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

Global phase portraits of the key pitchfork bifurcation

“…例如, 二次系统就有超过 2,000 多种不同拓扑结构的全局相图. 据此, 大部分已有文献都是对某些特殊系统的全局结构进行分析, 如 二次系统 [1][2][3][4][5][6][7][8] 、三次系统 [9][10][11][12][13][14][15][16] 、四次系统 [17][18][19][20] 、Liénard 系统 [21][22][23][24] 、Hamiltonian 系统 [25][26][27] 和 Lotka-Volterra 系统 [28] 等. 此外, 文献 [29] 详细介绍了如何利用平面多项式相图软件 P4 (planar polynomial phase portraits) 绘制全局相图.…”

Dynamics of locally linearizable complex two dimensional cubic Hamiltonian systems

“…As we knew, there are lots of known results on characterization of integrability of differential systems (Hamiltonian or not, see e.g. [11,12,14,15,18]. For integrable real Hamiltonian systems of 2-degrees of freedom, Arnold [2] presented global dynamics of linear Hamiltonian ones with an elliptic-elliptic singularity.…”

Phase portraits of linear type centers of polynomial Hamiltonian systems with Hamiltonian function of degree 5 of the form <inline-formula><tex-math id="M1">\begin{document}$ H = H_1(x)+H_2(y)$\end{document}</tex-math></inline-formula>

“…We can find examples of separable Hamiltonians as in (5) in classical books, for example [15] and [1], and more recently some specific examples can be found in relativistic potential or fluid kinetics. In fact Hamiltonian systems appear in relativistic mechanic studying constant periodic oscillators [19], in fluid kinetics the authors of [27] analyze a non trivial Hamiltonian system with separable variables, and in [23] is studied a mechanic Hamiltonian with rational potential. An important work is due to Guillamon and Pantazi in [16], where they present an algorithm to study the type of local phase portrait of the equilibrium points of a separable polynomial Hamiltonian and they analyze the phase portraits of some particular examples.…”

“…The eight finite equilibria of system (23) are the origin, e 2 = (0, r), e 3 = (ρ, 0), e 4 = (ρ, r), e 5 = (σ, 0), e 6 = (σ, r), e 7 = (τ, 0) and e 8 = (τ, r). Now we study the local phase portrait of each one, we know that the origin is a center.…”

“…For studying the global phase portraits, noting that the local phase portraits coincide with the ones of case I-iv, we consider the energy level h i = H(e i ), which in this case are h 2 = r 2 /10, h 3 = ρ 2 3ρ 2 − 5ρ(σ + τ ) + 10στ /(60στ ), h 5 = σ 2 (σ(3σ − 5τ ) − 5ρ(σ − 2τ ))/(60ρτ ), and h 8 = h 2 + τ 2 (−5τ (ρ + σ) + 10ρσ + 3τ 2 )/(60ρσ), the phase portraits of system (23) are topologically equivalent to one of the 67 phase portraits in Figures 22-26.…”