Phase Portraits of Quadratic Systems with Finite Multiplicity Three and OneCritical Point of Infinity

摘要

A quadratic system is an autonomous system of ordinary differential equations. Anapproach to investigate the phase portraits of multiplicity 3, being characterized by the most complicated infinite critical points that can occur in multiplicity 3, is presented. A multiple infinite critical point is indicated by M(sub p, q)(sup i), where i indicates the index of the point, p the number of finite critical points and q the number of infinite critical points that can bifurcate from it. If p is different from zero, the point is transversally nonhyperbolic, and for multiplicity 3 there is one such point and p = 1. The most complicated infinite critical point is then a M(sub 1,3)(sup i), a fourth order saddle node with index 0, and the only critical point at infinity possible in the systems to be studied. Investigation of these systems should serve as a stepping stone toward the study of systems in multiplicity 3 with critical point M(sub 1,2)(sup i) and finally with a M(sub 1,1)(sup i).