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

  • * Corresponding author: Reza Mazrooei-Sebdani

    * Corresponding author: Reza Mazrooei-Sebdani 
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  • 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.

    Mathematics Subject Classification: Primary: 37J12, 37N20; Secondary: 37J40, 37J35.

    Citation:

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  • 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

    Figure 2.  Changes of $ \tilde{H} $ respect to initial conditions

    Figure 3.  Changes of $ \tilde{H} $ respect to distance

    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 $

    Figure 5.  $ (\pi_j(T),\pi_{j+1}(T)) $ for all $ j = 5,k_ge 9,k_ge 11 $

    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 $

    Figure 7.  Changes of $ \tilde{H} $ respect to $ \pi_1(T) $

    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} $

    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 $

    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} $

    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} $

    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 $
     | Show Table
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    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 $
     | Show Table
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    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 $ $\pi_6$ $-\pi_5$ $0$ $0$
    $\pi_2$ $0$ $0$ $0$ $0$ $0$ $0$ $\pi_8$ $-\pi_7$
    $\pi_3$ $0$ $0$ $0$ $0$ $-\pi_6$ $\pi_5$ $0$ $0$
    $\pi_4$ $0$ $0$ $0$ $0$ $0$ $0$ $-\pi_8$ $\pi_7$
    $\pi_5$ $-\pi_6$ $0$ $\pi_6$ $0$ $0$ $\frac{1}{2}(\pi_1-\pi_3)$ $0$ $0$
    $\pi_6$ $\pi_5$ $0$ $-\pi_5$ $0$ $-\frac{1}{2}(\pi_1-\pi_3)$ $0$ $0$ $0$
    $\pi_7$ $0$ $-\pi_8$ $0$ $\pi_8$ $0$ $0$ $0$ $\frac{1}{2}(\pi_2-\pi_4)$
    $\pi_8$ $0$ $\pi_7$ $0$ $-\pi_7$ $0$ $0$ $-\frac{1}{2}(\pi_2-\pi_4)$ $0$
    $\pi_{9}$ $-2\pi_{10}$ $\pi_{10}$ $0$ $0$ $-\frac{1}{2}\pi_{18}$ $-\frac{1}{2}\pi_{17}$ $\frac{1}{2}\pi_{12}$ $-\frac{1}{2}\pi_{11}$
    $\pi_{10}$ $2\pi_9$ $-\pi_9$ $0$ $0$ $\frac{1}{2}\pi_{17}$ $-\frac{1}{2}\pi_{18}$ $-\frac{1}{2}\pi_{11}$ $-\frac{1}{2}\pi_{12}$
    $\pi_{11}$ $-2\pi_{12}$ $0$ $0$ $\pi_{12}$ $-\frac{1}{2}\pi_{20}$ $-\frac{1}{2}\pi_{19}$ $\frac{1}{2}\pi_{10}$ $\frac{1}{2}\pi_{9}$
    $\pi_{12}$ $2\pi_{11}$ $0$ $0$ $-\pi_{11}$ $\frac{1}{2}\pi_{19}$ $-\frac{1}{2}\pi_{20}$ $-\frac{1}{2}\pi_{9}$ $\frac{1}{2}\pi_{10}$
    $\pi_{13}$ $0$ $\pi_{14}$ $-2\pi_{14}$ $0$ $-\frac{1}{2}\pi_{18}$ $\frac{1}{2}\pi_{17}$ $\frac{1}{2}\pi_{16}$ $-\frac{1}{2}\pi_{15}$
