi_1, ~\pi_5, ~\pi_6~with\\\sigma=0~and~\rho\neq0, ~2\gamma\rho+(\nu_1-\nu_3)\pi_1-2\gamma\pi_1^2=0\end{array}$
$E_5=(\frac{\nu_2-2\nu_3}{9\gamma}, \pi_5, \pi_6, \pi_9, \pi_{10}, \pi_{11}, \pi_{12})$ $\begin{array}{l} \forall~\pi_5, \pi_6, \pi_9, \pi_{10}, \pi_{11}, \pi_{12}~with\\\rho=\pi_1^2\neq0, ~\sigma\neq0~and\\ ~\pi_5\pi_9\pi_{12}-\pi_5\pi_{10}\pi_{11}+\pi_6\pi_9\pi_{11}+\pi_6\pi_{10}\pi_{12}=0\end{array}$
$E_6=(\pi_1, \pi_5, \pi_6, \pi_9, \pi_{10}, \pi_{11}, \pi_{12})$ $\begin{array}{*{20}{l}} {\forall {\pi _1},{\pi _9},{\pi _{10}},{\pi _{11}},{\pi _{12}}with}\\ {\rho = \pi _1^2 \ne 0 , \sigma \ne 0,2\gamma \tau - ({\nu _2} - 2{\nu _3})\pi _1^2 + \gamma \pi _1^3 = 0\;and}\\ {{\pi _5} = \frac{{{\pi _1}({\pi _9}{\pi _{11}} + {\pi _{10}}{\pi _{12}})[3\gamma \tau - ({\nu _4} - 2{\nu _3})\pi _1^2 + 6\gamma \pi _1^3]}}{{4\gamma \sigma (\pi _1^3 - \tau )}},}\\ {{\pi _6} = \frac{{{\pi _1}({\pi _9}{\pi _{12}} - {\pi _{10}}{\pi _{11}})[3\gamma \tau - ({\nu _4} - 2{\nu _3})\pi _1^2 + 6\gamma \pi _1^3]}}{{4\gamma \sigma (\pi _1^3 - \tau )}}} \end{array}$
$E_7=(\pi_1, \pi_5, \pi_6, \pi_9, \pi_{10}, \pi_{11}, \pi_{12})$ $\begin{array}{l}\forall~\pi_1, ~\pi_9, ~\pi_{10}, ~\pi_{11}, ~\pi_{12}~with~\rho=\pi_1^2\neq0, ~\sigma\neq0~and\\ \gamma(3\pi_1^3-\tau^2)\pi_9^4+[4\gamma\pi_1^6-(\nu_2-2\nu_3)\pi_1^5+6\gamma\pi_{10}^2\pi_1^3\\ +\tau(\nu_2-2\nu_3)\pi_1^2]\pi_9^2+4\gamma(\pi_{10}^2-\tau)\pi_1^6+[-(\nu_2-2\nu_3)\pi_{10}^2\\ -(2\nu_3-\nu_4)\tau]\pi_1^5+3\gamma(\pi_{10}^2-\tau)(\pi_{10}^2+\tau)\pi_1^3\\ +\pi_{10}^2\tau(\nu_2-\nu_4)\pi_1^2-\gamma\pi_{10}^2\tau(\pi_{10}^2-\tau)=0~and\\ \pi_5 = \frac{\pi_1(\pi_9\pi_{11}+\pi_{10}\pi_{12})[3\gamma\sigma+2\gamma\tau-(\nu_2-2\nu_3)\pi_1^2+4\gamma\pi_1^3]}{4\gamma\tau(\pi_1^3-\sigma)}, \\ \pi_6 = \frac{\pi_1(\pi_9\pi_{12}-\pi_{10}\pi_{12})[3\gamma\sigma+2\gamma\tau-(\nu_2-2\nu_3)\pi_1^2+4\gamma\pi_1^3]}{4\gamma(\pi_{11}^2+\pi_{12}^2)(\pi_1^3-\sigma)}\end{array}$
$E_8=(\pi_1, \pi_5, \pi_6, \pi_9, \pi_{10}, \pi_{11}, \pi_{12})$ $\begin{array}{l}\forall~\pi_1, ~\pi_9, ~\pi_{10}, ~\pi_{11}, ~\pi_{12}~with~\rho\neq\pi_1^2~and~\sigma\neq0~where\\12\rho\gamma^2\pi_1^8-8\gamma\rho(\nu_1-\nu_3)\pi_1^7+[-24\gamma^2\rho^2+2\gamma\sigma(2\nu_1-\nu_2)\\+\rho(\nu_1-\nu_3)^2]\pi_1^6+8\gamma\rho^2(\nu_1-\nu_3)\pi_1^5+[6(2\rho^3+\sigma^2)\\-4\gamma\rho\sigma(\nu_1-\nu_2+\nu_3)]\pi_1^4-[12\gamma^2\rho\sigma^2+2\gamma\rho^2\sigma(\nu_2-2\nu_3)]\pi_1^2\\+6\gamma^2\rho^2\sigma^2=0, \\72\gamma\rho^3\pi_1^{10}-108\gamma^2\rho(\nu_1-\nu_3)\pi_1^9+[-216\gamma^3\rho^2\\+24\gamma^2\sigma(2\nu_1-\nu_4)+54\gamma\rho(\nu_1-\nu_3)^2]\pi_1^8\\+[216\gamma^2\rho^2(\nu_1-\nu_3)-12\gamma\sigma(2\nu_1-\nu_4)]\pi_1^7\\+[72\gamma^3(3\rho^3+2\sigma^2)-24\gamma^2\rho\sigma(4\nu_1+2\nu_3-3\nu_4)\\-54\gamma\rho^2(\nu_1-\nu_3)^2]\pi_1^6+[-12\gamma^2(9\rho^3+2\sigma^2)\\+24\gamma\rho\sigma(\nu_1+\nu_3-\nu_4)](\nu_1-\nu_3)\pi_1^5\\+[-72\gamma^3(\rho^4+6\rho\sigma^2)-24\gamma^2\rho^2\sigma(2\nu_1+4\nu_3-3\nu_4)]\pi_1^4\\+[48\gamma^2\rho\sigma^2-12\gamma\rho^2\sigma(2\nu_3-\nu_4)](\nu_1-\nu_3)\pi_1^3\\+[432\gamma^3\rho^2\sigma^2-24\gamma^2\rho^3\sigma(2\nu_3-\nu_4)]\pi_1^2\\-24\gamma^2\rho^2\sigma^2(\nu_1-\nu_3)\pi_1-144\gamma^3\rho^3\sigma^2=0~and\\\pi_{11} = -\frac{\pi_1^2(\pi_5\pi_9+\pi_6\pi_{10})[2\gamma\pi_1^2-2\gamma\rho-(\nu_1-\nu_3)\pi_1]}{\gamma(\pi_1^2-\rho)\sigma}, \\\pi_{12} = -\frac{\pi_1^2(\pi_5\pi_{10}-\pi_6\pi_9)[2\gamma\pi_1^2-2\gamma\rho-(\nu_1-\nu_3)\pi_1]}{\gamma(\pi_1^2-\rho)\sigma} \end{array}$
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