The connection between the dynamical properties of 3D systems and the image of the energy-Casimir mapping

Figure 1.  $ Q_0\in \Omega^o\cap \partial U $ with $ U = \mathcal{EC}(B_{r_0}(P_0)) $: $ P_0 $ is an equilibrium state

Figure 2.  $ Q_0\in\partial\Omega $ and $ \mathcal{EC}^{-1}(Q_0) = \{P_0\}\cup D $ are composed of equilibrium states

Figure 3.  $ Q_0\in \partial \Omega, L\cap\Sigma = \emptyset, Q\in L: $ $ \Gamma_{h}\subset \mathcal{EC}^{-1}(Q) $ is a periodic orbit

Figure 4.  The image of the energy-Casimir mapping of the system (6)

Figure 5.  The correspondence between the dynamic behavior of (6) and $ \Omega $ for $ \beta = \frac{1}{4}, c_0 = -0.8 $: $ \mathcal{EC}^{-1}(Q_0) = \{P_0\} $, $ \mathcal{EC}^{-1}(Q_1) = \Gamma_{h_1} $, $ \mathcal{EC}^{-1}(Q_2) = \{P_2\} $

Figure 6.  The correspondence between the dynamic behavior of (6) and $ \Omega $ for $ \beta = \frac{1}{4}, c_0 = -4 $: $ \mathcal{EC}^{-1}(Q_0) = \{P_0\} $, $ \mathcal{EC}^{-1}(Q_1) = \Gamma_{h_1} $, $ \mathcal{EC}^{-1}(Q_2) = \Gamma_{h_2} $, $ \mathcal{EC}^{-1}(Q_3) = \{P_3\}\cup\Gamma^+_{h_3}\cup\Gamma^-_{h_3} $, $ \mathcal{EC}^{-1}(Q_4) = \Gamma^+_{h_4}\cup\Gamma^-_{h_4} $, $ \mathcal{EC}^{-1}(Q_5) = \{P^+_5, P^-_5\} $

Figure 7.  The correspondence between the dynamic behavior of (8) and $ \Omega (h_0 = 3) $: $ \mathcal{EC}^{-1}(Q_0) = \{P_0^+, P_0^-\} $, $ \mathcal{EC}^{-1}(Q_1) = \Gamma_{c_1}^+\cup\Gamma_{c_1}^- $, $ \mathcal{EC}^{-1}(Q_2) = \Gamma_1\cup\Gamma_2\cup\Gamma_3\cup\Gamma_4\cup\{P_2^+, P_2^-\} $, $ \mathcal{EC}^{-1}(Q_3) = \Gamma_{c_3}^+\cup \Gamma_{c_3}^- $