Although some interest was already given before that time, in the fifties of this century, real impetus was given to the development of the qualitative theory of quadratic vector fields. In fact, approximately eight hundred papers have been published on this subject, see Reyn \cite {R3}. One of the main problems in the qualitative theory of quadratic vector fields is the classification of all structurally stable ones. This problem has been open for more than twenty years. In this paper we solve this problem completely modulo limit cycles and give all possible phase portraits for such structurally stable quadratic vector fields. The main result of this paper is the completion the study of all structurally stable planar quadratic polynomial vector fields without limit cycles considering the three most common criteria of structural stability. In this sense we will prove that there are exactly 44 structurally stable planar quadratic vector fields without limit cycles with respect to perturbations in the set of all planar quadratic vector fields extended to the Poincar\'e sphere. If we consider perturbations in the set of all planar polynomial vector fields extended to the real plane, we obtain 24 structurally stable planar quadratic vector fields without limit cycles. If we consider $C^r$ perturbations in the set of all planar vector fields under the Whitney $C^r$--topology extended to the real plane, we also obtain 24 structurally stable planar quadratic vector fields without limit cycles. These numbers are given modulo orientation. Moreover, we show that the structurally quadratic vector fields with limit cycles have the same phase portraits as those without limit cycles if we identify the region(s) bounded by the outermost limit cycle(s) to a single point(s).
Type of paper: Topological research of phase portraits.
This paper is the first one which classified potential phase portraits, in this case of codimension 0, and opened the way for higher codimensions and the possibility to obtain all potential phase portraits of quadratic systems modulo limit cycles.
The idea of producing all potential phase portraits came from Shi Songli who created a list of 65 potential phase portraits for structurally stable phase portraits of quadratic systems modulo limit cycles. However his list had two repetitions and 7 missed cases in order to produce a complete list of all potential phase portraits od codimension 0.
After Shi Songli's list, Ye Yanqian and others already started to realize that not all potential phase portraits would be realizable and started to discard phase portraits from the list of Shi Songli.
In this paper, the authors improved the list of Shi Songli which was not complete, gathered all the proofs of impossibility already done, collected all phase portraits already found by previous mathematicians, and found the last case to complete the list of realizable cases, concretely \(QS10_{13}^{(0)}\).
In this paper, and some others using the same technique, the authors did not pay attention to some geometrical features that force that some phase portraits may break some rules as the number of contact points with the flow. They were mainly interested in producing all the potential topologically distinct phase portraits and not them to be accurately real pictures. Anyway, if a phase portrait has proved to be realizable, whether with the given shape or with another topologicaly equivalent, it will exist.