34 research outputs found
Basic algebro-geometric concepts in the study of planar polynomial vector fields
In this work we show that basic algebro-geometric concepts such as the concept of intersection multiplicity of projective curves at a point in the complex projective plane, are needed in the study of planar polynomial vector fields and in particular in summing up the information supplied by bifurcation diagrams of global families of polynomial systems. Algebro-geometric concepts are helpful in organizing and unifying in more intrinsic ways this information
Geometric and algebraic classification of quadratic differential systems with invariant hyperbolas
Let QSH be the whole class of non-degenerate planar quadratic differential systems possessing at least one invariant hyperbola. We classify this family of systems, modulo the action of the group of real affine transformations and time rescaling, according to their geometric properties encoded in the configurations of invariant hyperbolas and invariant straight lines which these systems possess. The classification is given both in terms of algebraic geometric invariants and also in terms of affine invariant polynomials and it yields a total of 205 distinct such configurations. We have 162 configurations for the subclass QSH(η>0) of systems which possess three distinct real singularities at infinity, and 43 configurations for the subclass QSH(η=0) of systems which possess either exactly two distinct real singularities at infinity or the line at infinity filled up with singularities. The algebraic classification, based on the invariant polynomials, is also an algorithm which makes it possible to verify for any given real quadratic differential system if it has invariant hyperbolas or not and to specify its configuration of invariant hyperbolas and straight lines
Geometric and algebraic classification of quadratic differential systems with invariant hyperbolas
Let QSH be the whole class of non-degenerate planar
quadratic differential systems possessing at least one invariant
hyperbola. We classify this family of systems, modulo the
action of the group of real affine transformations and time
rescaling, according to their geometric properties encoded in the
configurations of invariant hyperbolas and invariant straight
lines which these systems possess. The classification is given
both in terms of algebraic geometric invariants and also in terms
of affine invariant polynomials. It yields a total of 205
distinct such configurations. We have 162 configurations for
the subclass QSH of systems which possess three
distinct real singularities at infinity in ,
and 43 configurations for the subclass QSH of systems
which possess either exactly two distinct real singularities at
infinity or the line at infinity filled up with singularities.
The algebraic classification, based on the invariant polynomials,
is also an algorithm which makes it possible to verify for any given
real quadratic differential system if it has invariant hyperbolas or
not and to specify its configuration of invariant hyperbolas and
straight lines