Classification of global phase portraits and bifurcation diagrams of Hamiltonian systems with rational potential

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• 作者：Y.P. Martí ; nez^a ; ^{ymartinez@ubiobio.cl" class="auth_mail" title="E-mail the corresponding author} ; C. Vidal^b ; ¹ ; ^{clvidal@ubiobio.cl" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 34C07 ; secondary ; 34C08
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：5 December 2016
• 年：2016
• 卷：261
• 期：11
• 页码：5923-5948
• 全文大小：720 K

文摘

In this paper we study the global dynamics of the Hamiltonian systems $\dot{x}=H_y(x,y)$, $\dot{y}=-H_x(x,y)$, where the Hamiltonian function H   has the particular form H(x,y)=y²/2+P(x)/Q(x)$H(x,y)=y^2/2+P(x)/Q(x)$, a10f06042898a02cf73f84" title="Click to view the MathML source">P(x),Q(x)∈R[x]$P(x),Q(x)\in \mathbb{R}[x]$ are polynomials, in particular H is the sum of the kinetic and a rational potential energies. Firstly, we provide the normal forms by a suitable μ  -symplectic change of variables. Then, the global topological classification of the phase portraits of these systems having canonical forms in the Poincaré disk in the cases where degree(P)=0,1,2$\text{degree}(P)=0,1,2$ and degree(Q)=0,1,2$\text{degree}(Q)=0,1,2$ are studied as a function of the parameters that define each polynomial. We use a blow-up technique for finite equilibrium points and the Poincaré compactification for the infinite equilibrium points. Finally, we show some applications.