Journal Article10.1017/S0308210500000342
Plane polynomial vector fields with prescribed invariant curves
43
TL;DR: The main result of as discussed by the authors is the determination of all plane polynomial vector fields that admit a prescribed collection of algebraic curves as invariant sets, and the main result is that all these vector fields admit algebraic integrating factors.
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Abstract: The main result of this paper is the determination of all plane polynomial vector fields that admit a prescribed collection of algebraic curves as invariant sets. As an application, the polynomial vector fields admitting certain types of algebraic integrating factors are characterized.
read more
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Citations
Multiplicity of invariant algebraic curves in polynomial vector fields
TL;DR: Several natural definitions of multiplicity for invariant algebraic curves in polynomial vector fields are given and it is shown that there is a natural equivalence between the algebraic viewpoint and the geometric viewpoint.
Algebraic aspects of integrability for polynomial systems
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TL;DR: In this article, an introductory survey to the Darboux integrability theory of planar complex and real polynomial differential systems is presented, which contains some improvements to the classical theory.
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On the Poincaré Problem
TL;DR: In this paper, the authors derive some restrictions on the possible degrees of algebraic invariant curves and the possible form of integrating factors for plane polynomial vector fields whose stationary points at infinity satisfy a certain genericity condition.
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First derivative of the period function with applications
TL;DR: In this paper, the authors extend the use of the Lie bracket to the determination of the monotonicity character of the period function, and give several examples and a special method which simplifies the computations when a first integral is known.
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A Survey on the Inverse Integrating Factor
Isaac A. García,Maite Grau +1 more
TL;DR: The relation between limit cycles of planar differential systems and the inverse integrating factor was first shown in an article of Giacomini, Llibre and Viano appeared in 1996.
53
References
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Applications of Lie Groups to Differential Equations
Peter J. Olver
- 01 Jan 1986
TL;DR: In this paper, the Cauchy-Kovalevskaya Theorem has been used to define a set of invariant solutions for differential functions in a Lie Group.
9K
A note on the kernel of a locally nilpotent derivation
Gene Freudenburg,Gene Freudenburg +1 more
- 01 Jan 1996
TL;DR: In this paper, it was shown that for n ≥ 2, every locally nilpotent derivation on the polynomial ring k[X1,..., Xn] annihilates a variable.
Real Polynomial Systems of Degree n with n + 1 Line Invariants
TL;DR: In this paper, the authors studied the number of limit cycles for special classes of real polynomial systems of differential equations on the plane and showed that the relative positions of the n + 1 line invariants and their types (i.e., real or complex) influence the limit cycles.
9
Invariant algebraic curves and conditions for a centre
TL;DR: In this article, conditions for the existence of a centre in two-dimensional systems are considered along the lines of Darboux, and they are used in the search for maximal numbers of bifurcating limit cycles.
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