scholarly journals Algebraic Limit Cycles on Quadratic Polynomial Differential Systems

2018 ◽  
Vol 61 (2) ◽  
pp. 499-512 ◽  
Author(s):  
Jaume Llibre ◽  
Claudia Valls
Keyword(s):  
Limit Cycle ◽  
Limit Cycles ◽  
Algebraic Curve ◽  
Positive Answer ◽  
Quadratic Polynomial ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Invariant Algebraic Curve ◽  
Algebraic Limit Cycles ◽  
Algebraic Limit

AbstractAlgebraic limit cycles in quadratic polynomial differential systems started to be studied in 1958, and a few years later the following conjecture appeared: quadratic polynomial differential systems have at most one algebraic limit cycle. We prove that a quadratic polynomial differential system having an invariant algebraic curve with at most one pair of diametrically opposite singular points at infinity has at most one algebraic limit cycle. Our result provides a partial positive answer to this conjecture.

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 The Cubic Polynomial Differential Systems with two Circles as Algebraic Limit Cycles

2018 ◽  
Vol 18 (1) ◽  
pp. 183-193 ◽  
Author(s):  
Jaume Giné ◽  
Jaume Llibre ◽  
Claudia Valls
Keyword(s):  
Limit Cycles ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Cubic Polynomial ◽  
Algebraic Limit Cycles ◽  
Algebraic Limit

AbstractIn this paper we characterize all cubic polynomial differential systems in the plane having two circles as invariant algebraic limit cycles.

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.

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