scholarly journals Liouvillian first integrals for a class of generalized Liénard polynomial differential systems

2016 ◽  
Vol 146 (6) ◽  
pp. 1195-1210 ◽  
Author(s):  
Jaume Llibre ◽  
Clàudia Valls
Keyword(s):  
Algebraic Curve ◽  
First Integrals ◽  
Differential Systems ◽  
Polynomial Differential Systems ◽  
Invariant Algebraic Curve

We study the existence of Liouvillian first integrals for the generalized Liénard polynomial differential systems of the form xʹ = y, yʹ = –g(x) – f(x)y, where f(x) = 3Q(x)Qʹ(x)P(x) + Q(x)2Pʹ(x) and g(x) = Q(x)Qʹ(x)(Q(x)2P(x)2 – 1) with P,Q ∈ ℂ[x]. This class of generalized Liénard polynomial differential systems has the invariant algebraic curve (y + Q(x)P(x))2 – Q(x)2 = 0 of hyperelliptic type.

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 Integrability and Limit Cycles via First Integrals

Symmetry ◽  
2021 ◽  
Vol 13 (9) ◽  
pp. 1736
Author(s):  
Jaume Llibre
Keyword(s):  
Phase Portrait ◽  
Differential System ◽  
Limit Cycles ◽  
Applied Mathematics ◽  
First Integral ◽  
First Integrals ◽  
Differential Systems ◽  
Polynomial Differential Systems

In many problems appearing in applied mathematics in the nonlinear ordinary differential systems, as in physics, chemist, economics, etc., if we have a differential system on a manifold of dimension, two of them having a first integral, then its phase portrait is completely determined. While the existence of first integrals for differential systems on manifolds of a dimension higher than two allows to reduce the dimension of the space in as many dimensions as independent first integrals we have. Hence, to know first integrals is important, but the following question appears: Given a differential system, how to know if it has a first integral? The symmetries of many differential systems force the existence of first integrals. This paper has two main objectives. First, we study how to compute first integrals for polynomial differential systems using the so-called Darboux theory of integrability. Furthermore, second, we show how to use the existence of first integrals for finding limit cycles in piecewise differential systems.

scholarly journals Centers and limit cycles for a generalized kukles differential systems of degree 8

2021 ◽  
Author(s):  
Loubna Damene ◽  
Rebiha Benterki
Keyword(s):  
Limit Cycles ◽  
Main Tool ◽  
Upper Bounds ◽  
First Integrals ◽  
Averaging Theory ◽  
Phase Portraits ◽  
Differential Systems ◽  
Polynomial Differential Systems

Abstract In this paper we provide all the global phase portraits of the generalized kukles differential systems x= y; y = x + ax8 + bx6y2 + cx4y4 + dx2y6 + ey8; symmetric with respect to the x{axis, with a2 + b2 + c2 + d2 + e2 6= 0, and by using the averaging theory up to seven order, we give the upper bounds of limit cycles which can bifurcate from its center when we perturb it inside the class of all polynomial differential systems of degree 8. The main tool used for proving these results is based in the first integrals of the systems which form the discontinuous piecewise differential systems.

scholarly journals Liouvillian First Integrals for Generalized Riccati Polynomial Differential Systems

2015 ◽  
Vol 15 (4) ◽  
Author(s):  
Jaume Llibre ◽  
Claudia Valls
Keyword(s):  
First Integrals ◽  
Differential Systems ◽  
Polynomial Differential Systems

AbstractWe study the existence and non-existence of Liouvillian first integrals for the generalized Riccati polynomial differential systems of the form xʹ = y, yʹ = a(x)y

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