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Global phase portraits and bifurcation diagrams for reversible equivariant Hamiltonian systems of linear plus quartic homogeneous polynomials

  • * Corresponding author: Yulin Zhao

    * Corresponding author: Yulin Zhao
This research is supported by the NSF of China (No.11971495 and No.11801582)
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  • This paper is devoted to the complete classification of global phase portraits for reversible equivariant Hamiltonian systems of linear plus quartic homogeneous polynomials. Such system is affinely equivalent to one of five normal forms by an algebraic classification of its infinite singular points. Then, we classify the global phase portraits of these normal forms on the Poincaré disc. There are exactly $ 13 $ different global topological structures on the Poincaré disc. Finally we provide the bifurcation diagrams for the corresponding global phase portraits.

    Mathematics Subject Classification: Primary: 34C05, 34C20; Secondary: 34C14.

    Citation:

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  • Figure 1.  Phase portraits of system (3)

    Figure 2.  Bifurcation diagram of system $\left({{\bf{I}}{\bf{.5}}} \right)$

    Figure 3.  The local phase portrait of the system (8) at the origin

    Figure 4.  Local phase portrait of system $\left({{\bf{I}}{\bf{.1}}} \right)$ on the Poincaré disk

    Figure 5.  The local phase portrait of system (13) at origin for $ c = 0 $

    Figure 6.  The local phase portrait of the system (14) at the origin

    Figure 7.  All the local phase portraits of system $\left({{\bf{I}}{\bf{.2}}} \right)$ on the Poincaré disk

    Figure 8.  The local phase portrait of the system (15) at the origin for $ a\leq0 $

    Figure 9.  All the local phase portraits of system $\left({{\bf{I}}{\bf{.3}}} \right)$ on the Poincaré disk

    Figure 10.  The local phase portraits of system (17) at $ p_1^{\pm} $ for $ a<0 $

    Figure 11.  The local phase portraits of system $\left({{\bf{I}}{\bf{.4}}} \right)$ on the Poincaré disk

    Figure 12.  The local phase portraits of system $\left({{\bf{I}}{\bf{.5}}} \right)$ with $ \Delta>0 $ on the Poincaré disk

    Table 1.  Algebraic classification of system (6)

    $b$ $\omega(z)$ = 0ConditionsRoots of $\omega(z)$Linear change Normal forms
    $b = 0$ Linear equation $a = 0, c\neq0$No roots $\left(x, y\right)\mapsto\left(c^{-1/3}x, c^{-1/3}y\right)$ $({\bf{I.1}})$
    $a\neq0, c\in\mathbb{R}$One root $\left(x, y\right)\mapsto\left(a^{-1/3}x, a^{-1/3}y\right)$ $({\bf{I.2}})$
    $b\neq0$ Quadratic equation $a\in\mathbb{R}, c = 0$ $0$ is a root $\left(x, y\right)\mapsto\left(b^{-1/3}x, b^{-1/3}y\right)$ $({\bf{I.3}})$
    $a\in\mathbb{R}, c\neq0, \Delta = 0$Multiple root $\left(x, y\right)\mapsto\left(b^{-1/3}x, b^{-1/3}y\right)$ $({\bf{I.4}})$
    $a\in\mathbb{R}, c\neq0, \Delta\neq0$Two simple roots $\left(x, y\right)\mapsto\left(b^{-1/3}x, b^{-1/3}y\right)$ $({\bf{I.5}})$
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