Institutul de Matematică şi Informatică "Vladimir Andrunachievici"
Universitatea de Stat din MoldovaRO EN
The bifurcation diagram of the configurations of invariant lines of total multiplicity exactly three in quadratic vector fields
Authors: Bujac Cristina, Schlomiuk D., Vulpe Nicolae
Keywords: quadratic differential system, invariant line, singularity, configuration of invariant lines, group action, polynomial invariant
– 1.65 Mb
Keywords: quadratic differential system, invariant line, singularity, configuration of invariant lines, group action, polynomial invariant
Abstract
We denote by ${\mbox{\boldmath $QSL$}}_3$ the family of quadratic differential systems possessing invariant straight lines, finite and infinite, of total multiplicity exactly three. In a sequence of papers the complete study of quadratic systems with invariant lines of total multiplicity at least four was achieved. In addition three more families of quadratic systems possessing invariant lines of total multiplicity at least three were also studied, among them the Lotka-Volterra family. However there were still systems in ${\mbox{\boldmath ${\mbox{\boldmath $QSL$}}$}}_3$ missing from all these studies. The goals of this article are: to complete the study of the geometric configurations of invariant lines of ${\mbox{\boldmath ${\mbox{\boldmath $QSL$}}$}}_3$ by studying all the remaining cases and to give the full classification of this family modulo their configurations of invariant lines together with their bifurcation diagram. The family ${\mbox{\boldmath ${\mbox{\boldmath $QSL$}}$}}_3$ has a total of 81 distinct configurations of invariant lines. This classification is done in affine invariant terms and we also present the bifurcation diagram of these configurations in the 12-parameter space of coefficients of the systems. This diagram provides an algorithm for deciding for any given system whether it belongs to ${\mbox{\boldmath $QSL$}}_3$ and in case it does, by producing its configuration of invariant straight lines.Cristina Bujac
Institute of Mathematics and Computer Science,
State University of Moldova
E-mail:
Dana Schlomiuk
D´epartement de Math´ematiques et de Statistiques
Universit´e de Montr´eal
E-mail:
Nicolae Vulpe
Institute of Mathematics and Computer Science,
State University of Moldova
E-mail:
Institute of Mathematics and Computer Science,
State University of Moldova
E-mail:
Dana Schlomiuk
D´epartement de Math´ematiques et de Statistiques
Universit´e de Montr´eal
E-mail:
Nicolae Vulpe
Institute of Mathematics and Computer Science,
State University of Moldova
E-mail:
DOI
https://doi.org/10.56415/basm.y2023.i1.p42Fulltext
– 1.65 Mb
Contents
- Criteria for the nonexistence of periodic orbits in planar differential systems
- Some families of quadratic systems with at most one limit cycle
- Time-Reversibility and Ivariants of Some 3-dim Systems
- A survey on local integrability and its regularity
- The bifurcation diagram of the configurations of invariant lines of total multiplicity exactly three in quadratic vector fields
- Counting configurations of limit cycles and centers