    $\pi_{14}$ $0$ $-\pi_{13}$ $2\pi_{13}$ $0$ $\frac{1}{2}\pi_{17}$ $\frac{1}{2}\pi_{18}$ $-\frac{1}{2}\pi_{15}$ $-\frac{1}{2}\pi_{16}$
    $\pi_{15}$ $0$ $0$ $-2\pi_{16}$ $\pi_{16}$ $-\frac{1}{2}\pi_{20}$ $\frac{1}{2}\pi_{19}$ $\frac{1}{2}\pi_{14}$ $\frac{1}{2}\pi_{13}$
    $\pi_{16}$ $0$ $0$ $2\pi_{15}$ $-\pi_{15}$ $\frac{1}{2}\pi_{19}$ $\frac{1}{2}\pi_{20}$ $-\frac{1}{2}\pi_{13}$ $\frac{1}{2}\pi_{14}$
    $\{ \downarrow , \rightarrow \}$ $\pi_9$ $\pi_{10}$ $\pi_{11}$ $\pi_{12}$ $\pi_{13}$ $\pi_{14}$ $\pi_{15}$ $\pi_{16}$
    $\pi_1$ $2\pi_{10}$ $-2\pi_9$ $2\pi_{12}$ $-2\pi_{11}$ $0$ $0$ $0$ $0$
    $\pi_2$ $-\pi_{10}$ $\pi_9$ $0$ $0$ $-\pi_{14}$ $\pi_{13}$ $0$ $0$
    $\pi_3$ $0$ $0$ $0$ $0$ $2\pi_{14}$ $-2\pi_{13}$ $2\pi_{16}$ $-2\pi_{15}$
    $\pi_4$ $0$ $0$ $-\pi_{12}$ $\pi_{11}$ $0$ $0$ $-\pi_{16}$ $\pi_{15}$
    $\pi_5$ $\frac{1}{2}\pi_{18}$ $-\frac{1}{2}\pi_{17}$ $\frac{1}{2}\pi_{20}$ $-\frac{1}{2}\pi_{19}$ $\frac{1}{2}\pi_{18}$ $-\frac{1}{2}\pi_{17}$ $\frac{1}{2}\pi_{20}$ $-\frac{1}{2}\pi_{19}$
    $\pi_6$ $\frac{1}{2}\pi_{17}$ $\frac{1}{2}\pi_{18}$ $\frac{1}{2}\pi_{19}$ $\frac{1}{2}\pi_{20}$ $-\frac{1}{2}\pi_{17}$ $-\frac{1}{2}\pi_{18}$ $-\frac{1}{2}\pi_{19}$ $-\frac{1}{2}\pi_{20}$
    $\pi_7$ $-\frac{1}{2}\pi_{12}$ $\frac{1}{2}\pi_{11}$ $-\frac{1}{2}\pi_{10}$ $\frac{1}{2}\pi_{9}$ $-\frac{1}{2}\pi_{16}$ $\frac{1}{2}\pi_{17}$ $-\frac{1}{2}\pi_{14}$ $\frac{1}{2}\pi_{13}$
    $\pi_8$ $\frac{1}{2}\pi_{11}$ $\frac{1}{2}\pi_{12}$ $-\frac{1}{2}\pi_{9}$ $-\frac{1}{2}\pi_{10}$ $\frac{1}{2}\pi_{15}$ $\frac{1}{2}\pi_{16}$ $-\frac{1}{2}\pi_{13}$ $-\frac{1}{2}\pi_{14}$
    $\pi_{9}$ $0$ $\pi_1(\pi_1-4\pi_2)$ $4 \pi_1 \pi_8$ $-4 \pi_1 \pi_7$ $2 \pi_5 \pi_6$ $(\pi_5^{2}-\pi_6^{2})$ $0$ $0$
    $\pi_{10}$ $-\pi_1(\pi_1-4\pi_2)$ $0$ $4 \pi_1 \pi_7$ $4 \pi_1 \pi_8$ $-(\pi_5^{2}-\pi_6^{2})$ $2 \pi_5 \pi_6$ $0$ $0$
    $\pi_{11}$ $-4\pi_1 \pi_8$ $-4\pi_1 \pi_7$ $0$ $\pi_1(\pi_1-4\pi_4)$ $0$ $0$ $2 \pi_5 \pi_6$ $(\pi_5^{2}-\pi_6^{2})$
    $\pi_{12}$ $4\pi_1 \pi_7$ $-4\pi_1 \pi_8$ $-\pi_1(\pi_1-4\pi_4)$ $0$ $0$ $0$ $-(\pi_5^{2}-\pi_6^{2})$ $2 \pi_5 \pi_6$
    $\pi_{13}$ $-2 \pi_5 \pi_6$ $(\pi_5^{2}-\pi_6^{2})$ $0$ $0$ $0$ $-\pi_3(4\pi_2-\pi_3)$ $4\pi_3 \pi_8$ $-4\pi_3 \pi_7$
    $\pi_{14}$ $-(\pi_5^{2}-\pi_6^{2})$ $-2 \pi_5 \pi_6$ $0$ $0$ $\pi_3(4\pi_2-\pi_3)$ $0$ $4\pi_3 \pi_7$ $4\pi_3 \pi_8$
    $\pi_{15}$ $0$ $0$ $-2 \pi_5 \pi_6$ $(\pi_5^{2}-\pi_6^{2})$ $-4\pi_3 \pi_8$ $-4\pi_3 \pi_7$ $0$ $-\pi_3(4\pi_4-\pi_3)$
    $\pi_{16}$ $0$ $0$ $-(\pi_5^{2}-\pi_6^{2})$ $-2 \pi_5 \pi_6$ $4\pi_3 \pi_7$ $-4\pi_3 \pi_8$ $\pi_3(4\pi_4-\pi_3)$ $0$
     | Show Table
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    Table 4.  The manifolds of equilibria of type OEE

