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

Reza Mazrooei-Sebdani; Zahra Yousefi

doi:10.3934/dcdsb.2020255

Article Contents

2021, Volume 26, Issue 7: 3737-3765. Doi: 10.3934/dcdsb.2020255

This issue Previous Article Analysis of non-Markovian effects in generalized birth-death models Next Article Dynamics of a nonlocal diffusive logistic model with free boundaries in time periodic environment

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
^* Corresponding author: Reza Mazrooei-Sebdani

Received: November 2019

Revised: June 2020

Early access: August 2020

Published: July 2021

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Abstract

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.

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

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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

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

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

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