Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The results in this paper show that the cubic vector fields ẋ = -y + M(x, y) - y(x2 + y2), ẏ = x + N(x, y) + x( x2 + y2), 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)x2 + α2λy2 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 five for γλ   0. Furthermore, the global bifurcation diagram is analyzed.
['Magdalena Caubergh', 'Joan Torregrosa']
I am writing a research paper on a idea Global phase portraits of some reversible cubic centers with noncollinear singularities
[{"from": "human", "value": "I am writing a research paper on a idea Global phase portraits of some reversible cubic centers with noncollinear singularities"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The results in this paper show that the cubic vector fields \u1e8b = -y + M(x, y) - y(x2 + y2), \u1e8f = x + N(x, y) + x( x2 + y2), 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) = -\u03b3xy, N(x, y) = (\u03b3 - \u03bb)x2 + \u03b12\u03bby2 with \u03b1, \u03b3 \u2208 \u211d and \u03bb \u2260 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 five for \u03b3\u03bb   0. Furthermore, the global bifurcation diagram is analyzed."}]