Global Phase Portraits of Some Reversible Cubic Centers With Noncollinear Singularities

Abstract

Abstract. The results in this paper show that the cubic vector fieldsẋ = −y + M (x, y) − y(x 2 + y 2 ),ẏ = x + N (x, y) + x(x 2 + y 2 ), where M, N are quadratic homogeneous polynomials, having simultaneously a center at the origin and at infinity, have at least 61 and at most 68 topologically different phase portraits. To this end the reversible subfamily defined by M (x, y) = −γxy, N (x, y) = (γ − λ)x 2 + α 2 λy 2 with α, γ ∈ R and λ = 0, is studied in detail and it is shown to have at least 48 and at most 55 topologically different phase portraits. In particular, there are exactly 5 for γλ < 0 and at least 46 for γλ > 0. Furthermore, the global bifurcation diagram is analyzed.

Centers and Limit Cycles of Vector Fields Defined on Invariant Spheres

“…Note that π preserves closed curves and contact between curves contained on its domain of definition. We say that p ∈ S 2 ρ is an equilibrium point of center type of X| S 2 ρ if π(p) = q and q is an equilibrium point of center type of Y (x) defined in (6). Moreover, as π(0, ρ, 0) = (0, 0) we can assume that (0, 0) is an equilibrium point of the planar projected system (6).…”

“…Proof of Theorem 2. Using Lemma 12 we can consider system (2) restricted to the sphere S 2 1 and its projection Y defined in (6). We restrict our attention to the equilibrium point of (2) which is located at the origin after projection.…”

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.…”

Global Phase Portraits of Some Reversible Cubic Centers With Collinear or Infinitely Many Singularities

“…not only considering the particular case when all singularities are collinear, it is convenient to introduce new parameters as well as different techniques. An elaborate analysis of Case 5 is organized in a forthcoming paper, see [Caubergh et al, 2011b].…”

“…In the rest of the paper we only focus on vector fields X R (a,b,c) for which (a, b, c) belongs to P 1 ∪ P 2 . For (a, b, c) belonging to P 3 an elaborate analysis of the global phase portraits is worked out in [Caubergh et al, 2011b]. Ending section 2 we find that essentially three cases have to be considered in classifying the global phase portraits of X R (a,b,c) when (a, b, c) belongs to P 1 ∪ P 2 .…”