Electron. J. Differential Equations,
Vol. 2017 (2017), No. 295, pp. 1122.
Geometric and algebraic classification of quadratic
differential systems with invariant hyperbolas
Regilene D. S. Oliveira, Alex C. Rezende, Dana Schlomiuk, Nicolae Vulpe
Abstract:
Let QSH be the whole class of nondegenerate 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.
Submitted February 9, 2017. Published November 28, 2017.
Math Subject Classifications: 34C23, 34A34.
Key Words: Quadratic differential systems; algebraic solution;
configuration of algebraic solutions; affine invariant polynomials;
group action.
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Regilene D. S. Oliveira
Instituto de Ciências Matemáticas e de Computação
Universidade de São Paulo, Brazil
email: regilene@icmc.usp.br


Alex C. Rezende
Universidade Federal de Santa Maria
Campus Palmeira das Missões, Brazil
email: alexcrezende@gmail.com


Dana Schlomiuk
Département de Mathématiques et de Statistiques
Université de Montréal, Canada
email: dasch@dms.umontreal.ca


Nicolae Vulpe
Institute of Mathematics and Computer Science
Academy of Sciences of Moldova
email: nvulpe@gmail.com

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