    No. Relative Equilibria Features Conditions and Parameters
    $\begin{array}{l}(\alpha, 0, \frac{\sqrt{a_3a_7}}{a_7}\alpha, \mp \frac{a_7}{a_5} \gamma, -\frac{a_7}{a_5}\beta, \pm \gamma, \beta), \\(\alpha, 0, - \frac{\sqrt{a_3a_7}}{a_7}\alpha, \mp \frac{a_7}{a_5} \gamma, -\frac{a_7}{a_5}\beta, \pm \gamma, \beta)\end{array}$ $\begin{array}{l}\pi_{10}\pi_{12}\neq0, \\ \pi_5=0, \\ \pi_6 \neq 0\end{array}$ $\begin{array}{l}a_3a_7>0, \\{(a_3a_5^2\alpha+2a_7^3)[2a_5^2a_7\alpha^2(2\eta-\alpha)-(a_3a_5^2\alpha+2a_7^3)\beta^2]}\geq0, \\ \gamma=\frac{\sqrt{(a_3a_5^2\alpha+2a_7^3)[2a_5^2a_7\alpha^2(2\eta-\alpha)-(a_3a_5^2\alpha+2a_7^3)\beta^2]}}{a_3a_5^2\alpha+2a_7^3}, \\ \forall~\alpha, ~\beta\end{array}$
    $\begin{array}{l}(\alpha, \frac{\sqrt{-a_3a_7}}{a_7}\alpha, 0, \mp \frac{a_7}{a_5} \gamma, -\frac{a_7}{a_5}\beta, \pm \gamma, \beta), \\(\alpha, - \frac{\sqrt{-a_3a_7}}{a_7}\alpha, 0, \mp \frac{a_7}{a_5} \gamma, -\frac{a_7}{a_5}\beta, \pm \gamma, \beta)\end{array}$ $\begin{array}{l}\pi_{10}\pi_{12}\neq0, \\ \pi_5 \neq 0, \\ \pi_6=0, \\ \pi_1 \neq \frac{-2a_7^3}{a_3a_5^2}, \\{(a_3a_5^2\alpha+2a_7^3)[2a_5^2a_7\alpha^2(2\eta-\alpha)-(a_3a_5^2\alpha+2a_7^3)\beta^2]}\geq0\end{array}$ $\begin{array}{l}a_3a_7<0, \\ \gamma=\frac{\sqrt{(a_3a_5^2\alpha+2a_7^3)[2a_5^2a_7\alpha^2(2\eta-\alpha)-(a_3a_5^2\alpha+2a_7^3)\beta^2]}}{a_3a_5^2\alpha+2a_7^3}, \\ \forall~\alpha, ~\beta\end{array}$
    $(\frac{-2a_7^3}{a_3a_5^2}, \pm 2a_7^2 \frac{\sqrt{-a_3a_7}}{a_3a_5^2}, 0, 0, -\frac{a_7}{a_5}\beta, 0, \beta)$ $\begin{array}{l}\pi_{10}\pi_{12}\neq0, \\ \pi_5 \neq 0, \\ \pi_6=0, \\ \pi_1 = \frac{-2a_7^3}{a_3a_5^2}\end{array}$ $\begin{array}{l}a_3a_7<0, \\ \forall~\beta\end{array}$
    $(\varrho, \gamma_1, \alpha, \gamma_2, -\frac{a_3}{a_1}\gamma_3, \gamma_4, \gamma_3)$ $\begin{array}{l}\pi_{10}\pi_{12}\neq0, \\ \pi_5 \neq 0, \\ \pi_6 \neq 0\end{array}$ $\begin{array}{l}|a_5\alpha|\leq|a_3|\varrho, \\ \varrho=\frac{4}{3}\frac{\eta a_5^2}{a_5^2+a_3^2}, \\ \gamma_1 = \pm \frac{\sqrt{a_3^2\varrho^2-a_5^2\alpha^2}}{a_5}, \\ \gamma_2 = \frac{2 a_5 \gamma_4 \pm \sqrt{2\varrho^3(a_5^2+a_7^2)}}{2 a_7}, \\ \gamma_3 = \mp \frac{a_5a_7 \sqrt{2} \gamma_1 \alpha}{a_3 \sqrt{\varrho(a_5^2+a_7^2)}}, \\ \gamma_4 = \pm \frac{2a_5a_7\alpha^2-a_3(a_5-a_1)\varrho^2}{a_3\sqrt{2\varrho(a_5^2+a_7^2)}}, \\ \forall~\alpha\end{array}$
     | Show Table
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    Table 5.  Equilibria with π10 = π12 = 0

