Bifurcation Diagrams and Moduli Spaces of Planar Quadratic Vector Fields with Invariant Lines of Total Multiplicity Four and Having Exactly Three Real Singularities at Infinity

摘要

In this article we consider the class ${{bf QSL}_{bf4}^{bf3sboldsymbolinfty}}$ of all real quadratic differential systems ${frac{{rm{d}}x}{{rm{d}}t}=p(x,y),frac{{rm{d}}y}{{rm{d}}t}=q(x,y)}$ with gcd(p, q) = 1, having invariant lines of total multiplicity four and three real distinct infinite singularities. Firstly we construct compactified canonical forms for the class ${{bf QSL}_{bf4}^{bf3sboldsymbolinfty}}$ so as to include limit points in the 12-dimensional parameter space of the set ${{bf QSL}_{bf4}^{bf3sboldsymbolinfty}}$ . We next construct the bifurcation diagrams for these compactified canonical forms. These diagrams contain many repetitions of phase portraits and we show that these are due to many symmetries under the group action. To retain the essence of the dynamics we finally construct the moduli spaces under the action of the group of affine transformations and time homotheties and we place the phase portraits in these moduli spaces. The final diagrams retain only the necessary information to capture the dynamics under the motion in the parameter space as well as under the group action.