JOAN C. ARTÉS, ROBERT E. KOOIJ AND JAUME LLIBRE
Abstract
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).
1991 Mathematics subject classifications: 34D30, 58F10.