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 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 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 Cofactors and equilibria for polynomial vector fields

2014 ◽  
Vol 144 (4) ◽  
pp. 753-760
Author(s):  
Antoni Ferragut ◽  
Jaume Llibre
Keyword(s):  
Vector Fields ◽  
Algebraic Curves ◽  
Equilibrium Points ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Differential Systems ◽  
Existence Of Equilibrium ◽  
Invariant Algebraic Curves ◽  
Exponential Factors

We present a relationship between the existence of equilibrium points of differential systems and the cofactors of the invariant algebraic curves and the exponential factors of the system.

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