New results on the structurally unstable quadratic differential systems of codimensions one and two [ Back ]

Date:
28.01.19   
Times:
15:30
Place:
UAB - Dept. Matemàtiques (C1/-128)
Speaker:
Alex Carlucci Rezende
University:
Universidade Federal de São Carlos

Abstrat

In this talk we present the new results concerning the structurally unstable quadratic differential systems of codimension one and two. In 1998, Artés, Kooij and Llibre proved that there exist 44 structurally stable topologically distinct phase portraits in the Poincaré disc of quadratic vector fields modulo limit cycles, and, in 2018, Artés, Llibre and Rezende showed the existence of at least 204 (at most 211) structurally unstable topologically distinct phase portraits of codimension-one quadratic systems, modulo limit cycles. Now, we begin to study the codimension-two quadratic systems. Combining the groups of codimension-one quadratic vector fields one to each other, we obtain ten new groups. We first consider group $AA$ obtained by the coalescence of two finite singular points, yielding either a triple saddle, or a triple node, or a cusp point, or two saddle-nodes. We obtain all the possible topological phase portraits of group $AA$ and prove their realization. We got 34 new topologically distinct phase portraits in the Poincaré disc modulo limit cycles.