Phase portraits of quadratic polynomial vector fields having a rational first integral of degree 3

Llibre, Jaume; Oliveira, Regilene

doi:10.1016/j.na.2008.07.012

Cited by 16 publications

(28 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…We must mention that the corrections to the paper [10] made in [11] have been possible due to the present work.…”

Section: Introductionmentioning

confidence: 84%

“…Again we shall use a preliminary work, see [10,11] where they are characterized the phase portraits of the non-degenerate quadratic systems having a rational first integral of degree 3. Then, again using the algebraic invariant theory, we classified all the non-degenerate quadratic systems of R 12 having a rational first integral of degree 3.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Quadratic systems with a rational first integral of degree three: a complete classification in the coefficient space ℝ12

Artés

Llibre

Vulpe

2010

Rend. Circ. Mat. Palermo

Self Cite

View full text Add to dashboard Cite

A quadratic polynomial differential system can be identified with a single point of R 12 through its coefficients. The phase portrait of the quadratic systems having a rational first integral of degree 3 have been studied using normal forms. Here using the algebraic invariant theory, we characterize all the non-degenerate quadratic polynomial differential systems in R 12 having a rational first integral of degree 3. We show that there are only 31 different topological phase portraits in the Poincaré disc associated to this family of quadratic systems up to a reversal of the sense of their orbits, and we provide representatives of every class modulo an affine change of variables and a rescaling of the time variable. Moreover, each one of these 31 representatives is determined by a set of algebraic invariant conditions and we provide for it a first integral.

show abstract

“…We must mention that the corrections to the paper [10] made in [11] have been possible due to the present work.…”

Section: Introductionmentioning

confidence: 84%

Section: Introductionmentioning

confidence: 99%

Quadratic systems with a rational first integral of degree three: a complete classification in the coefficient space ℝ12

Artés

Llibre

Vulpe

2010

Rend. Circ. Mat. Palermo

Self Cite

View full text Add to dashboard Cite

show abstract

“…In the sequel we provide three examples of application. More examples of families where our approach can be used can be found for instance in [15,17,20,21].…”

Section: Applicationsmentioning

confidence: 99%

Detection of Special Curves Via the Double Resultant

Ferragut

García-Saldaña

Gasull

2015

Qual. Theory Dyn. Syst.

View full text Add to dashboard Cite

Abstract. We introduce several applications of the use of the double resultant through some examples of computation of different nature: special level sets of rational first integrals for rational discrete dynamical systems; remarkable values of rational first integrals of polynomial vector fields; bifurcation values in phase portraits of polynomial vector fields; and the different topologies of the offset of curves.

show abstract

“…• A rational first integral of degree 2 or rational first integral of degree 3, [15], [16], [38], [39].…”

Section: 2mentioning

confidence: 99%

Topological and polynomial invariants, moduli spaces, in classification problems of polynomial vector fields

Schlomiuk¹

2014

Publicacions Matemàtiques

View full text Add to dashboard Cite

Abstract:We describe the origin and evolution of ideas on topological and polynomial invariants and their interaction, in problems of classification of polynomial vector fields. The concept of moduli space is discussed in the last section and we indicate its value in understanding the dynamics of families of such systems. Our interest here is in the concepts and the way they interact in the process of topologically classifying polynomial vector fields. We survey the literature giving an ample list of references and we illustrate the ideas on the testing ground of families of quadratic vector fields. In particular, the role of polynomial invariants is illustrated in the proof of our theorem in the section next to last. These concepts have proven their worth in a number of classification results, among them the most recent work on the geometric classification of the whole class of quadratic vector fields, according to their configurations of infinite singularities. An analog work including both finite and infinite singularities of the whole quadratic class, joint work with J. C. Artés, J. Llibre, and N. Vulpe, is in progress.

show abstract

Phase portraits of quadratic polynomial vector fields having a rational first integral of degree 3

Cited by 16 publications

References 6 publications

Quadratic systems with a rational first integral of degree three: a complete classification in the coefficient space ℝ12

Quadratic systems with a rational first integral of degree three: a complete classification in the coefficient space ℝ12

Detection of Special Curves Via the Double Resultant

Topological and polynomial invariants, moduli spaces, in classification problems of polynomial vector fields

Contact Info

Product

Resources

About