On the inverse problems of the polynomial vector fields in ℝ3

2020 ◽  
Vol 23 (8) ◽  
pp. 1585-1599
Author(s):  
Najmeh Khajoei ◽  
Mohammad Reza Molaei
Keyword(s):  
Inverse Problems ◽  
Vector Fields ◽  
Polynomial Vector ◽  
Polynomial Vector Fields

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.

scholarly journals Morphisms and inverse problems for Darboux integrating factors

2013 ◽  
Vol 143 (6) ◽  
pp. 1291-1302 ◽  
Author(s):  
Jaume Llibre ◽  
Chara Pantazi ◽  
Sebastian Walcher
Keyword(s):  
Inverse Problems ◽  
Affine Plane ◽  
Vector Fields ◽  
General Setting ◽  
Polynomial Vector ◽  
Reflection Groups ◽  
Polynomial Vector Fields ◽  
Integrating Factor ◽  
Structural Insight ◽  
Integrating Factors

Polynomial vector fields which admit a prescribed Darboux integrating factor are quite well understood when the geometry of the underlying curve is non-degenerate. In the general setting, morphisms of the affine plane may remove degeneracies of the curve, and thus allow more structural insight. In the present paper we establish some properties of integrating factors subjected to morphisms, and we discuss in detail one particular class of morphisms related to finite reflection groups. The results indicate that degeneracies for the underlying curve generally impose additional restrictions on vector fields admitting a given integrating factor.

Limit cycles of three-dimensional polynomial vector fields

Nonlinearity ◽  
2004 ◽  
Vol 18 (1) ◽  
pp. 175-209 ◽  
Author(s):  
Marcin Bobie ski ◽  
Henryk o a dek
Keyword(s):  
Limit Cycles ◽  
Vector Fields ◽  
Three Dimensional ◽  
Polynomial Vector ◽  
Polynomial Vector Fields

scholarly journals Multiplicity of polynomials on trajectories of polynomial vector fields in $C^3$

1998 ◽  
Vol 44 (1) ◽  
pp. 109-121 ◽  
Author(s):  
Andrei Gabrielov ◽  
Frédéric Jean ◽  
Jean-Jacques Risler
Keyword(s):  
Vector Fields ◽  
Polynomial Vector ◽  
Polynomial Vector Fields
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