    No. Relative Equilibria Features Conditions and Parameters Types
    $(\frac{4}{3}\eta, 0, 0, \pm\frac{4a_1 \eta\sqrt{6(a_1^2+a_3^2)\eta}}{9(a_1^2+a_3^2)}, 0, \pm\frac{4a_3 \sqrt{6} \eta^{2}}{9 \sqrt{(a_1^2+a_3^2) \eta}}, 0)$ $\begin{array}{l}\pi_5=\pi_6=0, \\\pi_{10}= \pi_{12}=0\end{array}$ $\begin{array}{l}EEE, ~EEH, \\EEO\end{array}$
    $\begin{array}{l}e_j= \big( \frac{4}{3}\frac{\eta \alpha_1}{\beta_3}, \pm\frac{4}{3} \eta\frac{\sqrt{\alpha_1\alpha_2}}{\alpha_3} , 0, +\frac{4 \sqrt{6}}{9}\frac{(a_3-a_7)\alpha_1 \eta }{\alpha_3\sqrt{\alpha_3}}, 0, -\frac{4 \sqrt{6}}{9}\frac{(a_1-a_5)\alpha_1 \eta }{\alpha_3\sqrt{\alpha_3}}, 0\big), ~~ j = 1, 2\\e_j= \big( \frac{4}{3}\frac{\eta \alpha_1}{\alpha_3}, \pm\frac{4}{3} \eta\frac{\sqrt{\alpha_1\alpha_2}}{\alpha_3} , 0, -\frac{4 \sqrt{6}}{9}\frac{(a_3-a_7)\alpha_1 \eta }{\alpha_3\sqrt{\alpha_3}}, 0, +\frac{4 \sqrt{6}}{9}\frac{(a_1-a_5)\alpha_1 \eta }{\alpha_3\sqrt{\alpha_3}}, 0\big), ~~ j = 3, 4 \end{array}$ $\begin{array}{l}\pi_{10}= \pi_{12}=0, \\\pi_6=0, \\\pi_9(a_1-a_5)+\pi_{11}(a_3-a_7)=0\end{array}$ $\begin{array}{l}a_1 \neq a_5, \\a_3 \neq a_7, \\\alpha_1 = a_5^2+a_7^2\\-a_1a_5-a_3a_7\geq0, \\\alpha_2 = a_1^2+a_3^2\\-a_1a_5-a_3a_7\geq0, \\\alpha_3 = (a_1-a_5)^2\\+(a_3-a_7)^2>0\end{array}$ $\begin{array}{l}EHH, ~EEE, \\EHE, ~EOH, \\EOE, ~EOO, \\OOO\end{array}$
    $(\frac{4}{3}\frac{\eta a_7}{(a_7-a_3)}, \pm\frac{4}{3} \frac{\sqrt{-a_3a_7}\eta}{(a_7-a_3)}, 0, \pm\frac{4}{9}\frac{\sqrt{6 \eta a_7^2 }\eta}{ (a_7-a_3)}, 0, 0, 0)$ $\begin{array}{l}\pi_{10}= \pi_{12}=0, \\\pi_6=0, \\~\pi_9(a_1-a_5)+\pi_{11}(a_3-a_7)=0\end{array}$ $\begin{array}{l}a_1 = a_5, \\a_3 \neq a_7, \\a_3a_7<0\end{array}$ $EEH, ~EHE$
    $(\frac{4}{3}\frac{\eta a_5}{(a_5-a_1)}, \pm\frac{4}{3} \frac{\sqrt{-a_1a_5}\eta}{(a_5-a_1)}, 0, \pm\frac{4}{9}\frac{\sqrt{6 \eta a_5^2 }\eta}{ (a_5-a_1)}, 0, 0, 0)$ $\begin{array}{l}\pi_{10}= \pi_{12}=0, \\\pi_6=0, \\\pi_9(a_1-a_5)+\pi_{11}(a_3-a_7)=0\end{array}$ $\begin{array}{l}a_1 \neq a_5, \\a_3 =a_7, \\a_1a_5<0\end{array}$ $EEE, ~EHE$
