scholarly journals Classical Planar Algebraic Curves Realizable by Quadratic Polynomial Differential Systems

2017 ◽  
Vol 27 (09) ◽  
pp. 1750141 ◽  
Author(s):  
Isaac A. García ◽  
Jaume Llibre
Keyword(s):  
Algebraic Curves ◽  
Quadratic Polynomial ◽  
Darboux Integrability ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Invariant Algebraic Curves

In this paper, we show planar quadratic polynomial differential systems exhibiting as solutions some famous planar invariant algebraic curves. Also we place particular attention to the Darboux integrability of these differential systems.

scholarly journals Algebraic Limit Cycles Bifurcating from Algebraic Ovals of Quadratic Centers

2018 ◽  
Vol 28 (12) ◽  
pp. 1850145 ◽  
Author(s):  
Jaume Llibre ◽  
Yun Tian
Keyword(s):  
Limit Cycles ◽  
Algebraic Curves ◽  
Polynomial Systems ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Period Annulus ◽  
Invariant Algebraic Curves ◽  
Relevant Role ◽  
Algebraic Limit Cycles ◽  
Algebraic Limit

In the integrability of polynomial differential systems it is well known that the invariant algebraic curves play a relevant role. Here we will see that they can also play an important role with respect to limit cycles.In this paper, we study quadratic polynomial systems with an algebraic periodic orbit of degree [Formula: see text] surrounding a center. We show that there exists only one family of such systems satisfying that an algebraic limit cycle of degree [Formula: see text] can bifurcate from the period annulus of the mentioned center under quadratic perturbations.

scholarly journals Invariant Algebraic Curves of Generalized Liénard Polynomial Differential Systems

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 209
Author(s):  
Jaume Giné ◽  
Jaume Llibre
Keyword(s):  
Algebraic Curves ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Invariant Algebraic Curves

In this study, we focus on invariant algebraic curves of generalized Liénard polynomial differential systems x′=y, y′=−fm(x)y−gn(x), where the degrees of the polynomials f and g are m and n, respectively, and we correct some results previously stated.

scholarly journals The non-existence, existence and uniqueness of limit cycles for quadratic polynomial differential systems

2017 ◽  
Vol 149 (1) ◽  
pp. 1-14 ◽  
Author(s):  
Jaume Llibre ◽  
Xiang Zhang
Keyword(s):  
Differential System ◽  
Limit Cycles ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Quadratic Polynomial ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Polynomial Differential System ◽  
Uniqueness Of Limit Cycles

AbstractWe provide sufficient conditions for the non-existence, existence and uniqueness of limit cycles surrounding a focus of a quadratic polynomial differential system in the plane.

scholarly journals Detecting periodic orbits in some 3D chaotic quadratic polynomial differential systems

2015 ◽  
Vol 21 (1) ◽  
pp. 1-11 ◽  
Author(s):  
Durval José Tonon ◽  
Jaume Llibre ◽  
Rodrigo Donizete Euzébio ◽  
Tiago de Carvalho
Keyword(s):  
Periodic Orbits ◽  
Quadratic Polynomial ◽  
Differential Systems ◽  
Polynomial Differential Systems

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.

PHASE PORTRAITS OF THE QUADRATIC SYSTEMS WITH A POLYNOMIAL INVERSE INTEGRATING FACTOR

2009 ◽  
Vol 19 (03) ◽  
pp. 765-783 ◽  
Author(s):  
BARTOMEU COLL ◽  
ANTONI FERRAGUT ◽  
JAUME LLIBRE
Keyword(s):  
Quadratic Polynomial ◽  
Phase Portraits ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Polynomial Differential Systems ◽  
Integrating Factor ◽  
Inverse Integrating Factor

We classify the phase portraits of all planar quadratic polynomial differential systems having a polynomial inverse integrating factor.

scholarly journals Limit Cycles Bifurcating from a Family of Reversible Quadratic Centers via Averaging Theory

2020 ◽  
Vol 30 (04) ◽  
pp. 2050051
Author(s):  
Jaume Llibre ◽  
Arefeh Nabavi ◽  
Marzieh Mousavi
Keyword(s):  
Periodic Orbits ◽  
Limit Cycles ◽  
Quadratic Polynomial ◽  
Averaging Theory ◽  
Differential Systems ◽  
Quadratic Systems ◽  
Polynomial Differential Systems ◽  
Seventh Order ◽  
Whole Class

Consider the class of reversible quadratic systems [Formula: see text] with [Formula: see text]. These quadratic polynomial differential systems have a center at the point [Formula: see text] and the circle [Formula: see text] is one of the periodic orbits surrounding this center. These systems can be written into the form [Formula: see text] with [Formula: see text]. For all [Formula: see text] we prove that the averaging theory up to seventh order applied to this last system perturbed inside the whole class of quadratic polynomial differential systems can produce at most two limit cycles bifurcating from the periodic orbits surrounding the center (0,0) of that system. Up to now this result was only known for [Formula: see text] (see Li, 2002; Liu, 2012).

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