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Cubic Differential Systems with Invariant Straight Lines of Total Multiplicity Eight possessing One Infinite Singularity

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Cubic Differential Systems with Invariant Straight Lines of Total Multiplicity Eight possessing One Infinite Singularity

Content Provider	Springer Nature Link
Author	Bujac, Cristina Vulpe, Nicolae
Copyright Year	2016
Abstract	In this work we consider the problem of classifying all configurations of invariant lines of total multiplicity eight (including the line at infinity) of real planar cubic differential systems. The classification was initiated in Bujac and Vulpe (J Math Anal Appl 423:1025–1080, 2015) where the cubic systems which possess 4 distinct infinite singularities are considered and 17 such distinct configurations were detected. That work was continued in Bujac and Vulpe (Qual Theory Dyn Syst 14(1):109–137, 2015) where the classification was done for the family of cubic systems with three distinct infinite singularities and where the existence of 5 distinct configurations was proved. The cubic systems with 2 infinite singularities are considered in Bujac (Bul Acad Stiinte Repub Mold Mat 1:48–86, 2015) and Bujac and Vulpe (Classification of a subfamily of cubic differential systems with invariant straight lines of total multiplicity eight. Universitat Autonoma de Barselona, 35, 2015) where 25 configurations were constructed. This article deals with the last remaining subfamily of cubic systems. We prove here a classification theorem (Main Theorem) of real planar cubic vector fields which possess exactly one infinite singular point and eight invariant straight lines, including the line at infinity and including their multiplicities. We show that these systems could possess only 4 such configurations. We underline that the classifications of all the above mentioned families of cubic systems are taken modulo the action of the group of real affine transformations and time rescaling and are given in terms of invariant polynomials. The algebraic invariants and comitants allow one to verify for any given real cubic system whether or not it has invariant lines of total multiplicity eight, and to specify its configuration of lines endowed with their corresponding real singularities of this system. The calculations can be implemented on computer and the results can therefore be applied for any family of cubic systems, given in any normal form.
Starting Page	1
Ending Page	30
Page Count	30
File Format	PDF
ISSN	15755460
Journal	Qualitative Theory of Dynamical Systems
Volume Number	16
Issue Number	1
e-ISSN	16623592
Language	English
Publisher	Springer International Publishing
Publisher Date	2016-02-04
Publisher Place	Cham, Switzerland
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Cubic differential system Invariant line Singular point Configuration of invariant lines Group action Affine invariant polynomial Group actions and symmetry properties Moduli problems for differential geometric structures Symmetries, invariants Bifurcation Theory of limit cycles of polynomial and analytic vector fields Mathematics Dynamical Systems and Ergodic Theory Difference and Functional Equations
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Discrete Mathematics and Combinatorics

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