Planar quadratic vector fields with two or three finite singularities and a finite saddle connection on a straight line (Q845595)

Using techniques similar to those developed by \textit{W. A. Coppel} [Dynamics reported, Vol. 2, 61--88 (1989; Zbl 0674.34026)], the authors classify all possible configurations for a quadratic vector field in two cases when (i) it has two finite singularities that are hyperbolic saddles or (ii) it has three finite singularities (two hyperbolic saddles and a non-saddle). First, normal forms for planar quadratic vector fields which have two or three singularities and an invariant straight line passing through two finite saddles are obtained. Then, Poincaré compactification is used to describe singularities at infinity for planar quadratic vector fields with two or three finite singularities; the behavior of finite singularities is also discussed. The authors proceed with the complete description of the phase portraits of planar vector fields with two finite singularities which are saddles. They conclude by presenting the unfolding of phase spaces of vector fields with three finite singularities according to variation of parameters of the field in normal form.