Structurally unstable quadratic vector fields of codimension two: families possessing either a cusp point or two finite saddle-nodes.

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The goal of this work is to contribute to the classification of the phase portraits of planar quadratic differential systems according to their structural stability. Artés, Kooij and Llibre 7 (1998) proved that there exist 44 structurally stable topologically distinct phase portraits in the 8 Poincaré disc modulo limit cycles in this family, and Artés, Llibre and Rezende (2018) showed the 9 existence of at least 204 (at most 211) structurally unstable topologically distinct phase portraits 10 of codimension-one quadratic systems, modulo limit cycles. In this work we begin the classification 11 of planar quadratic systems of codimension two in the structural stability. Combining the groups 12 of codimension-one quadratic vector fields one to each other, we obtain ten new groups. Here 13 we consider group AA obtained by the coalescence of two finite singular points, yielding either a 14 triple saddle, or a triple node, or a cusp point, or two saddle-nodes. We obtain all the possible 15 topological phase portraits of group AA and prove their realization. We got 34 new topologically 16 distinct phase portraits in the Poincaré disc modulo limit cycles.

Equações diferenciais ordinárias, Teoria qualitativa