Liénard systems for quadratic systems with invariant algebraic curves

2011 ◽  
Vol 47 (10) ◽  
pp. 1435-1441 ◽  
Author(s):  
L. A. Cherkas
Keyword(s):  
Algebraic Curves ◽  
Quadratic Systems ◽  
Liénard Systems ◽  
Invariant Algebraic Curves

Classification of Invariant Algebraic Curves and Nonexistence of Algebraic Limit Cycles in Quadratic Systems from Family (I) of the Chinese Classification

2020 ◽  
Vol 30 (04) ◽  
pp. 2050056 ◽  
Author(s):  
Maria V. Demina ◽  
Claudia Valls
Keyword(s):  
Differential System ◽  
Limit Cycles ◽  
Algebraic Curves ◽  
Complete Classification ◽  
Quadratic Systems ◽  
Correct Proof ◽  
Invariant Algebraic Curves ◽  
Algebraic Limit Cycles ◽  
Algebraic Limit

We give the complete classification of irreducible invariant algebraic curves in quadratic systems from family [Formula: see text] of the Chinese classification, that is, of differential system [Formula: see text] with [Formula: see text]. In addition, we provide a complete and correct proof of the nonexistence of algebraic limit cycles for these equations.

Foliations on $$\mathbb {CP}^2$$ with a unique singular point without invariant algebraic curves

2019 ◽  
Vol 207 (1) ◽  
pp. 193-200
Author(s):  
Claudia R. Alcántara ◽  
Rubí Pantaleón-Mondragón
Keyword(s):  
Singular Point ◽  
Algebraic Curves ◽  
Invariant Algebraic Curves

Explicit travelling waves and invariant algebraic curves

Nonlinearity ◽  
2015 ◽  
Vol 28 (6) ◽  
pp. 1597-1606 ◽  
Author(s):  
Armengol Gasull ◽  
Hector Giacomini
Keyword(s):  
Algebraic Curves ◽  
Travelling Waves ◽  
Invariant Algebraic Curves

On the Poincaré problem and Liouvillian integrability of quadratic Liénard differential equations

2020 ◽  
Vol 150 (6) ◽  
pp. 3231-3251 ◽  
Author(s):  
Maria V. Demina ◽  
Claudia Valls
Keyword(s):  
Differential Equations ◽  
Algebraic Curves ◽  
First Integral ◽  
Complete Classification ◽  
Invariant Algebraic Curves ◽  
Liouvillian Integrability ◽  
Poincaré Problem

AbstractWe present the complete classification of irreducible invariant algebraic curves of quadratic Liénard differential equations. We prove that these equations have irreducible invariant algebraic curves of unbounded degrees, in contrast with what is wrongly claimed in the literature. In addition, we classify all the quadratic Liénard differential equations that admit a Liouvillian first integral.

scholarly journals Invariant algebraic curves for the cubic Liénard system with linear damping

2006 ◽  
Vol 130 (5) ◽  
pp. 428-441 ◽  
Author(s):  
J. Chavarriga ◽  
I.A. García ◽  
J. Llibre ◽  
H. Żołądek
Keyword(s):  
Algebraic Curves ◽  
Liénard System ◽  
Invariant Algebraic Curves ◽  
Lienard System ◽  
Linear Damping

scholarly journals Inverse problems for invariant algebraic curves: explicit computations

2009 ◽  
Vol 139 (2) ◽  
pp. 287-302 ◽  
Author(s):  
Colin Christopher ◽  
Jaume Llibre ◽  
Chara Pantazi ◽  
Sebastian Walcher
Keyword(s):  
Inverse Problems ◽  
Algebraic Curve ◽  
Affine Plane ◽  
Vector Fields ◽  
Algebraic Curves ◽  
General Setting ◽  
Polynomial Vector ◽  
Polynomial Vector Fields ◽  
Algorithmic Approach ◽  
Invariant Algebraic Curves

Given an algebraic curve in the complex affine plane, we describe how to determine all planar polynomial vector fields which leave this curve invariant. If all (finite) singular points of the curve are non-degenerate, we give an explicit expression for these vector fields. In the general setting we provide an algorithmic approach, and as an alternative we discuss sigma processes.

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