On the geometric structure of the class of planar quadratic differential systems

2002 ◽  
Vol 3 (1) ◽  
pp. 93-121 ◽  
Author(s):  
Robert Roussarie ◽  
Dana Schlomiuk
Keyword(s):  
Geometric Structure ◽  
Quadratic Differential ◽  
Differential Systems

Survey of results on quadratic differential systems

2021 ◽  
pp. 17-36
Author(s):  
Joan C. Artés ◽  
Jaume Llibre ◽  
Dana Schlomiuk ◽  
Nicolae Vulpe
Keyword(s):  
Quadratic Differential ◽  
Differential Systems

scholarly journals Phase portraits of Abel quadratic differential systems of second kind with symmetries

2018 ◽  
Vol 34 (2) ◽  
pp. 301-333
Author(s):  
Antoni Ferragut ◽  
Johanna D. García-Saldaña ◽  
Claudia Valls
Keyword(s):  
Quadratic Differential ◽  
Phase Portraits ◽  
Differential Systems

scholarly journals On the Integrable Chaplygin Type Hydrodynamic Systems and Their Geometric Structure

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 697
Author(s):  
Yarema Prykarpatskyy
Keyword(s):  
Geometric Structure ◽  
Algebraic Approach ◽  
Strong Relationship ◽  
Diffeomorphism Group ◽  
Differential Systems ◽  
One Dimensional ◽  
Infinite Hierarchy ◽  
Completely Integrable ◽  
Hydrodynamic Systems

A class of spatially one-dimensional completely integrable Chaplygin hydrodynamic systems was studied within framework of Lie-algebraic approach. The Chaplygin hydrodynamic systems were considered as differential systems on the torus. It has been shown that the geometric structure of the systems under analysis has strong relationship with diffeomorphism group orbits on them. It has allowed to find a new infinite hierarchy of integrable Chaplygin like hydrodynamic systems.

The Geometry of Quadratic Differential Systems with a Weak Focus of Third Order

2004 ◽  
Vol 56 (2) ◽  
pp. 310-343 ◽  
Author(s):  
Jaume Llibre ◽  
Dana Schlomiuk
Keyword(s):  
Limit Cycles ◽  
Quadratic Differential ◽  
Topological Classification ◽  
Phase Portraits ◽  
Third Order ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Weak Focus ◽  
Affine Invariants ◽  
Geometric Concepts

AbstractIn this article we determine the global geometry of the planar quadratic differential systems with a weak focus of third order. This class plays a significant role in the context of Hilbert's 16-th problem. Indeed, all examples of quadratic differential systems with at least four limit cycles, were obtained by perturbing a system in this family. We use the algebro-geometric concepts of divisor and zero-cycle to encode global properties of the systems and to give structure to this class. We give a theorem of topological classification of such systems in terms of integer-valued affine invariants. According to the possible values taken by them in this family we obtain a total of 18 topologically distinct phase portraits. We show that inside the class of all quadratic systems with the topology of the coefficients, there exists a neighborhood of the family of quadratic systems with a weak focus of third order and which may have graphics but no polycycle in the sense of [15] and no limit cycle, such that any quadratic system in this neighborhood has at most four limit cycles.

scholarly journals On the birth and death of algebraic limit cycles in quadratic differential systems

2020 ◽  
pp. 1-20
Author(s):  
JAUME LLIBRE ◽  
REGILENE OLIVEIRA ◽  
YULIN ZHAO
Keyword(s):  
Limit Cycles ◽  
Open Problem ◽  
Quadratic Differential ◽  
Quadratic Polynomial ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Algebraic Limit Cycles ◽  
Algebraic Limit

In 1958 started the study of the families of algebraic limit cycles in the class of planar quadratic polynomial differential systems. In the present we known one family of algebraic limit cycles of degree 2 and four families of algebraic limit cycles of degree 4, and that there are no limit cycles of degree 3. All the families of algebraic limit cycles of degree 2 and 4 are known, this is not the case for the families of degree higher than 4. We also know that there exist two families of algebraic limit cycles of degree 5 and one family of degree 6, but we do not know if these families are all the families of degree 5 and 6. Until today it is an open problem to know if there are algebraic limit cycles of degree higher than 6 inside the class of quadratic polynomial differential systems. Here we investigate the birth and death of all the known families of algebraic limit cycles of quadratic polynomial differential systems.

scholarly journals Geometry of quadratic differential systems in the neighborhood of infinity

2005 ◽  
Vol 215 (2) ◽  
pp. 357-400 ◽  
Author(s):  
Dana Schlomiuk ◽  
Nicolae Vulpe
Keyword(s):  
Quadratic Differential ◽  
Differential Systems
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