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

2006 ◽  
pp. 337-348
Author(s):  
GRZEGORZ ŚWIRSZCZ
Keyword(s):  
Vector Fields ◽  
Algebraic Curves ◽  
Invariant Algebraic Curves ◽  
Planar Vector Fields ◽  
Planar Vector

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.

scholarly journals A New Normal Form for Monodromic Nilpotent Singularities of Planar Vector Fields

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Antonio Algaba ◽  
Cristóbal García ◽  
Jaume Giné
Keyword(s):  
Normal Form ◽  
Vector Fields ◽  
New Technique ◽  
Center Problem ◽  
New Normal ◽  
Planar Vector Fields ◽  
Planar Vector ◽  
A New Technique

AbstractIn this work, we present a new technique for solving the center problem for nilpotent singularities which consists of determining a new normal form conveniently adapted to study the center problem for this singularity. In fact, it is a pre-normal form with respect to classical Bogdanov–Takens normal formal and it allows to approach the center problem more efficiently. The new normal form is applied to several examples.

Linearization of smooth planar vector fields around singular points via commuting flows

2008 ◽  
Vol 7 (6) ◽  
pp. 1415-1428 ◽  
Author(s):  
Isaac A. García ◽  
◽  
Jaume Giné ◽  
Susanna Maza ◽  
Keyword(s):  
Vector Fields ◽  
Singular Points ◽  
Commuting Flows ◽  
Planar Vector Fields ◽  
Planar Vector
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