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

Yuzhou Tian; Yulin Zhao

doi:10.3934/dcdsb.2020214

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2021, Volume 26, Issue 6: 2941-2956. Doi: 10.3934/dcdsb.2020214

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

Yuzhou Tian and
Yulin Zhao^,

School of Mathematics (Zhuhai), Sun Yat-Sen University, Zhuhai, 519082, China

^* Corresponding author: Yulin Zhao
^* Corresponding author: Yulin Zhao

Received: June 30, 2019

Early access: July 2020

Published: June 2021

This research is supported by the NSF of China (No.11971495 and No.11801582)

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Abstract

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.

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

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Figure 1. Phase portraits of system (3)

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

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

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

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

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

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

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

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

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

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

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

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Table 1. Algebraic classification of system (6)

$b$	$\omega(z)$ = 0	Conditions	Roots of $\omega(z)$	Linear change	Normal forms
$b = 0$	Linear equation	$a = 0, c\neq0$	No roots	$\left(x, y\right)\mapsto\left(c^{-1/3}x, c^{-1/3}y\right)$	$({\bf{I.1}})$
$b = 0$	Linear equation	$a\neq0, c\in\mathbb{R}$	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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