Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: In this article,  consider the class of all real quadratic differential systems , with gcd(p, q) = 1, having invariant lines of total multiplicity four and two complex and one real infinite singularities.  first construct compactified canonical forms for the class so as to include limit points in the 12-dimensional parameter space of this class.  next construct the bifurcation diagrams for these compactified canonical forms. These diagrams contain many repetitions of phase portraits and  show that these are due to many symmetries under the group action. To retain the essence of the dynamics  finally construct the quotient spaces under the action of the group G = Aff(2, ℝ) × ℝ* of affine transformations and time homotheties and  place the phase portraits in these quotient spaces. The final diagrams retain only the necessary information to capture the dynamics under the motion in the parameter space as ll as under this group action.  also present here necessary and sufficient conditions for an affine line to be invariant of multiplicity k for a quadratic system.
['Oxana Cerba Diaconescu', 'Dana Schlomiuk', 'Nicolae Vulpe']
I am writing a research paper on a idea Bifurcation Diagrams and Quotient Topological Spaces Under the Action of the Affine Group of a Family of Planar Quadratic Vector Fields
[{"from": "human", "value": "I am writing a research paper on a idea Bifurcation Diagrams and Quotient Topological Spaces Under the Action of the Affine Group of a Family of Planar Quadratic Vector Fields"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: In this article,  consider the class of all real quadratic differential systems , with gcd(p, q) = 1, having invariant lines of total multiplicity four and two complex and one real infinite singularities.  first construct compactified canonical forms for the class so as to include limit points in the 12-dimensional parameter space of this class.  next construct the bifurcation diagrams for these compactified canonical forms. These diagrams contain many repetitions of phase portraits and  show that these are due to many symmetries under the group action. To retain the essence of the dynamics  finally construct the quotient spaces under the action of the group G = Aff(2, \u211d) \u00d7 \u211d* of affine transformations and time homotheties and  place the phase portraits in these quotient spaces. The final diagrams retain only the necessary information to capture the dynamics under the motion in the parameter space as ll as under this group action.  also present here necessary and sufficient conditions for an affine line to be invariant of multiplicity k for a quadratic system."}]