Phase Portraits of Quadratic Systems with Finite Multiplicity Three and a M(i)1,2Type of Critical Point at Infinity

摘要

In this report, by a quadratic system is meant an autonomous system of ordinarydifferential equations x(dot) = a(00) + a(10)x + a(01)y + a(20)x(sup 2) + a(11)xy + a(02)y(sup 2) identically equal to p(x,y), y(dot) = b(00) + b(10)x + b(01)y + b(20)y(sup 2) + b(02)y(sup 2) identically equal to Q(x,y), where (dot) identically equal to d/dt and a(ij), b(ij) is a member of R, and P(x,y) and Q(x,y) are relatively prime real polynomials, which are not both linear. The authors study the class of quadratic systems with finite multiplicity three, consisting of systems with three elementary critical points, possibly complex or coinciding, and a M(i)(sub 1,2) type of critical point at infinity, being a point, which upon bifurcation such that only elementary critical points result, leaves one critical point in the finite part of the plane and two critical points at infinity.