Finding Integrating Factor or Inverse by Invariant Algebraic Curves

2020 ◽  
Vol 10 (06) ◽  
pp. 593-598
Author(s):  
红伟 李
Keyword(s):  
Algebraic Curves ◽  
Integrating Factor ◽  
Invariant Algebraic Curves

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.

AN ALGORITHM FOR FINDING INVARIANT ALGEBRAIC CURVES OF A GIVEN DEGREE FOR POLYNOMIAL PLANAR VECTOR FIELDS

2005 ◽  
Vol 15 (03) ◽  
pp. 1033-1044 ◽  
Author(s):  
GRZEGORZ ŚWIRSZCZ
Keyword(s):  
Differential Equations ◽  
Algebraic Curve ◽  
Vector Fields ◽  
Algebraic Curves ◽  
Polynomial System ◽  
Polynomial Systems ◽  
Invariant Algebraic Curves ◽  
Invariant Algebraic Curve ◽  
Planar Vector Fields ◽  
Planar Vector

Given a system of two autonomous ordinary differential equations whose right-hand sides are polynomials, it is very hard to tell if any nonsingular trajectories of the system are contained in algebraic curves. We present an effective method of deciding whether a given system has an invariant algebraic curve of a given degree. The method also allows the construction of examples of polynomial systems with invariant algebraic curves of a given degree. We present the first known example of a degree 6 algebraic saddle-loop for polynomial system of degree 2, which has been found using the described method. We also present some new examples of invariant algebraic curves of degrees 4 and 5 with an interesting geometry.

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.

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