    $\begin{array}{l}e_j= \big( \frac{4}{3}\frac{\eta \alpha_1}{\beta_3}, \pm\frac{4}{3} \eta\frac{\sqrt{\alpha_1\alpha_2}}{\alpha_3} , 0, +\frac{4 \sqrt{6}}{9}\frac{(a_3+a_7)\alpha_1 \eta }{\alpha_3\sqrt{\alpha_3}}, 0, -\frac{4 \sqrt{6}}{9}\frac{(a_1+a_5)\alpha_1 \eta }{\alpha_3\sqrt{\alpha_3}}, 0\big), ~~ j = 1, 2\\e_j= \big( \frac{4}{3}\frac{\eta \alpha_1}{\alpha_3}, \pm\frac{4}{3} \eta\frac{\sqrt{\alpha_1\alpha_2}}{\alpha_3} , 0, -\frac{4 \sqrt{6}}{9}\frac{(a_3+a_7)\alpha_1 \eta }{\alpha_3\sqrt{\alpha_3}}, 0, +\frac{4 \sqrt{6}}{9}\frac{(a_1+a_5)\alpha_1 \eta }{\alpha_3\sqrt{\alpha_3}}, 0\big), ~~ j = 3, 4\end{array}$ $\begin{array}{l}\pi_{10}= \pi_{12}=0, \\\pi_5=0, \\\pi_9(a_1+a_5)+\pi_{11}(a_3+a_7)=0\end{array}$ $\begin{array}{l}a_1 \neq -a_5, \\a_3 \neq -a_7, \\\alpha_1 = a_5^2+a_7^2\\+a_1a_5+a_3a_7 \geq 0, \\\alpha_2 = a_1^2+a_3^2\\+a_1a_5+a_3a_7 \geq 0, \\\alpha_3 = (a_1+a_5)^2\\+(a_3+a_7)^2 >0\end{array}$ $\begin{array}{l}EHH, ~EEE, \\EHE, ~EOH, \\EOE, ~EOO, \\OOO\end{array}$
    $(\frac{4}{3}\frac{\eta a_7}{(a_7+a_3)}, 0, \pm\frac{4}{3} \frac{\sqrt{a_3a_7}\eta}{(a_7+a_3)}, \pm\frac{4}{9}\frac{\sqrt{6 \eta a_7^2 }\eta}{ (a_7+a_3)}, 0, 0, 0)$ $\begin{array}{l}\pi_{10}= \pi_{12}=0, \\\pi_5=0, \\\pi_9(a_1+a_5)+\pi_{11}(a_3+a_7)=0\end{array}$ $\begin{array}{l}a_1 = -a_5, \\a_3 \neq -a_7, \\a_3a_7>0\end{array}$ $EHE, ~EEE$
    $(\frac{4}{3}\frac{\eta a_5}{(a_1+a_5)}, 0, \pm\frac{4}{3} \frac{\sqrt{a_1a_5}\eta}{(a_1+a_5)}, \pm\frac{4}{9}\frac{\sqrt{6 \eta a_5^2 }\eta}{ (a_5+a_1)}, 0, 0, 0)$ $\begin{array}{l}\pi_{10}= \pi_{12}=0, \\\pi_5=0, \\\pi_9(a_1+a_5)+\pi_{11}(a_3+a_7)=0\end{array}$ $\begin{array}{l}a_1 \neq -a_5, \\a_3=-a_7, \\a_1a_5>0\end{array}$ $EHE, ~EEE$
    $\begin{array}{l}(\frac{2\eta a_5}{(a_1+a_5)}, 0, \pm \frac{2\eta \sqrt{a_1a_5}}{(a_1+a_5)}, 0, 0, 0, 0), \\(-\frac{2\eta a_5}{(a_1-a_5)}, \pm \frac{2\eta \sqrt{-a_5a_1}}{(a_1-a_5)}, 0, 0, 0, 0, 0)\end{array}$ $\begin{array}{l}\pi_{10}= \pi_{12}=0, \\\pi_9(a_1+a_5)+\pi_{11}(a_3+a_7)=0, \\\pi_9(a_1-a_5)+\pi_{11}(a_3-a_7)=0\end{array}$ $OEE$
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    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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