On Families QSL $$_{\ge 2}$$ of Quadratic Systems with Invariant Lines of Total Multiplicity At Least 2

Bujac, Cristina; Schlomiuk, Dana; Vulpe, Nicolae

doi:10.1007/s12346-022-00659-x

Cited by 2 publications

(4 citation statements)

References 20 publications

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“…In this case g 2 − 1 = 0 and since for systems (4) with h = 0 we have K = 2g(g − 1)x 2 we consider two subcases: K = 0 and K = 0. for which we have Coefficient[B 2 (a, x, y), y 4 ] = −648d 4 y 4 and hence the condition B 2 = 0 implies d = 0. However in this case we obtain two parallel invariant lines a + cx − x 2 = 0 and this class of systems is already investigated in [5].…”

Section: Configurations Of Quadratic Systems That Are Limit Points Of...mentioning

confidence: 79%

“…3.13 are constructed. In a recent published article [5] the family of systems possessing two parallel invariant lines is considered and the configurations Config. 3.14 -Config.…”

Section: Configurations Of Invariant Lines For Quadratic Vector Field...mentioning

confidence: 99%

“…In [5] one more subfamily of QSL 3 was studied. More exactly, in [5] the authors made the study of the family QSL 2p of quadratic systems possessing one of the following defining properties: two parallel invariant lines or a unique affine line that is double, or an affine invariant line and the double line at infinity or the triple line at infinity.…”

Section: Introduction and The Statement Of The Main Theoremmentioning

confidence: 99%

“….72 if H 3 = 0.1.The subcase θ = 0. Considering(5) this condition gives h(g − 1) = 0 and since µ 0 = gh 2 we examine two possibilities: µ 0 = 0 and µ 0 = 0.1.The possibility µ 0 = 0. Then h = 0 and hence the condition θ = 0 yields g = 1.…”

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confidence: 99%

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The bifurcation diagram of the configurations of invariant lines of total multiplicity exactly three in quadratic vector fields

Bujac,

Schlomiuk,

Vulpe

2023

BASM

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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.

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Section: Configurations Of Quadratic Systems That Are Limit Points Of...mentioning

confidence: 79%

“…3.13 are constructed. In a recent published article [5] the family of systems possessing two parallel invariant lines is considered and the configurations Config. 3.14 -Config.…”

Section: Configurations Of Invariant Lines For Quadratic Vector Field...mentioning

confidence: 99%

Section: Introduction and The Statement Of The Main Theoremmentioning

confidence: 99%

mentioning

confidence: 99%

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The bifurcation diagram of the configurations of invariant lines of total multiplicity exactly three in quadratic vector fields

Bujac,

Schlomiuk,

Vulpe

2023

BASM

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Global Analysis of Riccati Quadratic Differential Systems

Artés,

Llibre,

Schlomiuk

et al. 2024

Int. J. Bifurcation Chaos

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In this paper, we study the family of quadratic Riccati differential systems. Our goal is to obtain the complete topological classification of this family on the Poincaré disk compactification of the plane. The family was partially studied before but never from a truly global viewpoint. Our approach is global and we use geometry to achieve our goal. The geometric analysis we perform is via the presence of two invariant parallel straight lines in any generic Riccati system. We obtain a total of 119 topologically distinct phase portraits for this family. Furthermore, we give the complete bifurcation diagram in the 12-dimensional space of parameters of this family in terms of invariant polynomials, meaning that it is independent of the normal forms in which the systems may be presented. This bifurcation diagram provides an algorithm to decide for any given quadratic system in any form it may be presented, whether it is a Riccati system or not, and in case it is to provide its phase portrait.

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On Families QSL $$_{\ge 2}$$ of Quadratic Systems with Invariant Lines of Total Multiplicity At Least 2

Cited by 2 publications

References 20 publications

The bifurcation diagram of the configurations of invariant lines of total multiplicity exactly three in quadratic vector fields

The bifurcation diagram of the configurations of invariant lines of total multiplicity exactly three in quadratic vector fields

Global Analysis of Riccati Quadratic Differential Systems

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