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 [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 Poincaré 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 Cr perturbations in the set of all planar vector fields under the Whitney Cr-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).
- Quadratic Vector Field
- Structural Stability